local constraints in combinatorial optimizationmadhurt/talks/cci-talk.pdf · local constraints in...
TRANSCRIPT
![Page 1: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/1.jpg)
Local Constraints in CombinatorialOptimization
Madhur Tulsiani
Institute for Advanced Study
![Page 2: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/2.jpg)
Local Constraints in Approximation Algorithms
Linear Programming (LP) or Semidefinite Programming(SDP) based approximation algorithms impose constraintson few variables at a time.
When can local constraints help in approximating a globalproperty (eg. Vertex Cover, Chromatic Number)?
How does one reason about increasingly larger localconstraints?
Does approximation get better as constraints get larger?
![Page 3: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/3.jpg)
Local Constraints in Approximation Algorithms
Linear Programming (LP) or Semidefinite Programming(SDP) based approximation algorithms impose constraintson few variables at a time.
When can local constraints help in approximating a globalproperty (eg. Vertex Cover, Chromatic Number)?
How does one reason about increasingly larger localconstraints?
Does approximation get better as constraints get larger?
![Page 4: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/4.jpg)
Local Constraints in Approximation Algorithms
Linear Programming (LP) or Semidefinite Programming(SDP) based approximation algorithms impose constraintson few variables at a time.
When can local constraints help in approximating a globalproperty (eg. Vertex Cover, Chromatic Number)?
How does one reason about increasingly larger localconstraints?
Does approximation get better as constraints get larger?
![Page 5: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/5.jpg)
Local Constraints in Approximation Algorithms
Linear Programming (LP) or Semidefinite Programming(SDP) based approximation algorithms impose constraintson few variables at a time.
When can local constraints help in approximating a globalproperty (eg. Vertex Cover, Chromatic Number)?
How does one reason about increasingly larger localconstraints?
Does approximation get better as constraints get larger?
![Page 6: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/6.jpg)
LP/SDP Hierarchies
Various hierarchies give increasingly powerful programs at differentlevels (rounds).
Lovász-Schrijver (LS, LS+)Sherali-AdamsLasserre
LS(1)
LS(2)
...SA(1)
SA(2)
...
LS(1)+
LS(2)+
...
Las(1)
Las(2)
...
Can optimize over r th level in time nO(r). nth level is tight.
![Page 7: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/7.jpg)
LP/SDP Hierarchies
Various hierarchies give increasingly powerful programs at differentlevels (rounds).
Lovász-Schrijver (LS, LS+)Sherali-AdamsLasserre
LS(1)
LS(2)
...SA(1)
SA(2)
...
LS(1)+
LS(2)+
...
Las(1)
Las(2)
...
Can optimize over r th level in time nO(r). nth level is tight.
![Page 8: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/8.jpg)
LP/SDP Hierarchies
Various hierarchies give increasingly powerful programs at differentlevels (rounds).
Lovász-Schrijver (LS, LS+)Sherali-AdamsLasserre
LS(1)
LS(2)
...SA(1)
SA(2)
...
LS(1)+
LS(2)+
...
Las(1)
Las(2)
...
Can optimize over r th level in time nO(r). nth level is tight.
![Page 9: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/9.jpg)
LP/SDP Hierarchies
Powerful computational model capturing most known LP/SDPalgorithms within constant number of levels.
Lower bounds rule out large and natural class of algorithms.
Performance measured by considering integrality gap at variouslevels.
Integrality Gap =Optimum of Relaxation
Integer Optimum(for maximization)
![Page 10: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/10.jpg)
LP/SDP Hierarchies
Powerful computational model capturing most known LP/SDPalgorithms within constant number of levels.
Lower bounds rule out large and natural class of algorithms.
Performance measured by considering integrality gap at variouslevels.
Integrality Gap =Optimum of Relaxation
Integer Optimum(for maximization)
![Page 11: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/11.jpg)
LP/SDP Hierarchies
Powerful computational model capturing most known LP/SDPalgorithms within constant number of levels.
Lower bounds rule out large and natural class of algorithms.
Performance measured by considering integrality gap at variouslevels.
Integrality Gap =Optimum of Relaxation
Integer Optimum(for maximization)
![Page 12: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/12.jpg)
Why bother?
UG-Hardness
NP-Hardness
LP/SDPHierarchies
- Conditional- All polytime algorithms
- Unconditional- Restricted class of algorithms
![Page 13: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/13.jpg)
What Hierarchies want
Example: Maximum Independent Set for graph G = (V , E)
minimize∑
u
xu
subject to xu + xv ≤ 1 ∀ (u, v) ∈ Exu ∈ [0, 1]
Hope: x1, . . . , xn is convex combination of 0/1 solutions.
1/3 1/3
1/3
= 13×
1 0
0
+ 13×
0 0
1
+ 13×
0 1
0
Hierarchies add variables for conditional/joint probabilities.
![Page 14: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/14.jpg)
What Hierarchies want
Example: Maximum Independent Set for graph G = (V , E)
minimize∑
u
xu
subject to xu + xv ≤ 1 ∀ (u, v) ∈ Exu ∈ [0, 1]
Hope: x1, . . . , xn is convex combination of 0/1 solutions.
1/3 1/3
1/3
= 13×
1 0
0
+ 13×
0 0
1
+ 13×
0 1
0
Hierarchies add variables for conditional/joint probabilities.
![Page 15: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/15.jpg)
What Hierarchies want
Example: Maximum Independent Set for graph G = (V , E)
minimize∑
u
xu
subject to xu + xv ≤ 1 ∀ (u, v) ∈ Exu ∈ [0, 1]
Hope: x1, . . . , xn is marginal of distribution over 0/1 solutions.
1/3 1/3
1/3
= 13×
1 0
0
+ 13×
0 0
1
+ 13×
0 1
0
Hierarchies add variables for conditional/joint probabilities.
![Page 16: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/16.jpg)
What Hierarchies want
Example: Maximum Independent Set for graph G = (V , E)
minimize∑
u
xu
subject to xu + xv ≤ 1 ∀ (u, v) ∈ Exu ∈ [0, 1]
Hope: x1, . . . , xn is marginal of distribution over 0/1 solutions.
1/3 1/3
1/3
= 13×
1 0
0
+ 13×
0 0
1
+ 13×
0 1
0
Hierarchies add variables for conditional/joint probabilities.
![Page 17: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/17.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 18: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/18.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 19: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/19.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 20: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/20.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 21: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/21.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 22: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/22.jpg)
Lovász-Schrijver in action
r th level optimizes over distributions conditioned on r variables.
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 23: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/23.jpg)
The Lovàsz-Schrijver Hierarchy
Start with a 0/1 integer program and a relaxation P. Definetigher relaxation LS(P).
Hope: Fractional (x1, . . . , xn) = E [(z1, . . . , zn)] for integral(z1, . . . , zn)
Restriction: x = (x1, . . . , xn) ∈ LS(P) if ∃Y satisfying(think Yij = E [zizj ] = P [zi ∧ zj ])
Y = Y T
Yii = xi ∀iYi
xi∈ P,
x− Yi
1− xi∈ P ∀i
Y 0
Above is an LP (SDP) in n2 + n variables.
![Page 24: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/24.jpg)
The Lovàsz-Schrijver Hierarchy
Start with a 0/1 integer program and a relaxation P. Definetigher relaxation LS(P).
Hope: Fractional (x1, . . . , xn) = E [(z1, . . . , zn)] for integral(z1, . . . , zn)
Restriction: x = (x1, . . . , xn) ∈ LS(P) if ∃Y satisfying(think Yij = E [zizj ] = P [zi ∧ zj ])
Y = Y T
Yii = xi ∀iYi
xi∈ P,
x− Yi
1− xi∈ P ∀i
Y 0
Above is an LP (SDP) in n2 + n variables.
![Page 25: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/25.jpg)
The Lovàsz-Schrijver Hierarchy
Start with a 0/1 integer program and a relaxation P. Definetigher relaxation LS(P).
Hope: Fractional (x1, . . . , xn) = E [(z1, . . . , zn)] for integral(z1, . . . , zn)
Restriction: x = (x1, . . . , xn) ∈ LS(P) if ∃Y satisfying(think Yij = E [zizj ] = P [zi ∧ zj ])
Y = Y T
Yii = xi ∀iYi
xi∈ P,
x− Yi
1− xi∈ P ∀i
Y 0
Above is an LP (SDP) in n2 + n variables.
![Page 26: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/26.jpg)
The Lovàsz-Schrijver Hierarchy
Start with a 0/1 integer program and a relaxation P. Definetigher relaxation LS(P).
Hope: Fractional (x1, . . . , xn) = E [(z1, . . . , zn)] for integral(z1, . . . , zn)
Restriction: x = (x1, . . . , xn) ∈ LS(P) if ∃Y satisfying(think Yij = E [zizj ] = P [zi ∧ zj ])
Y = Y T
Yii = xi ∀iYi
xi∈ P,
x− Yi
1− xi∈ P ∀i
Y 0
Above is an LP (SDP) in n2 + n variables.
![Page 27: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/27.jpg)
The Lovàsz-Schrijver Hierarchy
Start with a 0/1 integer program and a relaxation P. Definetigher relaxation LS(P).
Hope: Fractional (x1, . . . , xn) = E [(z1, . . . , zn)] for integral(z1, . . . , zn)
Restriction: x = (x1, . . . , xn) ∈ LS(P) if ∃Y satisfying(think Yij = E [zizj ] = P [zi ∧ zj ])
Y = Y T
Yii = xi ∀iYi
xi∈ P,
x− Yi
1− xi∈ P ∀i
Y 0
Above is an LP (SDP) in n2 + n variables.
![Page 28: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/28.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 29: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/29.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 30: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/30.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 31: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/31.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 32: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/32.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 33: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/33.jpg)
Action replay
1/3
1/3
1/3
1/3
1/3
1/3
1/2
1/2
1/2
0
1/2
1/2
0
0
0
1
0
0
2/3× 1/3×
1/2×0
0
1
0
0
0
1/2×?
?
?
0
?
?
![Page 34: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/34.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints:
∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 35: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/35.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints:
∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 36: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/36.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints:
∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 37: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/37.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints:
∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 38: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/38.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints: ∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 39: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/39.jpg)
The Sherali-Adams Hierarchy
Start with a 0/1 integer linear program.
Add “big variables” XS for |S| ≤ r(think XS = E
[∏i∈S zi
]= P [All vars in S are 1])
Constraints: ∑i
aizi ≤ b
E
[(∑i
aizi
)· z5z7(1− z9)
]≤ E [b · z5z7(1− z9)]
∑i
ai · (Xi,5,7 − Xi,5,7,9) ≤ b · (X5,7 − X5,7,9)
LP on nr variables.
![Page 40: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/40.jpg)
Sherali-Adams ≈ Locally Consistent Distributions
Using 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1
0 ≤ X1,2 ≤ 10 ≤ X1 − X1,2 ≤ 10 ≤ X2 − X1,2 ≤ 10 ≤ 1− X1 − X2 + X1,2 ≤ 1
X1, X2, X1,2 define a distribution D(1, 2) over 0, 12.
D(1, 2, 3) and D(1, 2, 4) must agree with D(1, 2).
SA(r) =⇒ LCD(r). If each constraint has at most k vars,LCD(r+k) =⇒ SA(r)
![Page 41: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/41.jpg)
Sherali-Adams ≈ Locally Consistent Distributions
Using 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1
0 ≤ X1,2 ≤ 10 ≤ X1 − X1,2 ≤ 10 ≤ X2 − X1,2 ≤ 10 ≤ 1− X1 − X2 + X1,2 ≤ 1
X1, X2, X1,2 define a distribution D(1, 2) over 0, 12.
D(1, 2, 3) and D(1, 2, 4) must agree with D(1, 2).
SA(r) =⇒ LCD(r). If each constraint has at most k vars,LCD(r+k) =⇒ SA(r)
![Page 42: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/42.jpg)
Sherali-Adams ≈ Locally Consistent Distributions
Using 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1
0 ≤ X1,2 ≤ 10 ≤ X1 − X1,2 ≤ 10 ≤ X2 − X1,2 ≤ 10 ≤ 1− X1 − X2 + X1,2 ≤ 1
X1, X2, X1,2 define a distribution D(1, 2) over 0, 12.
D(1, 2, 3) and D(1, 2, 4) must agree with D(1, 2).
SA(r) =⇒ LCD(r). If each constraint has at most k vars,LCD(r+k) =⇒ SA(r)
![Page 43: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/43.jpg)
Sherali-Adams ≈ Locally Consistent Distributions
Using 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1
0 ≤ X1,2 ≤ 10 ≤ X1 − X1,2 ≤ 10 ≤ X2 − X1,2 ≤ 10 ≤ 1− X1 − X2 + X1,2 ≤ 1
X1, X2, X1,2 define a distribution D(1, 2) over 0, 12.
D(1, 2, 3) and D(1, 2, 4) must agree with D(1, 2).
SA(r) =⇒ LCD(r). If each constraint has at most k vars,LCD(r+k) =⇒ SA(r)
![Page 44: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/44.jpg)
Sherali-Adams ≈ Locally Consistent Distributions
Using 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1
0 ≤ X1,2 ≤ 10 ≤ X1 − X1,2 ≤ 10 ≤ X2 − X1,2 ≤ 10 ≤ 1− X1 − X2 + X1,2 ≤ 1
X1, X2, X1,2 define a distribution D(1, 2) over 0, 12.
D(1, 2, 3) and D(1, 2, 4) must agree with D(1, 2).
SA(r) =⇒ LCD(r). If each constraint has at most k vars,LCD(r+k) =⇒ SA(r)
![Page 45: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/45.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35
= P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 46: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/46.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35
= P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 47: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/47.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35
= P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 48: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/48.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35
= P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 49: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/49.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35
= P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 50: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/50.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35 = P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 51: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/51.jpg)
The Lasserre Hierarchy
Start with a 0/1 integer quadratic program.
Think “big" variables ZS =∏
i∈S zi .
Associated psd matrix Y (moment matrix)
YS1,S2 = EˆZS1 · ZS2
˜= E
24 Yi∈S1∪S2
zi
35 = P[All vars in S1 ∪ S2 are 1]
(Y 0) + original constraints + consistency constraints.
![Page 52: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/52.jpg)
The Lasserre hierarchy (constraints)
Y is psd. (i.e. find vectors US satisfying YS1,S2 =˙US1 , US2
¸)
YS1,S2 only depends on S1 ∪ S2. (YS1,S2 = P[All vars in S1 ∪ S2 are 1])
Original quadratic constraints as inner products.
SDP for Independent Set
maximizeXi∈V
˛Ui
˛2
subject to˙Ui, Uj
¸= 0 ∀ (i, j) ∈ E˙
US1 , US2
¸=
˙US3 , US4
¸∀ S1 ∪ S2 = S3 ∪ S4˙
US1 , US2
¸∈ [0, 1] ∀S1, S2
![Page 53: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/53.jpg)
The Lasserre hierarchy (constraints)
Y is psd. (i.e. find vectors US satisfying YS1,S2 =˙US1 , US2
¸)
YS1,S2 only depends on S1 ∪ S2. (YS1,S2 = P[All vars in S1 ∪ S2 are 1])
Original quadratic constraints as inner products.
SDP for Independent Set
maximizeXi∈V
˛Ui
˛2
subject to˙Ui, Uj
¸= 0 ∀ (i, j) ∈ E˙
US1 , US2
¸=
˙US3 , US4
¸∀ S1 ∪ S2 = S3 ∪ S4˙
US1 , US2
¸∈ [0, 1] ∀S1, S2
![Page 54: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/54.jpg)
The Lasserre hierarchy (constraints)
Y is psd. (i.e. find vectors US satisfying YS1,S2 =˙US1 , US2
¸)
YS1,S2 only depends on S1 ∪ S2. (YS1,S2 = P[All vars in S1 ∪ S2 are 1])
Original quadratic constraints as inner products.
SDP for Independent Set
maximizeXi∈V
˛Ui
˛2
subject to˙Ui, Uj
¸= 0 ∀ (i, j) ∈ E˙
US1 , US2
¸=
˙US3 , US4
¸∀ S1 ∪ S2 = S3 ∪ S4˙
US1 , US2
¸∈ [0, 1] ∀S1, S2
![Page 55: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/55.jpg)
And if you just woke up . . .
LS(1)
LS(2)
...
Yi
xi,
x− Yi
1− xi
SA(1)
SA(2)
...
XS
LS(1)+
LS(2)+
...
Y 0
Las(1)
Las(2)
...
US
![Page 56: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/56.jpg)
And if you just woke up . . .
LS(1)
LS(2)
...
Yi
xi,
x− Yi
1− xi
SA(1)
SA(2)
...
XS
LS(1)+
LS(2)+
...
Y 0
Las(1)
Las(2)
...
US
![Page 57: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/57.jpg)
And if you just woke up . . .
LS(1)
LS(2)
...
Yi
xi,
x− Yi
1− xi
SA(1)
SA(2)
...XS
LS(1)+
LS(2)+
...Y 0
Las(1)
Las(2)
...
US
![Page 58: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/58.jpg)
Local Distributions
![Page 59: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/59.jpg)
Ω(log n) level LS gap for Vertex Cover [ABLT’06]
Random Sparse Graphs
Girth = Ω(log n)
|VC| ≥ (1− ε)n
(1/2 + ε, . . . , 1/2 + ε) survives at Ω(log n) levels
1/2 + ε
v = 1
![Page 60: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/60.jpg)
Ω(log n) level LS gap for Vertex Cover [ABLT’06]
Random Sparse Graphs
Girth = Ω(log n)
|VC| ≥ (1− ε)n
(1/2 + ε, . . . , 1/2 + ε) survives at Ω(log n) levels
1/2 + ε
v = 1
![Page 61: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/61.jpg)
Ω(log n) level LS gap for Vertex Cover [ABLT’06]
Random Sparse Graphs
Girth = Ω(log n)
|VC| ≥ (1− ε)n
(1/2 + ε, . . . , 1/2 + ε) survives at Ω(log n) levels
1/2 + ε
v = 1
![Page 62: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/62.jpg)
Ω(log n) level LS gap for Vertex Cover [ABLT’06]
Random Sparse Graphs
Girth = Ω(log n)
|VC| ≥ (1− ε)n
(1/2 + ε, . . . , 1/2 + ε) survives at Ω(log n) levels
1/2 + ε
v = 1
![Page 63: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/63.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 64: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/64.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 65: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/65.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 66: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/66.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 67: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/67.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 68: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/68.jpg)
Creating conditional distributions
Tree1/2
1/2
= 12×
0
1
+ 12×
1
0
Locally a tree1/2 + ε
1/2 + ε
= ( 12 − ε)×
0
1
+ ( 12 + ε)×
1
1 w.p.4ε/(1 + 2ε)
0 o.w.
1/2 + εv
1
4ε
1− 4ε
8ε
1− 8ε
O(1/ε · log(1/ε))
![Page 69: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/69.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 70: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/70.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 71: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/71.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 72: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/72.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 73: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/73.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 74: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/74.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 75: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/75.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 76: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/76.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 77: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/77.jpg)
Cheat while you can
easy
doable
Valid distributions exist for O(log n) levels.
Can be extended to Ω(n) levels (but needs other ideas). [STT’07]
Similar ideas also useful in constructing metrics which are locally `1 (butnot globally). [CMM’07]
![Page 78: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/78.jpg)
Local Satisfiability for Expanding CSPs
![Page 79: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/79.jpg)
CSP Expansion
MAX k-CSP: m constraints on k -tuples of (n) boolean variables.Satisfy maximum. e.g. MAX 3-XOR (linear equations mod 2)
z1 + z2 + z3 = 0 z3 + z4 + z5 = 1 · · ·
Expansion: Every set S of constraints involves at least β|S|variables (for |S| < αm).(Used extensively in proof complexity e.g. [BW01], [BGHMP03])
Cm
...
C1
zn
...
z1
In fact, γ|S| variables appearing in only one constraint in S.
![Page 80: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/80.jpg)
CSP Expansion
MAX k-CSP: m constraints on k -tuples of (n) boolean variables.Satisfy maximum. e.g. MAX 3-XOR (linear equations mod 2)
z1 + z2 + z3 = 0 z3 + z4 + z5 = 1 · · ·
Expansion: Every set S of constraints involves at least β|S|variables (for |S| < αm).(Used extensively in proof complexity e.g. [BW01], [BGHMP03])
Cm
...
C1
zn
...
z1
In fact, γ|S| variables appearing in only one constraint in S.
![Page 81: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/81.jpg)
CSP Expansion
MAX k-CSP: m constraints on k -tuples of (n) boolean variables.Satisfy maximum. e.g. MAX 3-XOR (linear equations mod 2)
z1 + z2 + z3 = 0 z3 + z4 + z5 = 1 · · ·
Expansion: Every set S of constraints involves at least β|S|variables (for |S| < αm).(Used extensively in proof complexity e.g. [BW01], [BGHMP03])
Cm
...
C1
zn
...
z1
In fact, γ|S| variables appearing in only one constraint in S.
![Page 82: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/82.jpg)
CSP Expansion
MAX k-CSP: m constraints on k -tuples of (n) boolean variables.Satisfy maximum. e.g. MAX 3-XOR (linear equations mod 2)
z1 + z2 + z3 = 0 z3 + z4 + z5 = 1 · · ·
Expansion: Every set S of constraints involves at least β|S|variables (for |S| < αm).(Used extensively in proof complexity e.g. [BW01], [BGHMP03])
Cm
...
C1
zn
...
z1
In fact, γ|S| variables appearing in only one constraint in S.
![Page 83: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/83.jpg)
CSP Expansion
MAX k-CSP: m constraints on k -tuples of (n) boolean variables.Satisfy maximum. e.g. MAX 3-XOR (linear equations mod 2)
z1 + z2 + z3 = 0 z3 + z4 + z5 = 1 · · ·
Expansion: Every set S of constraints involves at least β|S|variables (for |S| < αm).(Used extensively in proof complexity e.g. [BW01], [BGHMP03])
Cm
...
C1
zn
...
z1
In fact, γ|S| variables appearing in only one constraint in S.
![Page 84: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/84.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 85: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/85.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 86: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/86.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 87: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/87.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 88: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/88.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 89: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/89.jpg)
Local Satisfiability
C1
C2
C3
z1
z2
z3
z4
z5
z6
• Take γ = 0.9
• Can show any three 3-XOR constraints aresimultaneously satisfiable.
• Can take γ ≈ (k − 2) and any αn constraints.
• Just require E[C(z1, . . . , zk )] over any k − 2vars to be constant.
Ez1...z6 [C1(z1, z2, z3) · C2(z3, z4, z5) · C3(z4, z5, z6)]
= Ez2...z6 [C2(z3, z4, z5) · C3(z4, z5, z6) · Ez1 [C1(z1, z2, z3)]]
= Ez4,z5,z6 [C3(z4, z5, z6) · Ez3 [C2(z3, z4, z5)] · (1/2)]
= 1/8
![Page 90: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/90.jpg)
Sherali-Adams LP for CSPs
Variables: X(S,α) for |S| ≤ t , partial assignments α ∈ 0, 1S
maximizemX
i=1
Xα∈0,1Ti
Ci(α)·X(Ti ,α)
subject to X(S∪i,α0) + X(S∪i,α1) = X(S,α) ∀i /∈ S
X(S,α) ≥ 0
X(∅,∅) = 1
X(S,α) ∼ P[Vars in S assigned according to α]
Need distributions D(S) such that D(S1), D(S2) agree onS1 ∩ S2.
Distributions should “locally look like" supported on satisfyingassignments.
![Page 91: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/91.jpg)
Sherali-Adams LP for CSPs
Variables: X(S,α) for |S| ≤ t , partial assignments α ∈ 0, 1S
maximizemX
i=1
Xα∈0,1Ti
Ci(α)·X(Ti ,α)
subject to X(S∪i,α0) + X(S∪i,α1) = X(S,α) ∀i /∈ S
X(S,α) ≥ 0
X(∅,∅) = 1
X(S,α) ∼ P[Vars in S assigned according to α]
Need distributions D(S) such that D(S1), D(S2) agree onS1 ∩ S2.
Distributions should “locally look like" supported on satisfyingassignments.
![Page 92: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/92.jpg)
Sherali-Adams LP for CSPs
Variables: X(S,α) for |S| ≤ t , partial assignments α ∈ 0, 1S
maximizemX
i=1
Xα∈0,1Ti
Ci(α)·X(Ti ,α)
subject to X(S∪i,α0) + X(S∪i,α1) = X(S,α) ∀i /∈ S
X(S,α) ≥ 0
X(∅,∅) = 1
X(S,α) ∼ P[Vars in S assigned according to α]
Need distributions D(S) such that D(S1), D(S2) agree onS1 ∩ S2.
Distributions should “locally look like" supported on satisfyingassignments.
![Page 93: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/93.jpg)
Sherali-Adams LP for CSPs
Variables: X(S,α) for |S| ≤ t , partial assignments α ∈ 0, 1S
maximizemX
i=1
Xα∈0,1Ti
Ci(α)·X(Ti ,α)
subject to X(S∪i,α0) + X(S∪i,α1) = X(S,α) ∀i /∈ S
X(S,α) ≥ 0
X(∅,∅) = 1
X(S,α) ∼ P[Vars in S assigned according to α]
Need distributions D(S) such that D(S1), D(S2) agree onS1 ∩ S2.
Distributions should “locally look like" supported on satisfyingassignments.
![Page 94: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/94.jpg)
Obtaining integrality gaps for CSPs
Cm
...
C1
zn
...
z1
Want to define distribution D(S) for set S of variables.
Find set of constraints C such that G − C − S remains expanding.D(S) = uniform over assignments satisfying C
Remaining constraints “independent" of this assignment.
![Page 95: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/95.jpg)
Obtaining integrality gaps for CSPs
Cm
...
C1
zn
...
z1
Want to define distribution D(S) for set S of variables.
Find set of constraints C such that G − C − S remains expanding.D(S) = uniform over assignments satisfying C
Remaining constraints “independent" of this assignment.
![Page 96: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/96.jpg)
Obtaining integrality gaps for CSPs
Cm
...
C1
zn
...
z1
Want to define distribution D(S) for set S of variables.
Find set of constraints C such that G − C − S remains expanding.D(S) = uniform over assignments satisfying C
Remaining constraints “independent" of this assignment.
![Page 97: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/97.jpg)
Obtaining integrality gaps for CSPs
Cm
...
C1
zn
...
z1
Want to define distribution D(S) for set S of variables.
Find set of constraints C such that G − C − S remains expanding.D(S) = uniform over assignments satisfying C
Remaining constraints “independent" of this assignment.
![Page 98: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/98.jpg)
Obtaining integrality gaps for CSPs
Cm
...
C1
zn
...
z1
Want to define distribution D(S) for set S of variables.
Find set of constraints C such that G − C − S remains expanding.D(S) = uniform over assignments satisfying C
Remaining constraints “independent" of this assignment.
![Page 99: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/99.jpg)
Vectors for Linear CSPs
![Page 100: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/100.jpg)
A “new look” Lasserre
Start with a −1, 1 quadratic integer program.(z1, . . . , zn) → ((−1)z1 , . . . , (−1)zn)
Define big variables ZS =∏
i∈S(−1)zi .
Consider the psd matrix Y
YS1,S2 = E[ZS1 · ZS2
]= E
∏i∈S1∆S2
(−1)zi
Write program for inner products of vectors WS s.t.YS1,S2 = 〈WS1 , WS2〉
![Page 101: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/101.jpg)
A “new look” Lasserre
Start with a −1, 1 quadratic integer program.(z1, . . . , zn) → ((−1)z1 , . . . , (−1)zn)
Define big variables ZS =∏
i∈S(−1)zi .
Consider the psd matrix Y
YS1,S2 = E[ZS1 · ZS2
]= E
∏i∈S1∆S2
(−1)zi
Write program for inner products of vectors WS s.t.YS1,S2 = 〈WS1 , WS2〉
![Page 102: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/102.jpg)
A “new look” Lasserre
Start with a −1, 1 quadratic integer program.(z1, . . . , zn) → ((−1)z1 , . . . , (−1)zn)
Define big variables ZS =∏
i∈S(−1)zi .
Consider the psd matrix Y
YS1,S2 = E[ZS1 · ZS2
]= E
∏i∈S1∆S2
(−1)zi
Write program for inner products of vectors WS s.t.YS1,S2 = 〈WS1 , WS2〉
![Page 103: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/103.jpg)
A “new look” Lasserre
Start with a −1, 1 quadratic integer program.(z1, . . . , zn) → ((−1)z1 , . . . , (−1)zn)
Define big variables ZS =∏
i∈S(−1)zi .
Consider the psd matrix Y
YS1,S2 = E[ZS1 · ZS2
]= E
∏i∈S1∆S2
(−1)zi
Write program for inner products of vectors WS s.t.YS1,S2 = 〈WS1 , WS2〉
![Page 104: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/104.jpg)
Gaps for 3-XOR
SDP for MAX 3-XOR
maximizeX
Ci≡(zi1+zi2
+zi3=bi )
1 + (−1)bi˙Wi1,i2,i3, W∅
¸2
subject to˙WS1 , WS2
¸=
˙WS3 , WS4
¸∀ S1∆S2 = S3∆S4
|WS | = 1 ∀S, |S| ≤ r
[Schoenebeck’08]: If width 2r resolution does not derivecontradiction, then SDP value =1 after r levels.
Expansion guarantees there are no width 2r contradictions.
![Page 105: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/105.jpg)
Gaps for 3-XOR
SDP for MAX 3-XOR
maximizeX
Ci≡(zi1+zi2
+zi3=bi )
1 + (−1)bi˙Wi1,i2,i3, W∅
¸2
subject to˙WS1 , WS2
¸=
˙WS3 , WS4
¸∀ S1∆S2 = S3∆S4
|WS | = 1 ∀S, |S| ≤ r
[Schoenebeck’08]: If width 2r resolution does not derivecontradiction, then SDP value =1 after r levels.
Expansion guarantees there are no width 2r contradictions.
![Page 106: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/106.jpg)
Gaps for 3-XOR
SDP for MAX 3-XOR
maximizeX
Ci≡(zi1+zi2
+zi3=bi )
1 + (−1)bi˙Wi1,i2,i3, W∅
¸2
subject to˙WS1 , WS2
¸=
˙WS3 , WS4
¸∀ S1∆S2 = S3∆S4
|WS | = 1 ∀S, |S| ≤ r
[Schoenebeck’08]: If width 2r resolution does not derivecontradiction, then SDP value =1 after r levels.
Expansion guarantees there are no width 2r contradictions.
![Page 107: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/107.jpg)
Schonebeck’s construction
z1 + z2 + z3 = 1 mod 2 =⇒ (−1)z1+z2 = −(−1)z3
=⇒ W1,2 = −W3
Equations of width 2r divide |S| ≤ r into equivalence classes. Chooseorthogonal eC for each class C.
No contradictions ensure each S ∈ C can be uniquely assigned ±eC .
Relies heavily on constraints being linear equations.
WS1= eC1
WS2= −eC2
WS3= eC2
![Page 108: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/108.jpg)
Schonebeck’s construction
z1 + z2 + z3 = 1 mod 2 =⇒ (−1)z1+z2 = −(−1)z3
=⇒ W1,2 = −W3
Equations of width 2r divide |S| ≤ r into equivalence classes. Chooseorthogonal eC for each class C.
No contradictions ensure each S ∈ C can be uniquely assigned ±eC .
Relies heavily on constraints being linear equations.
WS1= eC1
WS2= −eC2
WS3= eC2
![Page 109: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/109.jpg)
Schonebeck’s construction
z1 + z2 + z3 = 1 mod 2 =⇒ (−1)z1+z2 = −(−1)z3
=⇒ W1,2 = −W3
Equations of width 2r divide |S| ≤ r into equivalence classes. Chooseorthogonal eC for each class C.
No contradictions ensure each S ∈ C can be uniquely assigned ±eC .
Relies heavily on constraints being linear equations.
WS1= eC1
WS2= −eC2
WS3= eC2
![Page 110: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/110.jpg)
Schonebeck’s construction
z1 + z2 + z3 = 1 mod 2 =⇒ (−1)z1+z2 = −(−1)z3
=⇒ W1,2 = −W3
Equations of width 2r divide |S| ≤ r into equivalence classes. Chooseorthogonal eC for each class C.
No contradictions ensure each S ∈ C can be uniquely assigned ±eC .
Relies heavily on constraints being linear equations.
WS1= eC1
WS2= −eC2
WS3= eC2
![Page 111: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/111.jpg)
Schonebeck’s construction
z1 + z2 + z3 = 1 mod 2 =⇒ (−1)z1+z2 = −(−1)z3
=⇒ W1,2 = −W3
Equations of width 2r divide |S| ≤ r into equivalence classes. Chooseorthogonal eC for each class C.
No contradictions ensure each S ∈ C can be uniquely assigned ±eC .
Relies heavily on constraints being linear equations.
WS1= eC1
WS2= −eC2
WS3= eC2
![Page 112: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/112.jpg)
Reductions
![Page 113: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/113.jpg)
Spreading the hardness around (Reductions) [T]
If problem A reduces to B, can we sayIntegrality Gap for A =⇒ Integrality Gap for B?
Reductions are (often) local algorithms.
Reduction from integer program A to integer program B. Eachvariable z ′i of B is a boolean function of few (say 5) variableszi1 , . . . , zi5 of A.
To show: If A has good vector solution, so does B.
![Page 114: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/114.jpg)
Spreading the hardness around (Reductions) [T]
If problem A reduces to B, can we sayIntegrality Gap for A =⇒ Integrality Gap for B?
Reductions are (often) local algorithms.
Reduction from integer program A to integer program B. Eachvariable z ′i of B is a boolean function of few (say 5) variableszi1 , . . . , zi5 of A.
To show: If A has good vector solution, so does B.
![Page 115: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/115.jpg)
Spreading the hardness around (Reductions) [T]
If problem A reduces to B, can we sayIntegrality Gap for A =⇒ Integrality Gap for B?
Reductions are (often) local algorithms.
Reduction from integer program A to integer program B. Eachvariable z ′i of B is a boolean function of few (say 5) variableszi1 , . . . , zi5 of A.
To show: If A has good vector solution, so does B.
![Page 116: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/116.jpg)
Spreading the hardness around (Reductions) [T]
If problem A reduces to B, can we sayIntegrality Gap for A =⇒ Integrality Gap for B?
Reductions are (often) local algorithms.
Reduction from integer program A to integer program B. Eachvariable z ′i of B is a boolean function of few (say 5) variableszi1 , . . . , zi5 of A.
To show: If A has good vector solution, so does B.
![Page 117: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/117.jpg)
A generic transformation
A Bf
z ′i = f (zi1 , . . . , zi5)
U′z′i
=∑
S⊆i1,...,i5
f (S) ·WS
![Page 118: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/118.jpg)
A generic transformation
A Bf
z ′i = f (zi1 , . . . , zi5)
U′z′i
=∑
S⊆i1,...,i5
f (S) ·WS
![Page 119: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/119.jpg)
What can be proved
NP-hard UG-hard Gap Levels
MAX k-CSP 2k
2√
2k2k
k+o(k)2k
2k Ω(n)
Independent
Setn
2(log n)3/4+εn
2c1√
log n log log n2c2√
log n log log n
Approximate
Graph Coloringl vs. 2
125 log2 l l vs. 2l/2
4l2 Ω(n)
Chromatic
Numbern
2(log n)3/4+εn
2c1√
log n log log n2c2√
log n log log n
Vertex Cover 1.36 2 - ε 1.36 Ω(nδ)
![Page 120: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/120.jpg)
The FGLSS Construction
Reduces MAX k-CSP to Independent Set in graph GΦ.
z1 + z2 + z3 = 1
001 010
100111
z3 + z4 + z5 = 0
110 011
000101
Need vectors for subsets of vertices in the GΦ.
Every vertex (or set of vertices) in GΦ is an indicator function!
U(z1,z2,z3)=(0,0,1) =1
8(W∅ + W1 + W2 −W3 + W1,2 −W2,3 −W1,3 −W1,2,3)
![Page 121: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/121.jpg)
The FGLSS Construction
Reduces MAX k-CSP to Independent Set in graph GΦ.
z1 + z2 + z3 = 1
001 010
100111
z3 + z4 + z5 = 0
110 011
000101
Need vectors for subsets of vertices in the GΦ.
Every vertex (or set of vertices) in GΦ is an indicator function!
U(z1,z2,z3)=(0,0,1) =1
8(W∅ + W1 + W2 −W3 + W1,2 −W2,3 −W1,3 −W1,2,3)
![Page 122: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/122.jpg)
The FGLSS Construction
Reduces MAX k-CSP to Independent Set in graph GΦ.
z1 + z2 + z3 = 1
001 010
100111
z3 + z4 + z5 = 0
110 011
000101
Need vectors for subsets of vertices in the GΦ.
Every vertex (or set of vertices) in GΦ is an indicator function!
U(z1,z2,z3)=(0,0,1) =1
8(W∅ + W1 + W2 −W3 + W1,2 −W2,3 −W1,3 −W1,2,3)
![Page 123: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/123.jpg)
The FGLSS Construction
Reduces MAX k-CSP to Independent Set in graph GΦ.
z1 + z2 + z3 = 1
001 010
100111
z3 + z4 + z5 = 0
110 011
000101
Need vectors for subsets of vertices in the GΦ.
Every vertex (or set of vertices) in GΦ is an indicator function!U(z1,z2,z3)=(0,0,1) =
1
8(W∅ + W1 + W2 −W3 + W1,2 −W2,3 −W1,3 −W1,2,3)
![Page 124: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/124.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) =
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 125: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/125.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) =
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 126: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/126.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) = ?
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 127: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/127.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) = Uv1 ⊗ Uv2 ⊗ Uv3
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 128: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/128.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) = Uv1 ⊗ Uv2 ⊗ Uv3
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 129: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/129.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) = Uv1 ⊗ Uv2 ⊗ Uv3
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 130: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/130.jpg)
Graph Products
v1
w1
×v2
w2
×v3
w3
U(v1,v2,v3) = Uv1 ⊗ Uv2 ⊗ Uv3
Similar transformation for sets (project to each copy of G).
Intuition: Independent set in product graph is product ofindependent sets in G.
Together give a gap of n2O(
√log n log log n)
.
![Page 131: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/131.jpg)
A few problems
![Page 132: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/132.jpg)
Problem 1: Size vs. Rank
All previous bounds are on the number of levels (rank).
What if there is a program that uses poly(n) constraints (size),but takes them from up to level n?
If proved, is this kind of hardness closed under local reductions?
![Page 133: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/133.jpg)
Problem 1: Size vs. Rank
All previous bounds are on the number of levels (rank).
What if there is a program that uses poly(n) constraints (size),but takes them from up to level n?
If proved, is this kind of hardness closed under local reductions?
![Page 134: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/134.jpg)
Problem 1: Size vs. Rank
All previous bounds are on the number of levels (rank).
What if there is a program that uses poly(n) constraints (size),but takes them from up to level n?
If proved, is this kind of hardness closed under local reductions?
![Page 135: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/135.jpg)
Problem 2: Generalize Schoenebeck’s technique
Technique seems specialized for linear equations.
Breaks down even if there are few local contradictions (whichdoesn’t rule out a gap).
We have distributions, but not vectors for other type of CSPs.
What extra constraints do vectors capture?
![Page 136: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/136.jpg)
Problem 2: Generalize Schoenebeck’s technique
Technique seems specialized for linear equations.
Breaks down even if there are few local contradictions (whichdoesn’t rule out a gap).
We have distributions, but not vectors for other type of CSPs.
What extra constraints do vectors capture?
![Page 137: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/137.jpg)
Problem 2: Generalize Schoenebeck’s technique
Technique seems specialized for linear equations.
Breaks down even if there are few local contradictions (whichdoesn’t rule out a gap).
We have distributions, but not vectors for other type of CSPs.
What extra constraints do vectors capture?
![Page 138: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/138.jpg)
Problem 2: Generalize Schoenebeck’s technique
Technique seems specialized for linear equations.
Breaks down even if there are few local contradictions (whichdoesn’t rule out a gap).
We have distributions, but not vectors for other type of CSPs.
What extra constraints do vectors capture?
![Page 139: Local Constraints in Combinatorial Optimizationmadhurt/Talks/cci-talk.pdf · Local Constraints in Combinatorial Optimization Madhur Tulsiani Institute for Advanced Study. Local Constraints](https://reader034.vdocuments.site/reader034/viewer/2022052102/603c6d8ba65f9240f444d976/html5/thumbnails/139.jpg)
Thank You
Questions?