extended formulations and information complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf ·...
TRANSCRIPT
![Page 1: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/1.jpg)
Extended Formulations and Information Complexity
Ankur Moitra Massachusetts Institute of Technology
Dagstuhl, March 2014
![Page 2: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/2.jpg)
The Permutahedron
![Page 3: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/3.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
![Page 4: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/4.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have?
![Page 5: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/5.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have? exponentially many!
![Page 6: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/6.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have?
e.g. S [n], Σi in S xi ≥ 1 + 2 + … + |S| = |S|(|S|+1)/2
exponentially many!
![Page 7: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/7.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have?
e.g. S [n], Σi in S xi ≥ 1 + 2 + … + |S| = |S|(|S|+1)/2
Let Q = {A| A is doubly-stochastic}
exponentially many!
![Page 8: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/8.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have?
e.g. S [n], Σi in S xi ≥ 1 + 2 + … + |S| = |S|(|S|+1)/2
Let Q = {A| A is doubly-stochastic}
Then P is the projection of Q: P = {A t | A in Q }
exponentially many!
![Page 9: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/9.jpg)
The Permutahedron
Let t = [1, 2, 3, … n], P = conv{π(t) | π is permutation}
How many facets of P have?
e.g. S [n], Σi in S xi ≥ 1 + 2 + … + |S| = |S|(|S|+1)/2
Let Q = {A| A is doubly-stochastic}
Then P is the projection of Q: P = {A t | A in Q }
Yet Q has only O(n2) facets
exponentially many!
![Page 10: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/10.jpg)
Extended Formulations
The extension complexity (xc) of a polytope P is the minimum number of facets of Q so that P = proj(Q)
![Page 11: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/11.jpg)
Extended Formulations
The extension complexity (xc) of a polytope P is the minimum number of facets of Q so that P = proj(Q)
e.g. xc(P) = Θ(n logn) for permutahedron
![Page 12: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/12.jpg)
Extended Formulations
The extension complexity (xc) of a polytope P is the minimum number of facets of Q so that P = proj(Q)
e.g. xc(P) = Θ(n logn) for permutahedron
xc(P) = Θ(logn) for a regular n-gon, but Ω(√n) for its perturbation
![Page 13: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/13.jpg)
Extended Formulations
The extension complexity (xc) of a polytope P is the minimum number of facets of Q so that P = proj(Q)
e.g. xc(P) = Θ(n logn) for permutahedron
xc(P) = Θ(logn) for a regular n-gon, but Ω(√n) for its perturbation
In general, P = {x | y, (x,y) in Q} E
![Page 14: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/14.jpg)
Extended Formulations
The extension complexity (xc) of a polytope P is the minimum number of facets of Q so that P = proj(Q)
e.g. xc(P) = Θ(n logn) for permutahedron
xc(P) = Θ(logn) for a regular n-gon, but Ω(√n) for its perturbation
In general, P = {x | y, (x,y) in Q} E
…analogy with quantifiers in Boolean formulae
![Page 15: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/15.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
![Page 16: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/16.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
Through EFs, we can reduce # facets exponentially!
![Page 17: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/17.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
Through EFs, we can reduce # facets exponentially!
Hence, we can run standard LP solvers instead of the ellipsoid algorithm
![Page 18: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/18.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
Through EFs, we can reduce # facets exponentially!
Hence, we can run standard LP solvers instead of the ellipsoid algorithm EFs often give, or are based on new combinatorial insights
![Page 19: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/19.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
Through EFs, we can reduce # facets exponentially!
Hence, we can run standard LP solvers instead of the ellipsoid algorithm EFs often give, or are based on new combinatorial insights
e.g. Birkhoff-von Neumann Thm and permutahedron
![Page 20: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/20.jpg)
Applications of EFs
In general, P = {x | y, (x,y) in Q} E
Through EFs, we can reduce # facets exponentially!
Hence, we can run standard LP solvers instead of the ellipsoid algorithm EFs often give, or are based on new combinatorial insights
e.g. Birkhoff-von Neumann Thm and permutahedron e.g. prove there is low-cost object, through its polytope
![Page 21: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/21.jpg)
Explicit, Hard Polytopes?
![Page 22: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/22.jpg)
Explicit, Hard Polytopes? Definition: TSP polytope:
P = conv{1F| F is the set of edges on a tour of Kn}
![Page 23: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/23.jpg)
Explicit, Hard Polytopes? Definition: TSP polytope:
P = conv{1F| F is the set of edges on a tour of Kn} (If we could optimize over this polytope, then P = NP)
![Page 24: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/24.jpg)
Explicit, Hard Polytopes?
Can we prove unconditionally there is no small EF?
Definition: TSP polytope:
P = conv{1F| F is the set of edges on a tour of Kn} (If we could optimize over this polytope, then P = NP)
![Page 25: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/25.jpg)
Explicit, Hard Polytopes?
Can we prove unconditionally there is no small EF?
Definition: TSP polytope:
P = conv{1F| F is the set of edges on a tour of Kn}
Caveat: this is unrelated to proving complexity l.b.s
(If we could optimize over this polytope, then P = NP)
![Page 26: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/26.jpg)
Explicit, Hard Polytopes?
Can we prove unconditionally there is no small EF?
[Yannakakis ’90]: Yes, through the nonnegative rank
Definition: TSP polytope:
P = conv{1F| F is the set of edges on a tour of Kn}
Caveat: this is unrelated to proving complexity l.b.s
(If we could optimize over this polytope, then P = NP)
![Page 27: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/27.jpg)
An Abridged History
Theorem [Yannakakis ’90]: Any symmetric EF for TSP or matching has size 2Ω(n)
![Page 28: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/28.jpg)
An Abridged History
Theorem [Yannakakis ’90]: Any symmetric EF for TSP or matching has size 2Ω(n)
…but asymmetric EFs can be more powerful
![Page 29: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/29.jpg)
An Abridged History
Theorem [Yannakakis ’90]: Any symmetric EF for TSP or matching has size 2Ω(n)
…but asymmetric EFs can be more powerful
…
…
![Page 30: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/30.jpg)
An Abridged History
Theorem [Yannakakis ’90]: Any symmetric EF for TSP or matching has size 2Ω(n)
…but asymmetric EFs can be more powerful
…
…
Theorem [Fiorini et al ’12]: Any EF for TSP has size 2Ω(√n) (based on a 2Ω(n) lower bd for clique)
Approach: connections to non-deterministic CC
![Page 31: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/31.jpg)
An Abridged History II
Theorem [Braun et al ’12]: Any EF that approximates clique within n1/2-eps has size exp(neps)
Approach: Razborov’s rectangle corruption lemma
![Page 32: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/32.jpg)
An Abridged History II
Theorem [Braun et al ’12]: Any EF that approximates clique within n1/2-eps has size exp(neps)
Approach: Razborov’s rectangle corruption lemma
Theorem [Braverman, Moitra ’13]: Any EF that approximates clique within n1-eps has size exp(neps)
Approach: information complexity
![Page 33: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/33.jpg)
An Abridged History II
Theorem [Braun et al ’12]: Any EF that approximates clique within n1/2-eps has size exp(neps)
Approach: Razborov’s rectangle corruption lemma
Theorem [Braverman, Moitra ’13]: Any EF that approximates clique within n1-eps has size exp(neps)
Approach: information complexity
see also [Braun, Pokutta ’13]: reformulation using common information, applications to avg. case
![Page 34: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/34.jpg)
An Abridged History III
Theorem [Chan et al ’12]: Any EF that approximates MAXCUT within 2-eps has size nΩ(log n/loglog n)
Approach: reduction to Sherali-Adams
![Page 35: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/35.jpg)
An Abridged History III
Theorem [Chan et al ’12]: Any EF that approximates MAXCUT within 2-eps has size nΩ(log n/loglog n)
Approach: reduction to Sherali-Adams
Theorem [Rothvoss ’13]: Any EF for perfect matching has size 2Ω(n) (same for TSP)
Approach: hyperplane separation lower bound
![Page 36: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/36.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 37: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/37.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 38: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/38.jpg)
The Factorization Theorem
![Page 39: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/39.jpg)
The Factorization Theorem
How can we prove lower bounds on EFs?
![Page 40: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/40.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
![Page 41: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/41.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
Definition of the slack matrix…
![Page 42: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/42.jpg)
The Slack Matrix
![Page 43: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/43.jpg)
The Slack Matrix
P
![Page 44: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/44.jpg)
The Slack Matrix
P S(P)
vertex
face
t
![Page 45: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/45.jpg)
The Slack Matrix
P S(P)
vertex
face
t
vj
![Page 46: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/46.jpg)
The Slack Matrix
P S(P)
vertex
face
t
vj
<ai,x> ≤ bi
![Page 47: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/47.jpg)
The Slack Matrix
The entry in row i, column j is how slack the jth vertex is on the ith constraint
P S(P)
vertex
face
t
vj
<ai,x> ≤ bi
![Page 48: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/48.jpg)
The Slack Matrix
The entry in row i, column j is how slack the jth vertex is on the ith constraint
bi - <ai, vj>
P S(P)
vertex
face
t
vj
<ai,x> ≤ bi
![Page 49: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/49.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
Definition of the slack matrix…
![Page 50: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/50.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
Definition of the slack matrix…
Definition of the nonnegative rank…
![Page 51: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/51.jpg)
Nonnegative Rank
S =
![Page 52: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/52.jpg)
Nonnegative Rank
S M1 Mr = + + …
rank one, nonnegative
![Page 53: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/53.jpg)
Nonnegative Rank
S M1 Mr = + + …
Definition: rank+(S) is the smallest r s.t. S can be written as the sum of r rank one, nonneg. matrices
rank one, nonnegative
![Page 54: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/54.jpg)
Nonnegative Rank
S M1 Mr = + + …
Definition: rank+(S) is the smallest r s.t. S can be written as the sum of r rank one, nonneg. matrices
rank one, nonnegative
Note: rank+(S) ≥ rank(S), but can be much larger too!
![Page 55: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/55.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
![Page 56: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/56.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
xc(P) = rank+(S(P))
![Page 57: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/57.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
xc(P) = rank+(S(P))
Intuition: the factorization gives a change of variables that preserves the slack matrix!
![Page 58: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/58.jpg)
The Factorization Theorem
[Yannakakis ’90]:
How can we prove lower bounds on EFs?
Geometric Parameter
Algebraic Parameter
xc(P) = rank+(S(P))
Intuition: the factorization gives a change of variables that preserves the slack matrix!
Next we will give a method to lower bound rank+ via information complexity…
![Page 59: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/59.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 60: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/60.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 61: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/61.jpg)
The Rectangle Bound
=
rank one, nonnegative
S M1 Mr + + …
![Page 62: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/62.jpg)
The Rectangle Bound
=
rank one, nonnegative
M1 Mr + + …
![Page 63: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/63.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
Mr
![Page 64: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/64.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
Mr
![Page 65: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/65.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
![Page 66: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/66.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
The support of each Mi is a combinatorial rectangle
![Page 67: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/67.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
The support of each Mi is a combinatorial rectangle
rank+(S) is at least # rectangles needed to cover supp of S
![Page 68: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/68.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
rank+(S) is at least # rectangles needed to cover supp of S
![Page 69: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/69.jpg)
The Rectangle Bound
= + + …
rank one, nonnegative
rank+(S) is at least # rectangles needed to cover supp of S
Non-deterministic Comm. Complexity
![Page 70: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/70.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 71: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/71.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 72: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/72.jpg)
A Sampling Argument
= + + …
![Page 73: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/73.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
![Page 74: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/74.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
![Page 75: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/75.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
![Page 76: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/76.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
![Page 77: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/77.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 78: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/78.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 79: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/79.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 80: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/80.jpg)
A Sampling Argument
= + + …
This outputs a uniformly random sample from T
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 81: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/81.jpg)
A Sampling Argument
= + + …
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 82: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/82.jpg)
A Sampling Argument
= + + …
If r is too small, this procedure uses too little entropy!
T = { }, set of entries in S with same value
Choose Mi proportional to total value on T
Choose (a,b) in T proportional to relative value in Mi
![Page 83: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/83.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 84: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/84.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 85: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/85.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 86: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/86.jpg)
The Construction of [Fiorini et al]
S constraints:
vertices:
correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 87: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/87.jpg)
The Construction of [Fiorini et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 88: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/88.jpg)
The Construction of [Fiorini et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 89: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/89.jpg)
The Construction of [Fiorini et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
UNIQUE DISJ. Output ‘YES’ if a
and b as sets are disjoint, and ‘NO’ if a and b have one index
in common
correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 90: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/90.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
![Page 91: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/91.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
![Page 92: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/92.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
(1-aTb)2 = 1 – 2aTb + (aTb)2
![Page 93: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/93.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
(1-aTb)2 = 1 – 2aTb + (aTb)2
= 1 – 2 diag(b),aaT + bbT,aaT
![Page 94: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/94.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
(1-aTb)2 = 1 – 2aTb + (aTb)2
= 1 – 2 diag(b),aaT + bbT,aaT
1 ≥ 2diag(b) - bbT,aaT
![Page 95: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/95.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
(1-aTb)2 = 1 – 2aTb + (aTb)2
= 1 – 2 diag(b),aaT + bbT,aaT
1 ≥ 2diag(b) - bbT,aaT
What is the slack?
![Page 96: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/96.jpg)
The Construction of [Fiorini et al] correlation polytope: Pcorr= conv{aaT|a in {0,1}n }
Why is that (a sub-matrix of) the slack matrix?
(1-aTb)2 = 1 – 2aTb + (aTb)2
= 1 – 2 diag(b),aaT + bbT,aaT
1 ≥ 2diag(b) - bbT,aaT
What is the slack? (1-aTb)2
![Page 97: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/97.jpg)
A Hard Distribution
![Page 98: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/98.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
![Page 99: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/99.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 100: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/100.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
How does the sampling procedure specialize to this case? (Recall it generates (a,b) unif. from T)
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 101: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/101.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
Sampling Procedure:
How does the sampling procedure specialize to this case? (Recall it generates (a,b) unif. from T)
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 102: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/102.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
How does the sampling procedure specialize to this case? (Recall it generates (a,b) unif. from T)
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 103: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/103.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
How does the sampling procedure specialize to this case? (Recall it generates (a,b) unif. from T)
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 104: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/104.jpg)
A Hard Distribution Let T = {(a,b) | aTb = 0}, |T| = 3n
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
How does the sampling procedure specialize to this case? (Recall it generates (a,b) unif. from T)
Recall: Sa,b=(1-aTb)2, so Sa,b=1 for all pairs in T
![Page 105: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/105.jpg)
Entropy Accounting 101
![Page 106: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/106.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
![Page 107: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/107.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
≤ n log23
![Page 108: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/108.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
+ choose i
choose (a,b) conditioned on i
≤ n log23
![Page 109: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/109.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
log2r + choose i
choose (a,b) conditioned on i
≤ n log23
![Page 110: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/110.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
log2r (1-δ)n log23 + choose i
choose (a,b) conditioned on i
≤ n log23
![Page 111: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/111.jpg)
Entropy Accounting 101
Sampling Procedure:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
log2r (1-δ)n log23 + choose i
choose (a,b) conditioned on i
≤ n log23 (?)
![Page 112: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/112.jpg)
![Page 113: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/113.jpg)
Suppose that a-j and b-j are fixed
a b bj
aj
![Page 114: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/114.jpg)
Suppose that a-j and b-j are fixed
a b bj
aj
Mi restricted to (a-j,b-j)
![Page 115: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/115.jpg)
Suppose that a-j and b-j are fixed
a b bj
aj
Mi restricted to (a-j,b-j)
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b)
Mi(a,b)
Mi(a,b)
Mi(a,b)
(…bj=0…) (…bj=1…)
![Page 116: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/116.jpg)
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b)
Mi(a,b)
Mi(a,b)
Mi(a,b)
(…bj=0…) (…bj=1…)
![Page 117: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/117.jpg)
If aj=1, bj=1 then aTb = 1, hence Mi(a,b) = 0
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b)
Mi(a,b)
Mi(a,b)
Mi(a,b)
(…bj=0…) (…bj=1…)
![Page 118: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/118.jpg)
If aj=1, bj=1 then aTb = 1, hence Mi(a,b) = 0
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b)
Mi(a,b)
Mi(a,b)
(…bj=0…) (…bj=1…)
zero
![Page 119: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/119.jpg)
If aj=1, bj=1 then aTb = 1, hence Mi(a,b) = 0
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b)
Mi(a,b)
Mi(a,b)
(…bj=0…) (…bj=1…)
But rank(Mi)=1, hence there must be another zero in either the same row or column
zero
![Page 120: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/120.jpg)
If aj=1, bj=1 then aTb = 1, hence Mi(a,b) = 0
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b) Mi(a,b)
(…bj=0…) (…bj=1…)
But rank(Mi)=1, hence there must be another zero in either the same row or column
zero zero
![Page 121: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/121.jpg)
If aj=1, bj=1 then aTb = 1, hence Mi(a,b) = 0
(a1..j-1,aj=0,aj+1…n)
(a1..j-1,aj=1,aj+1…n)
Mi(a,b) Mi(a,b)
(…bj=0…) (…bj=1…)
But rank(Mi)=1, hence there must be another zero in either the same row or column
zero zero
H(aj,bj| i, a-j, b-j) ≤ 1 < log23
![Page 122: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/122.jpg)
Entropy Accounting 101
Generate uniformly random (a,b) in T:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
log2r + choose i
choose (a,b) conditioned on i
≤ n log23
![Page 123: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/123.jpg)
Entropy Accounting 101
Generate uniformly random (a,b) in T:
� Let Ri be the sum of Mi(a,b) over (a,b) in T and
let R be the sum of Ri
� Choose i with probability Ri/R
� Choose (a,b) with probability Mi(a,b)/Ri
Total Entropy:
log2r n + choose i
choose (a,b) conditioned on i
≤ n log23
![Page 124: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/124.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 125: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/125.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 126: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/126.jpg)
Approximate EFs [Braun et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
![Page 127: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/127.jpg)
Approximate EFs [Braun et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
Is there a K (with small xc) s.t. Pcorr K (C+1)Pcorr?
![Page 128: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/128.jpg)
Approximate EFs [Braun et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
Is there a K (with small xc) s.t. Pcorr K (C+1)Pcorr?
+ C
![Page 129: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/129.jpg)
Approximate EFs [Braun et al]
S constraints:
vertices: a in {0,1}n
b in {0,1}n
(1-aTb)2
New Goal: Output the answer to UDISJ with prob. at least ½ + 1/2(C+1)
Is there a K (with small xc) s.t. Pcorr K (C+1)Pcorr?
+ C
![Page 130: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/130.jpg)
![Page 131: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/131.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
![Page 132: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/132.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
There is a natural barrier at C = √n for proving l.b.s:
![Page 133: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/133.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
Claim: If UDISJ can be computed with prob. ½+1/2(C+1) using o(n/C2) bits, then UDISJ can be computed with prob. ¾ using o(n) bits
There is a natural barrier at C = √n for proving l.b.s:
![Page 134: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/134.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
Claim: If UDISJ can be computed with prob. ½+1/2(C+1) using o(n/C2) bits, then UDISJ can be computed with prob. ¾ using o(n) bits
Proof: Run the protocol O(C2) times and take the majority vote
There is a natural barrier at C = √n for proving l.b.s:
![Page 135: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/135.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
There is a natural barrier at C = √n for proving l.b.s:
![Page 136: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/136.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
Corollary [from K-S]: Computing UDISJ with probability ½+1/2(C+1) requires Ω(n/C2) bits
There is a natural barrier at C = √n for proving l.b.s:
![Page 137: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/137.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
Corollary [from K-S]: Computing UDISJ with probability ½+1/2(C+1) requires Ω(n/C2) bits
There is a natural barrier at C = √n for proving l.b.s:
Theorem [B-M]: Computing UDISJ with probability ½+1/2(C+1) requires Ω(n/C) bits
![Page 138: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/138.jpg)
Is the correlation polytope hard to approximate for large values of C?
Analogy: Is UDISJ hard to compute with prob. ½+1/2(C+1) for large values of C?
Theorem [B-M]: Any EF that approximates clique within n1-eps has size exp(neps)
There is a natural barrier at C = √n for proving l.b.s:
Theorem [B-M]: Computing UDISJ with probability ½+1/2(C+1) requires Ω(n/C) bits
![Page 139: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/139.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 140: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/140.jpg)
Outline
Part I: Tools for Extended Formulations
� Yannakakis’s Factorization Theorem
� The Rectangle Bound
� A Sampling Argument
Part II: Applications
� Correlation Polytope
� Approximating the Correlation Polytope
� Matching Polytope
![Page 141: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/141.jpg)
The Matching Polytope [Edmonds] PPM= conv{1M| M is a perfect matching in Kn}
![Page 142: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/142.jpg)
The Matching Polytope [Edmonds]
S
constraints:
vertices: 1M
U [n] with |U| = odd
PPM= conv{1M| M is a perfect matching in Kn}
![Page 143: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/143.jpg)
The Matching Polytope [Edmonds]
S
constraints:
vertices: 1M
U [n] with |U| = odd
|δ(U) M|-1
PPM= conv{1M| M is a perfect matching in Kn}
![Page 144: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/144.jpg)
The Matching Polytope [Edmonds]
S
constraints:
vertices: 1M
U [n] with |U| = odd
|δ(U) M|-1
Is there a small rectangle covering?
PPM= conv{1M| M is a perfect matching in Kn}
![Page 145: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/145.jpg)
The Matching Polytope [Edmonds]
S
constraints:
vertices: 1M
U [n] with |U| = odd
|δ(U) M|-1
Is there a small rectangle covering?
PPM= conv{1M| M is a perfect matching in Kn}
Yes! Just guess two edges in M, crossing the cut
![Page 146: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/146.jpg)
Hyperplane Separation Lemma [Rothvoss] attributed to [Fiorini]:
![Page 147: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/147.jpg)
Hyperplane Separation Lemma [Rothvoss] attributed to [Fiorini]:
Lemma: For slack matrix S, any matrix W:
rank+(S) ≥ S, W
||S||∞ α
where α = max W,R s.t. R is rank one, entries in [0,1]
![Page 148: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/148.jpg)
Hyperplane Separation Lemma [Rothvoss] attributed to [Fiorini]:
Lemma: For slack matrix S, any matrix W:
rank+(S) ≥ S, W
||S||∞ α
where α = max W,R s.t. R is rank one, entries in [0,1]
Proof:
W,S = Σ ||Ri||∞ W, Ri/||Ri||∞ ≤ α r ||S||∞
![Page 149: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/149.jpg)
Theorem [Rothvoss ’13]: Any EF for perfect matching has size 2Ω(n) (same for TSP)
![Page 150: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/150.jpg)
Theorem [Rothvoss ’13]: Any EF for perfect matching has size 2Ω(n) (same for TSP)
How do we choose W?
![Page 151: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/151.jpg)
Theorem [Rothvoss ’13]: Any EF for perfect matching has size 2Ω(n) (same for TSP)
WU,M =
-∞ if |δ(U) M| = 1
1/Q3 if |δ(U) M| = 3
-1/Qk if |δ(U) M| = k 0 else
How do we choose W?
![Page 152: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/152.jpg)
Theorem [Rothvoss ’13]: Any EF for perfect matching has size 2Ω(n) (same for TSP)
WU,M =
-∞ if |δ(U) M| = 1
1/Q3 if |δ(U) M| = 3
-1/Qk if |δ(U) M| = k 0 else
Proof is a substantial modification to Razborov’s rectangle corruption lemma
How do we choose W?
![Page 153: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/153.jpg)
Summary:
� Extended formulations and Yannakakis’ factorization theorem
![Page 154: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/154.jpg)
Summary:
� Extended formulations and Yannakakis’ factorization theorem
� Lower bound techniques: rectangle bound, information complexity, hyperplane separation
![Page 155: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/155.jpg)
Summary:
� Extended formulations and Yannakakis’ factorization theorem
� Lower bound techniques: rectangle bound, information complexity, hyperplane separation
� Applications: connections between correlation polytope and disjointness,
![Page 156: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/156.jpg)
Summary:
� Extended formulations and Yannakakis’ factorization theorem
� Lower bound techniques: rectangle bound, information complexity, hyperplane separation
� Applications: connections between correlation polytope and disjointness,
� Open question: Can we prove lower bounds against general SDPs?
![Page 157: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/157.jpg)
Any Questions? Summary:
� Extended formulations and Yannakakis’ factorization theorem
� Lower bound techniques: rectangle bound, information complexity, hyperplane separation
� Applications: connections between correlation polytope and disjointness,
� Open question: Can we prove lower bounds against general SDPs?
![Page 158: Extended Formulations and Information Complexitypeople.csail.mit.edu/moitra/docs/tutorialf.pdf · Extended Formulations The extension complexity (xc) of a polytope P is the minimum](https://reader031.vdocuments.site/reader031/viewer/2022041012/5ebf87cf9ea42d6535606c61/html5/thumbnails/158.jpg)
Thanks!
P vj