polynomial bounds for the grid-minor theorem
DESCRIPTION
Polynomial Bounds for the Grid-Minor Theorem. Chandra Chekuri University of Illinois at Urbana-Champaign. Julia Chuzhoy Toyota Technological Institute at Chicago. Grid Minor Theorem (Excluded Grid Theorem) [Robertson, Seymour ‘86]. Graph Minor Theory [Robertson – Seymour] - PowerPoint PPT PresentationTRANSCRIPT
Polynomial Bounds for the Grid-Minor Theorem
Julia ChuzhoyToyota Technological Institute at Chicago
Chandra ChekuriUniversity of Illinois at
Urbana-Champaign
Grid Minor Theorem (Excluded Grid Theorem)
[Robertson, Seymour ‘86]Graph Minor Theory [Robertson – Seymour]– Wagner’s conjecture: any infinite sequence of
finite graphs contains two graphs G,G’ where G is a minor of G’
– Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor
Grid Minor Theorem (Excluded Grid Theorem)
[Robertson, Seymour ‘86]Graph Minor Theory [Robertson – Seymour]– Wagner’s conjecture: any infinite sequence of
finite graphs contains two graphs G,G’ where G is a minor of G’
– Grid-Minor Theorem: if the treewidth of G is large, then G contains a large grid minor
Treewidth
Trees General Graphs
Tree Decomposition
d e
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a
f
g h
Example from Bodlaender’s talk
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fc
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Tree Decomposition
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Example from Bodlaender’s talk
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Tree Decomposition
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g h
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ga
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Example from Bodlaender’s talk
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d e
Tree Decomposition
d e
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f
g h
a
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g h
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ga
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Example from Bodlaender’s talk
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d e
Tree Decomposition
d e
b
c
a
f
g h
a
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g h
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ga
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b
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Example from Bodlaender’s talk
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d e
Tree Decomposition
d e
b
c
a
f
g h
a
f
g h
c
ga
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b
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a
Example from Bodlaender’s talk
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Tree Decomposition
d e
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a
f
g h
a
f
g h
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ga
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b
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a
Example from Bodlaender’s talk
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Tree Decomposition
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g h
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ga
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Example from Bodlaender’s talk
Decomposition width = max # of vertices in a bag -1
Treewidth: min width of any decomposition
Treewidth of Some Graphs
• Tree: 1• Cycle: 2• (√n×√n)-grid: √n• n-vertex expander: Ω(n)
Well-Linkedness
Well-Linkedness
A set T of vertices is well-linked in G iff for any two equal-sized subsets A,B of T, we can connect A to B with |A| disjoint paths.
Treewidth and Well-Linkedness
Thm. Let k be the maximum size of any well-linked set of vertices in G. Then:
k≤treewidth(G)≤4k.
Treewidth
Trees Small-Treewidth Graphs
Large-Treewidth Graphs
Grid-Minor Theorem[Robertson, Seymour]
If the treewidth of G is large, then it contains a large grid minor.
Grid-Minor Theorem[Robertson, Seymour]
If the treewidth of G is large, then it contains a large grid minor.
We can obtain the grid from G by a sequence of edge-deletion and edge-contraction operations
a size-4 grid
Minors by Embedding
Minors by Embedding
Grid-Minor Theorem[Robertson, Seymour]
If the treewidth of G is large, then it contains a large grid minor, so:• G contains many disjoint cycles• G contains many disjoint cycles of length 0
mod m• G contains a convenient routing structure• The size of the vertex cover in G is large• …
Applications
• Fixed parameter tractability• Erdos-Posa type results• Graph minor theory• …
If the treewidth of G is large, then it contains a large grid minor.
Grid-Minor Theorem
Grid-Minor TheoremIf the treewidth of G is k, then it contains a grid minor of size f(k).
• Easy to see that • [Robertson, Seymour ‘94]: • Conjecture [Robertson, Seymour ‘94]:
How large is f(k)?
Grid-Minor TheoremIf the treewidth of G is k, then it contains a grid minor of size f(k).• [Robertson, Seymour, Thomas ‘89]:• [Diestel, Gorbunov, Jensen, Thomassen ‘99] – simpler proof• [Kawarabayashi, Kobayashi ‘12], [Leaf, Seymour ‘12]:
• This talk:
Grid-Minor TheoremIf the treewidth of G is k, then it contains a grid minor of size f(k).• In some families of graphs f(k)=Ω(k)– Planar graphs [Robertson, Seymour, Thomas ‘94]– Bounded genus graphs [Demaine, Fomin,
Hajiaghayi, Thilikos ‘05]– Graphs excluding a fixed minor [Demaine,
Hajiaghayi ‘08]
Path-of-Sets System
A Path-of-Sets SystemC1 C2 C3 … Ch
• Each Ci is a connected cluster• The clusters are disjoint• Every consecutive pair of clusters connected by h paths• All blue paths are disjoint from each other and internally
disjoint from the clusters
…
A Path-of-Sets SystemC1 C2 C3 … Ch
• Each Ci is a connected cluster• The clusters are disjoint• Every consecutive pair of clusters connected by h paths• All blue paths are disjoint from each other and internally
disjoint from the clusters
h…
A Path-of-Sets SystemC1 C2 C3 … Ch
…
Ci
Interface vertex
The interface vertices are well-linked
inside Ci
A Path-of-Sets SystemC1 C2 C3 … Ch
…
Ci
The interface vertices are well-linked
inside Ci
A Path-of-Sets SystemC1 C2 C3 … Ch
Ci
The interface vertices are well-linked
inside Ci
A Path-of-Sets SystemC1 C2 C3 … Ch
The interface vertices are well-linked
inside Ci
Ci
A Path-of-Sets SystemC1 C2 C3 … Ch
The interface vertices are well-linked
inside Ci
Ci
A Path-of-Sets SystemC1 C2 C3 … Ch
h …
Thm [Leaf, Seymour ‘12]: Given a path-of-sets system, we can efficiently find a grid minor of size Ω(√h).
Corollary: enough to find a path-of-sets system with h=poly(k), where k is the treewidth.
From Path-of-Sets System to Grid Minor
Building the Grid
Building the Grid
Building the Grid
Building the Grid
Building the Grid
Building the Grid
C1
C2
C3
C4
C1 C2 C3 … Ch
P1P2P3
Ph
……
Direct vs Indirect Path
Direct path
Indirect path
Building the Grid
C1 C2 C3 … Ch
C1
C2
C3
C4For each Ci, we’ll be looking for a direct path connecting some consecutive pair
of horizontal paths
C1 C2 C3 … Ch
P1P2P3
Ph
……
Routing Inside ClustersCi
P1P2P3P4
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Routing Inside Clusters
P1
P2
P3
P4
Good scenario:The path graph for all Ci contains the same path
P1
P2 P3 P4
“Bad” scenario:
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Inside the Super-Clusters
Thm: for any n-vertex graph G,• Either there is a tree in G with Ω(√n) leaves• Or there is a 2-path in G of length Ω(√n)
Inside the Super-Clusters
Thm: for any n-vertex graph G,• Either there is a tree in G with Ω(√n) leaves• Or there is a 2-path in G of length Ω(√n)
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
• Cluster Ci is good if Hi has a tree with √h leaves.• Assume all clusters are good.
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
Routing Inside ClustersCi
P1P2P3P4
P1
P2 P3
P4
Path graph Hi for Ci
We say that Ci chooses the paths corresponding to the leaves of the tree.
Routing Inside Clusters
…
Routing Inside Clusters
…
Routing Inside Clusters
…
Routing Inside Clusters
If r is large enough, then some choice of √h will repeat h times.
…
r
Routing Inside Clusters
Routing Inside Clusters
Routing Inside Clusters
Routing Inside Clusters
Routing Inside Clusters
Re-connect the paths via even-indexed clusters, so all odd-indexed clusters choose the same paths!
Completing the Proof
Super-cluster
Completing the Proof
For each super-cluster Si:• Either build a large grid minor inside Si
• Or show that Si is a good cluster
Inside the Super-Clusters
P1P2P3P4
P1
P2 P3
P4 P1
P2 P3
P4 P1
P2 P3
P4
P1
P2 P3
P4
P1
P2 P3
P4
H: path-graph for the super-
cluster
Inside the Super-Clusters
P1P2P3P4
P1
P2 P3
P4
H: path-graph for the super-
cluster
• Either H contains a tree with many leaves
• Or it contains a long 2-path Can build a grid-minor directly
P1
P2 P3
P4
H1
P1
P2 P3
P4
H2
P1
P2 P3
P4
H4
P1
P2 P3
P4
H3
Inside the Super-Clusters
P1
P2 P3
P4
H: path-graph for the super-
cluster
P1
P2 P3
P4
H1
P1
P2 P3
P4
H2
P1
P2 P3
P4
H4
P1
P2 P3
P4
H3
v1 v2 v3 v4 … v√h
Want to show: this path appears in all Hi’sWill show: large sub-path appears in half the Hi’s
Inside the Super-Clusters
P1
P2 P3
P4
H: path-graph for the super-
cluster
P1
P2 P3
P4
H1
v1 v2 v3 v4 … v√h
Inside the Super-Clusters
P1
P2 P3
P4
H: path-graph for the super-
cluster
P1
P2 P3
P4
H1
v1 v2 v3 v4 … v√h
Inside the Super-Clusters
P1
P2 P3
P4
H: path-graph for the super-
cluster
P1
P2 P3
P4
H1
P1
P2 P3
P4
H2
P1
P2 P3
P4
H4
P1
P2 P3
P4
H3
v1 v2 v3 v4 … v√h
Completing the Proof
For each super-cluster Si:• Either build a large grid minor inside Si
• Or show that Si is a good cluster
Finding the Path-of-Sets System
C1 C2 C3 … Ch
…
Routing Problems
Node-Disjoint Paths (NDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via node-disjoint paths
Node-Disjoint Paths (NDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via node-disjoint paths
Node-Disjoint Paths (NDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via node-disjoint paths
Solution value: 2
Edge-Disjoint Paths (EDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via edge-disjoint paths
Edge-Disjoint Paths (EDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via edge-disjoint paths
Solution value: 3
Edge-Disjoint Paths (EDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via edge-disjoint paths
NDP is more general than EDP
Edge-Disjoint Paths (EDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Route as many pairs as possible via edge-disjoint paths
n – number of graph verticesk – number of demand pairsterminals – vertices participating in the demand pairs
EDP and NDP
• Efficient algorithm when k is constant [Robertson, Seymour ‘90].– running time: f(k)n2 [Kawarabayashi,Kobayashi,
Reed]• General k: both problems are NP-hard [Karp ’72]
An α-approximation algorithm:• efficient algorithm• always produces solutions of value at least
OPT/α.
Approximation Algorithm [Kolliopoulos, Stein ‘98]While there is a path P connecting any demand pair that has not been routed yet:• Add such a path of smallest length to the solution• (Delete from OPT all paths sharing vertices with P or
routing the same demand pair)
Analysis• If the length of P is less than – at most
paths are deleted from OPT.• If the length of P is more than – at most
paths remain in OPT.
Approximation Algorithm [Kolliopoulos, Stein ‘98]While there is a path P connecting any demand pair that has not been routed yet:• Add such a path of smallest length to the solution• (Delete from OPT all paths sharing vertices with P or
routing the same demand pair)
Analysis• If the length of P is less than – at most
paths are deleted from OPT.• If the length of P is more than – at most
paths remain in OPT.
-approximation
This algorithm gives an –approximation for EDP.An -approximation is known[Chekuri, Khanna, Shepherd ’06].
Approximation Status of NDP
• -approximation algorithm– Even on planar graphs– Even on grid graphs
• -hardness of approximation for any[Andrews, Zhang ‘05], [Andrews, C, Guruswami, Khanna, Talwar, Zhang ’10]
Open Problem: NDP on a Grid Graph
s1
t1
s2
t2
s3
t3
Open Problem: NDP on a Grid Graph
Open Problem: NDP on a Grid Graph
• -approximation algorithm [Chekuri, Khanna, Shepherd ’06].
• The problem is NP-hard
Ongoing work:• O(n1/4)-approximation [builds on Aggarwal, Kleinberg,
Williamson ‘96]• Hard to approximate up to some constant c.
Open Problem: EDP on Wall Graphs
EDP with Congestion (EDPwC)
• A factor- approximation algorithm with congestion c routes . demand pairs with congestion at most c.
optimum number of pairs with no congestion allowed
EDPwC• Congestion O(log n/log log n): constant approximation
[Raghavan, Thompson ’87]
• -approximation with congestion c [Azar, Regev ’01], [Baveja, Srinivasan ’00], [Kolliopoulos, Stein ‘04]
• polylog(n)-approximation with congestion poly(log log n) [Andrews ‘10]
• polylog(k)-approximation with congestion 14 [C, ‘11]
• polylog(k)-approximation with congestion 2 [C, Li, ‘12]
• polylog(k)-approximation with constant congestion for NDP [Chekuri, Ene ’13]
Edge-Disjoint Paths
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).Goal: Connect as many pairs as possible by edge-disjoint paths.• An instance is well-linked iff the set of all
terminals is well-linked in G.• Theorem [Chekuri, Khanna Shepherd ‘04]: an
α - approximation algorithm on well-linked instances gives an O(α log2k)-approximation on any instance.
terminals
• If an instance is well-linked, its treewidth is Ω(k)
• If the treewidth of G is k, can find a well-linked set of size
Algorithms for Edge-Disjoint Paths
well-linked instance
large crossbar
find the routing
graph of treewidth k
“similar” to path-of-sets
system
Crossbar
• Number of clusters poly(log k), not poly(k)• The paths are not disjoint from each other and from
the clusters, but cause a constant edge congestionWant: Path-of-sets systemCan get: Tree-of-sets system …
degree-3 tree
✔✔
Tree-of-Sets SystemA degree-3 tree with h vertices• Every vertex a connected cluster of G• Every edge – a collection of h paths in G– the blue paths are node-disjoint from each other
and internally disjoint from the clusters• For each cluster, its interface is well-linked.
h=kε
• If the tree has height h1/3 – done• Otherwise it has h1/3 leaves• Will build a path-of-sets system
on a subset of h1/3 leaves
Assumption: the tree has h1/3 leaves
High-Level Idea
High-Level Idea
Stage 1: connect every leaf to the root by many disjoint pathsStage 2: exploit these paths to build a path-of-sets system
Stage 1• h1/3 leaves• h parallel blue edges• each leaf gets h3/4
green paths
Stage 1• h1/3 leaves• h parallel blue edges• each leaf gets h3/4
green paths
Stage 1• h1/3 leaves• h parallel blue edges• each leaf gets h3/4
green paths
Stage 1• h1/3 leaves• h parallel blue edges• each leaf gets h2/3
green paths
Stage 2• Every leaf receives h2/3 flow units from the root• Will exploit these flows to build a path-of-sets system• Process the tree from top to bottom
Stage 2
Stage 2
A B
Stage 2
A B
Stage 2
Stage 2
Stage 2
Stage 2
A B C D
X
R
Stage 2
A B
A’B’
C D
C’D’
X
R
Stage 2
A B
A’B’
C D
C’D’
• h1/3 blue paths• intersect at most
h1/3 green paths from each set
R
X
Stage 2
A B
A’B’
C D
C’D’
B’
C
C’
• h1/3 leaves• each leaf had h2/3
green paths• want h1/3 parallel
paths in path-of-sets system
• tree height ≤ h1/3
R
X
Proof Summary
1. Path-of-sets system gives a large grid minor [Leaf, Seymour ‘12]
2. If G has large treewidth, can build a large tree-of-sets system: extension of [C ‘11], [C, Li ‘12], [Chekuri, Ene ‘12]
3. Can build a path-of-sets system from a tree-of-sets system
polylog(k)-approximation for Node-Disjoint Paths
with congestion 2
Bypassing the Grid-Minor Theorem?
Large-Treewidth Graph Decomposition [C, Chekuri ‘12]
treewidth ≥ r
treewidth ≥ r
treewidth ≥ r
treewidth ≥ r
GTreewidth k
h
Example of Use: Feedback Vertex Set
Feedback Vertex Set: given a graph G, select a min-cardinality subset U of vertices, such that G\U has no cycles.
k: size of feedback vertex setWant: a fixed-parameter tractable algorithm, with running time f(k)poly(n).
The Algorithm
• If treewidth of G at most g(k) – dynamic programming on the tree decomposition– running time:
• otherwise: G contains a grid minor of size , so feedback vertex set value more than k.
Can choose , running time . What is g(k)?
Typical use of grid-minor theorem in fixed parameter tractability algorithms.
Bi-dimentionality theory
• If treewidth of G at most g(k) – dynamic programming on the tree decomposition– running time:
• otherwise: G contains a grid minor of size , so feedback vertex set value more than k.
Can choose , running time .
The Algorithm
What is g(k)?
Large-Treewidth Graph Decomposition
treewidth ≥ 2
treewidth ≥ 2
treewidth ≥ 2
treewidth ≥ 2
G
k+1 feedback vertex set value at least k+1
• If treewidth of G at most g(k) – dynamic programming on the tree decomposition– running time:
• otherwise: G contains a grid minor of size , so feedback vertex set value more than k.
Can choose , running time .
The Algorithm
What is g(k)?
Conclusion
• First polynomial bound on grid minor size, , • Best current negative result:• Better upper/lower bounds?• Better/simpler constructions of path-of-sets or
tree-of-sets systems?
More Open Questions
• Approximability of NDP/EDP:– general graphs– planar graphs– grid/wall graphs
• Congestion minimization– -approximation [Raghavan,
Thompson ‘87]– -hard to approximate [Andrews, Zhang ‘07]
• Many more… Thank you!