polynomial bounds for the grid-minor theorem

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

b

c

a

f

g h

Example from Bodlaender’s talk

g hga

f

a

fc

c

d e

Tree Decomposition

d e

b

c

a

f

g h

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

Example from Bodlaender’s talk

c

d e

Tree Decomposition

d e

b

c

a

f

g h

a

f

g h

c

ga

f

b

c

a

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!