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]
– 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
a
g
h
b
c
d
f
e
Example from Bodlaender’s talk
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
a
g
h
b
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
Tree Decomposition
Decomposition
width
= max # of vertices in a bag -1
a
g
b
h
Treewidth: min width of any
decomposition
c
a
f
c
d
e
g
a
f
f
c
d
e
b
a
c
Example from Bodlaender’s talk
g
h
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 equalsized 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 welllinked 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
…
Grid-Minor Theorem
If the treewidth of G is large, then it contains a
large grid minor.
Grid-Minor Theorem
If the treewidth of G is k, then it contains a grid
minor of size f(k).
How large is f(k)?
• Easy to see that
• [Robertson, Seymour ‘94]:
• Conjecture [Robertson, Seymour ‘94]:
Grid-Minor Theorem
If 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 Theorem
If 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 System
C1
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 System
C1
C2
C3
…
Ch
…
h
•
•
•
•
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 System
C1
C2
C3
…
Ch
…
Ci
Interface
vertex
The interface
vertices are
well-linked
inside Ci
A Path-of-Sets System
C1
C2
C3
…
Ch
…
Ci
The interface
vertices are
well-linked
inside Ci
A Path-of-Sets System
C1
C2
C3
…
Ch
Ci
The interface
vertices are
well-linked
inside Ci
A Path-of-Sets System
C1
C2
C3
…
Ch
Ci
The interface
vertices are
well-linked
inside Ci
A Path-of-Sets System
C1
C2
C3
…
Ch
Ci
The interface
vertices are
well-linked
inside Ci
A Path-of-Sets System
C1
h
C2
C3
…
Ch
…
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
C4
C2
C3
C1
Ph
C3
…
…
…
P1
P2
P3
C2
Ch
Direct vs Indirect Path
Direct path
Indirect
path
Building the Grid
C1
C2
C4
For each Ci, we’ll be looking for a direct
path connecting some consecutive pair
of horizontal paths
C3
C1
Ph
C3
…
…
…
P1
P2
P3
C2
Ch
Routing Inside Clusters
Ci
P1
P2
P3
P4
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P4
P2
P3
Path graph Hi for Ci
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P4
P2
P3
Path graph Hi for Ci
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P4
P2
P3
Path graph Hi for Ci
Routing Inside Clusters
Good scenario:
The path graph for all
Ci contains the same
path
P1
P2
P3
P4
“Bad” scenario:
P1
P2 P3 P4
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P2
P4
P3
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 Clusters
Ci
P1
P2
P3
P4
P1
P4
P2
P3
Path graph Hi for Ci
• Cluster Ci is good if Hi has a tree with √h leaves.
• Assume all clusters are good.
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P2
P4
P3
Path graph Hi for Ci
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P2
P4
P3
Path graph Hi for Ci
Routing Inside Clusters
Ci
P1
P2
P3
P4
P1
P4
P2
P3
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
Supercluster
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
P1
P2
P3
P4
P1 P4
P1 P4
P1 P4
P1 P4
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
P2
P3
P2
P3
P2
P3
Inside the Super-Clusters
P•1 Either H contains a tree with many
P2 leaves
P•3 Or it contains a long 2-path
P4  Can build a grid-minor directly
P1 P4
P1 P4
P1 P4
P1 P4
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
H1
P2
P3
H2
P2
P3
H3
P2
P3
H4
Inside the Super-Clusters
v1
v2
v3
v4
…
v√h
P1 P4
Want to show: this path appears in all Hi’s
Will show: large sub-path appears in half the Hi’s
P1 P4
P1 P4
P1 P4
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
H1
P2
P3
H2
P2
P3
H3
P2
P3
H4
Inside the Super-Clusters
v1
v2
P1 P4
v3
v4
…
v√h
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
H1
Inside the Super-Clusters
v1
v2
P1 P4
v3
v4
…
v√h
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
H1
Inside the Super-Clusters
v1
v2
P1 P4
v3
v4
P1 P4
…
v√h
P1 P4
P1 P4
P1 P4
P2
P3
H: path-graph
for the supercluster
P2
P3
H1
P2
P3
H2
P2
P3
H3
P2
P3
H4
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 nodedisjoint paths
Node-Disjoint Paths (NDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible via nodedisjoint paths
Node-Disjoint Paths (NDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible via nodedisjoint 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 edgedisjoint paths
Edge-Disjoint Paths (EDP)
Input: Graph G, source-sink pairs (s1,t1),…,(sk,tk).
Goal: Route as many pairs as possible via edgedisjoint 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 edgedisjoint 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 edgedisjoint paths
n – number of graph vertices
k – number of demand pairs
terminals – 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
paths are deleted from OPT.
• If the length of P is more than
paths remain in OPT.
– at most
– at most
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 -approximation
to the solution
• (Delete from OPT all paths sharing vertices with P
or routing the same demand pair)
This algorithm gives Analysis
an
–
approximation for EDP.
• If the length of P is less than
– at most
An
-approximation is known
paths are deleted from OPT.
[Chekuri, Khanna, Shepherd ’06].
• If the length of P is more than
– at most
paths remain in OPT.
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
Open Problem: NDP on a Grid Graph
s1
s3
s2
t1
t2
t3
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.
terminals
• 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.
Algorithms for Edge-Disjoint Paths
graph of
treewidth k
well-linked
instance
“similar” to
path-of-sets
system
large
crossbar
find the
routing
• 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
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 congestion
Want: Path-of-sets system
Can get: Tree-of-sets system
…
✔
degree-3 tree
Tree-of-Sets System
h=kε
A 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
Assumption:
the
and internally disjoint from
1/3 the clusters
tree has h
leaves
• For each cluster, its interface is well-linked.
• 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
High-Level Idea
High-Level Idea
Stage 1: connect every
leaf to the root by
many disjoint paths
Stage 2: exploit these
paths to build a pathof-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
R
X
A
B
C
D
Stage 2
R
X
D’
A’
B’ C’
A
B
C
D
Stage 2
R
X
D’
A’
B’ C’
A
B
C
•
•
h1/3 blue paths
intersect at most
h1/3 green paths
D
from each set
Stage 2
•
•
R
•
X
D’
A’
•
h1/3 leaves
each leaf had h2/3
green paths
want h1/3 parallel
paths in path-of-sets
system
tree height ≤ h1/3
B’ C’
A
B
C
D
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 treeof-sets system
polylog(k)-approximation
for Node-Disjoint Paths
with congestion 2
Bypassing the Grid-Minor Theorem?
Large-Treewidth Graph Decomposition
[C, Chekuri ‘12]
G
Treewidth k
treewidth
≥r
treewidth
≥r
treewidth
≥r
h
treewidth
≥r
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 set
Want: 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
Typical time:
use of grid-minor theorem in fixed
– running
parameter
tractability
algorithms.
• otherwise:
G contains
a grid
minor of size
Bi-dimentionality
theory
, so feedback vertex set value more than k.
Can choose
What,isrunning
g(k)? time
.
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
What,isrunning
g(k)? time
.
Large-Treewidth Graph Decomposition
G
treewidth
≥2
treewidth
≥2
treewidth
≥2
k+1
treewidth
≥2
feedback vertex set
value at least k+1
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
What,isrunning
g(k)? time
.
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!