Subexponential Algorithms for Subgraph Isomorphism - PowerPoint Presentation

Slide1

Subexponential Algorithms for Subgraph Isomorphism(and related problems)on Minor-Free Graphs

Hans Bodlaender (U Utrecht, TU Eindhoven)Jesper Nederlof (TU Eindhoven) Tom van der Zanden (U Utrecht)

1Slide2

Subgraph IsomorphismGiven: a

pattern graph and host graph Question: d have a subgraph isomorphic to

Upper bound and (assuming ETH) matching lower bound

2Slide3

Complexity of Subgraph Isomorphism

Let and

and

General graphs:

time

lower bound [

Fomin

et al.

’

15]

Planar:

time [Dorn ‘09]

Connected and bounded degree: see previous talkThis talk:

time

This talk:

lower bound under ETH

3Slide4

Our ResultsTheorem

For any , and If is -Minor Free, and has treewidth

,

Subgraph

Isomorphism can be solved in

time.

Interesting special cases:

and

of bounded

treewidth,

and planar. Theorem

Assume the ETH. There is no

algorithm for

Subgraph Isomorphism, even when

and

are forests, with each component a caterpillar, or

is connected, has pathwidth 2 with one vertex of degree larger than

3 and is a tree

4Slide5

Similar resultsUpper and lower bound result hold for:Induced Subgraph Isomorphism

MinorInduced MinorTopological MinorShallow Minor (bounded radius)With small modifications to proofs and arguments5Slide6

Lower bound proofSimple proof (=> incomplete talk…) using

Theorem. (By Sparsification Lemma)Assume the ETH. There is no algorithm for 3-SAT with

clauses.

6Slide7

Numbers and stringsNumber variables

, and clauses Change each number to bitstring with

bits:

Take the number in binary

Add the number in binary with 0 and 1 switched

Add the string reversed:I.e.: 6 -> 110 001 100 011Note: strings are `incomparable’

7Slide8

G and P (partly on board)G: For each variable

a caterpillar for and a caterpillar for

P

For each variable

a caterpillar (“to eat the false literal”)

For each clause a caterpillar (“to make sure it is satisfied”)

paths

(“to eat unused clauses in true literals” )

8Slide9

End of proofInstance of 3-Sat is satisfiable

, if and only if P is isomorphic to a subgraph of GG and P has vertices, so

algorithm would break ETH

9Slide10

ModificationsConnected G and P:Add one vertex with edges (see board)

Minor, topological minor: same constructionInduced subgraph, induced minor: trivial modification to proof10Slide11

Outline of algorithmDynamic programming on a tree decomposition of

Partial solution: map of bag into + set of connected components Use isomorphism to reduce the number of partial solutions 

11Slide12

Partial Solution

Partial solution:

12Slide13

Partial Solution

Partial solution:

Consider connected components

13Slide14

Partial Solution

Partial solution:

Consider connected components

Either completely IN or completely OUT

14Slide15

Partial SolutionA partial solution can be characterized by

Image of options

List for each connected component of

if it is

in

or

out:

can be

options:

still too many

15

 Slide16

Partial SolutionA partial solution can be characterized by

Image of options

List for each connected component of

if it is

in

or

out: can be

options: still too manyIdea:

it does not matter which of theisomorphic components we have, onlyhow many

16

 Slide17

Cases For well chosen c: A

component is large if it has at least c log n vertices Small components are isomorphic ifThere is an isomorphism that also takes care of the adjacencies to

17

 Slide18

Info In DPFor each large component: store if it is

in or out large components:

For each equivalence class of small components:

Store how many in the class are

in

Equivalence test is sufficiently fast (only done on

subgraphs with

vertices)

Counting argument shows that the number of cases is bounded by

18Slide19

Number of isomorphisms (1)

Theorem [Amini et al. ‘12]: for any graph , there is a constant so that there are at most

non-isomorphic

-vertex

-Minor Free graphs

For small enough

, at most

19

This bounds the number of isomorphism classes except for how components are attached to

 Slide20

Small ComponentsComponent

itself: optionsWhat about incidence to

?

Each conn. comp.

is incident to

subset

Each vertex

is incident to

subset of

20Slide21

Small Components – Case 1Small Component with Big Neighborhood

Lemma [Gajarský et al.]: there areO(tw) small components with Big Neighborhood2O(tw

)

cases

21Slide22

Small Components – Case 2

Small Component with Small NeighborhoodLemma [Gajarský et al.]: there

are

O(

tw

) different possible small neighborhoods

22Slide23

Counting Small Components

distinct isomorphism classes:

distinct small

neighborhoods

Each vertex:

cases for adjacency to neighborhood

Total:

Small enough

:

23Slide24

Wrapping Up

cases for small components (

of each)

for large comps., large

nbh

.’s

cases for



algorithm for Subgraph Isomorphism on Minor-Free graphs

24Slide25

Consequences

algorithm for Subgraph Isomorphism on Minor-Free graphs

+ result

Fomin

et al. (previous talk)



algorithm for Subgraph Isomorphism on

Apex-Minor-Free graphs for connected patterns

25Slide26

Conclusions

algorithm and lower bound for Subgraph Isomorphism and other problems on H-minor free graphs (resolves some open problems)Can we do

in the disconnected case? Lower bound?

For which `

embeddings

’ does this work?

Algorithm has nice, simple, useful technique: speed up DP by isomorphism test

Isomorphic subproblems have the same answer

Counting shows there are sufficiently few

26