COUNTING SUBGRAPHS - PowerPoint Presentation

Slide1

COUNTING SUBGRAPHS

Radu Curticapean, Holger Dell, Dániel Marx

NEW INSIGHTS INTO

Saarland University,Cluster of Excellence (MMCI)

Institute for Computer Science and Control,Hungarian Academy of Sciences (MTA SZTAKI)Slide2

vertices

pattern

vertices

h

ost

count

subgraphs

F

of

G

isomorphic

to H

NOT NECESSARILY INDUCED

COUNTING SUBGRAPHSSlide3

-clique

-

cycle

-

matching

-star

with

vtx

-cover

of

size

any

[NP 85]

[BKK 14]

[BKK 14]

#W[1]-hard

refutes ETH

[CCFHJKX 04]

#W[1]-hard

[FG 04]

#W[1]-hard,

refutes ETH

vertices

pattern

vertices

h

ost

count

subgraphs

F

of

G

isomorphic

to

H

[BHKK 09]

[C 13, CM 14, CM 15]

ETH:

SAT on

-variable formulas

has no

time algorithm

parameterized problem X:

input comes with some parameter

X is FPT:

problem can be solved in time

is #W[1]-hard:

FPT-algorithm for #

-clique with

-oracle

and

all queries have parameter

for some

X most likely not FPT

 Slide4

vertices

pattern

vertices

h

ost

count

subgraphs

F

of

G

isomorphic

to H

poly-time

#W[1]-hard.

with vtx

-coverof

size

-star

-clique

-

cycle

-

matching

 Slide5

poly-time

#W[1]-hard

.

Input:

Graphs

and

Parameter:

Output:

maximum matching in

bounded

for

recursively

enumerable

unbounded

with

vtx

-cover

of

size

-star

[CM 14]

-clique

-

cycle

-

matching

classifies

w

is

poly

-time

assuming

despite

-intermediate cases!

[CTW 08]

 Slide6

poly-time

-

hard

finite

count

/find

induced

subgraphs

from

else

[CTW 08]

count/find

colorful subgraphs

from

has bounded

treewidth

else

[GSS 01][CM 14]

count

homomorphisms from

has bounded

treewidth

else

[G 07]

[DJ 04]

count

subgraphs from

has bounded

matchings

else

[CM 14]

tree-decomposition of H:

tree

with bags

at

Each

is in some bag

Each

is in some bag

If

appears in bags of

,

then

is connected in

width of

: max. bag size

treewidth of

: min. width over all

David Eppstein, Wikipedia

 Slide7

COUNTING

HOMOMORPHISMS

poly-time

-

hard

count

homomorphisms

from

has bounded

treewidth

else

[G 07]

[DJ 04]

time

via standard D

 Slide8

COUNTING

SUBGRAPH EMBEDDINGS

COUNTING

HOMOMORPHISMSSlide9

COUNTING

HOMOMORPHISMS

COUNTING

INJECTIVE HOMOMORPHISMSSlide10

COUNTING

HOMOMORPHISMS

COUNTING

INJECTIVE HOMOMORPHISMS

COUNTING LINEAR

COMBINATIONS OF

COUNTING LINEAR

COMBINATIONS OF

 Slide11

1

2

3

4

Define

for

partition

of

Contract every block of

in

.

Delete multi-

edges

,

keep

self

-loops.

12

3

4

12|3|4

34|1|2

341|2

1234

1|2|3|4

13|2|4

14|2|3

23|1|4

24|1|3

12|34

13|24

14|23

23|14

24|13

34|12

123|4

142|3

234|1

LINEAR COMBINATIONS OF

HOMOMORPHISMS

COUNTING

INJECTIVE HOMOMORPHISMS

Spasm

1

2

34

2

341

1

2

3

4Slide12

1|2|3|4

12|3|4

13|2|4

14|2|3

23|1|4

24|1|3

34|1|2

12|34

13|24

14|23

23|14

24|13

34|12

123|4

142|3

234|1

341|2

1234

1

2

3

4

LINEAR COMBINATIONS OF

HOMOMORPHISMS

COUNTING

INJECTIVE HOMOMORPHISMS

12

3

4

1

2

34

2

341

1

2

3

4Slide13

First

application

:

Counting

any

-

edge

subgraph

in time

best

known

upper

bound

on

treewidth

of

-

edge

graph

-

paths

(

before

: time

)

-

matchings

(

before

: time

)

Lovász

1967Slide14

Scott, Sorkin 2016

:

Graphs

with

edges have

time

poly

time

graphs,

each

with

edges

 Slide15

Second application:

Hardness of counting subgraphs

via fine-grained reductions

 Slide16

#

PartitionedSub

(

Pick

any

.

Given

oracle

for

this

,

we

can

compute

this

in

steps.

 Slide17

#

PartitionedSub

(

under #ETH

[Marx 2010]

even when fixing

and allowing only

under #ETH

even when fixing

and allowing only

Pick

any

.

Given

oracle

for

this

,

we

can

compute

this

in

steps.

has

-

matching

pick expander

with

 Slide18

always in time

.

Exact value of

for

fixed ?

exponent in running time

for #PartitionedSub

exponent in running time

for computing #H

Can

you

beat

treewidth

:

Clique

conjecture

:

if

Lovász

1967