Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised...
Transcript of Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised...
![Page 1: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/1.jpg)
Parameterised Subgraph Counting Problems
Kitty Meeks
University of Glasgow
University of Strathclyde, 27th May 2015
Joint work with Mark Jerrum (QMUL)
![Page 2: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/2.jpg)
What is a counting problem?
Decision problems
Given a graph G , does Gcontain a Hamilton cycle?
Given a bipartite graph G ,does G contain a perfectmatching?
![Page 3: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/3.jpg)
What is a counting problem?
Decision problems Counting problems
Given a graph G , does Gcontain a Hamilton cycle?
How many Hamilton cycles arethere in the graph G ?
Given a bipartite graph G ,does G contain a perfectmatching?
How many perfect matchingsare there in the bipartite graphG ?
![Page 4: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/4.jpg)
What is a parameterised counting problem?
Introduced by Flum and Grohe (2004)
Measure running time in terms of a parameter as well as thetotal input size
Examples:
How many vertex-covers of size k are there in G ?How many k-cliques are there in G ?Given a graph G of treewidth at most k , how many Hamiltoncycles are there in G ?
![Page 5: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/5.jpg)
What is a parameterised counting problem?
Introduced by Flum and Grohe (2004)
Measure running time in terms of a parameter as well as thetotal input size
Examples:
How many vertex-covers of size k are there in G ?
How many k-cliques are there in G ?Given a graph G of treewidth at most k , how many Hamiltoncycles are there in G ?
![Page 6: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/6.jpg)
What is a parameterised counting problem?
Introduced by Flum and Grohe (2004)
Measure running time in terms of a parameter as well as thetotal input size
Examples:
How many vertex-covers of size k are there in G ?How many k-cliques are there in G ?
Given a graph G of treewidth at most k , how many Hamiltoncycles are there in G ?
![Page 7: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/7.jpg)
What is a parameterised counting problem?
Introduced by Flum and Grohe (2004)
Measure running time in terms of a parameter as well as thetotal input size
Examples:
How many vertex-covers of size k are there in G ?How many k-cliques are there in G ?Given a graph G of treewidth at most k , how many Hamiltoncycles are there in G ?
![Page 8: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/8.jpg)
The theory of parameterised counting
Efficient algorithms: Fixed parameter tractable (FPT)Running time f (k) · nO(1)
Intractable problems: #W[1]-hardA #W[1]-complete problem: p-#Clique.
![Page 9: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/9.jpg)
The theory of parameterised counting
Efficient algorithms: Fixed parameter tractable (FPT)Running time f (k) · nO(1)
Intractable problems: #W[1]-hardA #W[1]-complete problem: p-#Clique.
![Page 10: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/10.jpg)
#W[1]-completeness
To show the problem Π′ (with parameter κ′) is #W[1]-hard,we give a reduction from a problem Π (with parameter κ) toΠ′.
An fpt Turing reduction from (Π, κ) to (Π′, κ′) is an algorithmA with an oracle to Π′ such that
1 A computes Π,2 A is an fpt-algorithm with respect to κ, and3 there is a computable function g : N→ N such that for all
oracle queries “Π′(y) =?” posed by A on input x we haveκ′(y) ≤ g(κ(x)).
In this case we write (Π, κ) ≤fptT (Π′, κ′).
![Page 11: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/11.jpg)
#W[1]-completeness
To show the problem Π′ (with parameter κ′) is #W[1]-hard,we give a reduction from a problem Π (with parameter κ) toΠ′.
An fpt Turing reduction from (Π, κ) to (Π′, κ′) is an algorithmA with an oracle to Π′ such that
1 A computes Π,2 A is an fpt-algorithm with respect to κ, and3 there is a computable function g : N→ N such that for all
oracle queries “Π′(y) =?” posed by A on input x we haveκ′(y) ≤ g(κ(x)).
In this case we write (Π, κ) ≤fptT (Π′, κ′).
![Page 12: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/12.jpg)
Subgraph Counting Model
Let Φ be a family (φ1, φ2, . . .) of functions, such that φk is amapping from labelled graphs on k-vertices to {0, 1}.
p-#Induced Subgraph With Property(Φ) (ISWP(Φ))Input: A graph G = (V ,E ) and an integer k .Parameter: k.Question: What is the cardinality of the set
{(v1, . . . , vk) ∈ V k : v1, . . . , vk all distinct,and φk(G [v1, . . . , vk ]) = 1}?
![Page 13: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/13.jpg)
Subgraph Counting Model
Let Φ be a family (φ1, φ2, . . .) of functions, such that φk is amapping from labelled graphs on k-vertices to {0, 1}.
p-#Induced Subgraph With Property(Φ) (ISWP(Φ))Input: A graph G = (V ,E ) and an integer k .Parameter: k.Question: What is the cardinality of the set
{(v1, . . . , vk) ∈ V k : v1, . . . , vk all distinct,and φk(G [v1, . . . , vk ]) = 1}?
![Page 14: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/14.jpg)
Examples
p-#Sub(H)
e.g. p-#Clique, p-#Path, p-#Cycle, p-#Matching
p-#Connected Induced Subgraph
p-#Planar Induced Subgraph
p-#Even Induced Subgraphp-#Odd Induced Subgraph
![Page 15: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/15.jpg)
Examples
p-#Sub(H)
e.g. p-#Clique, p-#Path, p-#Cycle, p-#Matching
p-#Connected Induced Subgraph
p-#Planar Induced Subgraph
p-#Even Induced Subgraphp-#Odd Induced Subgraph
![Page 16: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/16.jpg)
Examples
p-#Sub(H)
e.g. p-#Clique, p-#Path, p-#Cycle, p-#Matching
p-#Connected Induced Subgraph
p-#Planar Induced Subgraph
p-#Even Induced Subgraphp-#Odd Induced Subgraph
![Page 17: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/17.jpg)
Examples
p-#Sub(H)
e.g. p-#Clique, p-#Path, p-#Cycle, p-#Matching
p-#Connected Induced Subgraph
p-#Planar Induced Subgraph
p-#Even Induced Subgraphp-#Odd Induced Subgraph
![Page 18: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/18.jpg)
Complexity Questions
Is the corresponding decision problem in FPT?
Is there a fixed parameter algorithm for p-#InducedSubgraph With Property(Φ)?
Can we approximate p-#Induced Subgraph WithProperty(Φ) efficiently?
![Page 19: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/19.jpg)
Approximation Algorithms
An FPTRAS for a parameterised counting problem Π withparameter k is a randomised approximation scheme that takes aninstance I of Π (with |I | = n), and numbers ε > 0 and 0 < δ < 1,and in time f (k) · g(n, 1/ε, log(1/δ)) (where f is any function, andg is a polynomial in n, 1/ε and log(1/δ)) outputs a rationalnumber z such that
P[(1− ε)Π(I ) ≤ z ≤ (1 + ε)Π(I )] ≥ 1− δ.
![Page 20: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/20.jpg)
The Diagram of Everything
EXACT COUNTING
APPROXCOUNTING
DECISION
![Page 21: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/21.jpg)
The Diagram of Everything
EXACT COUNTING
APPROXCOUNTING
DECISION
![Page 22: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/22.jpg)
The Diagram of Everything
MULTICOLOURAPPROX
COUNTING
MULTICOLOUREXACT
COUNTING
EXACT COUNTING
APPROXCOUNTING
DECISIONMULTICOLOURDECISION
![Page 23: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/23.jpg)
The Diagram of Everything
MULTICOLOURAPPROX
COUNTING
MULTICOLOUREXACT
COUNTING
EXACT COUNTING
APPROXCOUNTING
DECISIONMULTICOLOURDECISION
![Page 24: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/24.jpg)
The Diagram of Everything
MULTICOLOURAPPROX
COUNTING
MULTICOLOUREXACT
COUNTING
EXACT COUNTING
APPROXCOUNTING
DECISIONMULTICOLOURDECISION
![Page 25: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/25.jpg)
Monotone properties I: p-#Sub(H)
Theorem (Arvind & Raman, 2002)
There is an FPTRAS for p-#Sub(H) whenever all graphs in Hhave bounded treewidth.
Theorem (Curticapean & Marx, 2014)
p-#Sub(H) is in FPT if all graphs in H have boundedvertex-cover number; otherwise p-#Sub(H) is #W[1]-complete.
![Page 26: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/26.jpg)
Monotone properties I: p-#Sub(H)
Theorem (Arvind & Raman, 2002)
There is an FPTRAS for p-#Sub(H) whenever all graphs in Hhave bounded treewidth.
Theorem (Curticapean & Marx, 2014)
p-#Sub(H) is in FPT if all graphs in H have boundedvertex-cover number; otherwise p-#Sub(H) is #W[1]-complete.
![Page 27: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/27.jpg)
Monotone properties II: properties with more than oneminimal element
Theorem (Jerrum & M.)
Let Φ be a monotone property, and suppose that there exists aconstant t such that, for every k ∈ N, all minimal graphs satisfyingφk have treewidth at most t. Then there is an FPTRAS forp-#Induced Subgraph With Property(Φ).
![Page 28: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/28.jpg)
Monotone properties II: properties with more than oneminimal element
Theorem (M.)
Suppose that there is no constant t such that, for every k ∈ N, allminimal graphs satisfying φk have treewidth at most t. Thenp-#Induced Subgraph With Property(Φ) is#W[1]-complete.
Theorem (Jerrum & M.)
p-#Connected Induced Subgraph is #W[1]-complete.
![Page 29: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/29.jpg)
Monotone properties II: properties with more than oneminimal element
Theorem (M.)
Suppose that there is no constant t such that, for every k ∈ N, allminimal graphs satisfying φk have treewidth at most t. Thenp-#Induced Subgraph With Property(Φ) is#W[1]-complete.
Theorem (Jerrum & M.)
p-#Connected Induced Subgraph is #W[1]-complete.
![Page 30: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/30.jpg)
Non-monotone properties
Theorem
Let Φ be a family (φ1, φ2, . . .) of functions φk from labelledk-vertex graphs to {0, 1} that are not identically zero, such thatthe function mapping k 7→ φk is computable. Suppose that
|{|E (H)| : |V (H)| = k and Φ is true for H}| = o(k2).
Then p-#Induced Subgraph With Property(Φ) is#W[1]-complete.
E.g. p-#Planar Induced Subgraph, p-#RegularInduced Subgraph
![Page 31: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/31.jpg)
Non-monotone properties
Theorem
Let Φ be a family (φ1, φ2, . . .) of functions φk from labelledk-vertex graphs to {0, 1} that are not identically zero, such thatthe function mapping k 7→ φk is computable. Suppose that
|{|E (H)| : |V (H)| = k and Φ is true for H}| = o(k2).
Then p-#Induced Subgraph With Property(Φ) is#W[1]-complete.
E.g. p-#Planar Induced Subgraph, p-#RegularInduced Subgraph
![Page 32: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/32.jpg)
Even induced subgraphs: FPT???
Theorem (Goldberg, Grohe, Jerrum & Thurley (2010); Lidl &Niederreiter (1983))
Given a graph G , there is a polynomial-time algorithm whichcomputes the number of induced subgraphs of G having an evennumber of edges.
![Page 33: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/33.jpg)
Even induced subgraphs: decision
Let G be a graph on n ≥ 22k vertices. Then:
If k ≡ 0 mod 4 or k ≡ 1 mod 4 then G contains a k-vertexsubgraph with an even number of edges.
If k ≡ 2 mod 4 then G contains a k-vertex subgraph with aneven number of edges unless G is a clique.
If k ≡ 3 mod 4 then G contains a k-vertex subgraph with aneven number of edges unless G is either a clique or thedisjoint union of two cliques.
![Page 34: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/34.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
![Page 35: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/35.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
![Page 36: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/36.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
·······
![Page 37: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/37.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
·······
![Page 38: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/38.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
4······
![Page 39: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/39.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
44·····
![Page 40: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/40.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
442····
![Page 41: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/41.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
4426···
![Page 42: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/42.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
44264··
![Page 43: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/43.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
442642·
![Page 44: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/44.jpg)
Even induced subgraphs: exact counting is#W[1]-complete
mask
underlying structure︷ ︸︸ ︷
0 1 1 1 0 0 0 11 1 0 0 0 0 1 11 0 1 0 0 1 0 11 0 0 1 1 0 0 10 0 0 1 0 1 1 10 0 1 0 1 0 1 10 1 0 0 1 1 0 11 1 1 1 1 1 1 1
·
#
#
#
#
#
#
#
#
=
4426422
![Page 45: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/45.jpg)
Inverting the matrix
Theorem
Let (P,≤) be a finite lattice and f : P → R a function. Set S to be theupward closure of the support of f , that is,
S = {x ∈ P : ∃y ∈ P with y ≤ x and f (y) 6= 0},
and suppose that S = {x1, . . . , xn}. Let A = (aij)1≤i,j≤n be the matrixgiven by aij = f (xi ∧ xj). Then
det(A) =n∏
i=1
∑xj≤xixj∈S
f (xj)µ(xj , xi ).
Based on results of Rajarama Bhat (1991) and Haukkanen (1996).
![Page 46: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/46.jpg)
Even induced subgraphs: an FPTRAS
Lemma
Suppose that, for each k and any graph G on n vertices, thenumber of k-vertex (labelled) subgraphs of G that satisfy φk iseither
1 zero, or
2 at least1
g(k)p(n)
(n
k
),
where p is a polynomial and g is a computable function.
Then there exists an FPTRAS for p-#ISWP(Φ).
![Page 47: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/47.jpg)
Even induced subgraphs: an FPTRAS
Theorem
Let k ≥ 3 and let G be a graph on n ≥ 22k vertices. Then eitherG contains no even k-vertex subgraph or else G contains at least
1
22k2k2n2
(n
k
)even k-vertex subgraphs.
![Page 48: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/48.jpg)
Even induced subgraphs: an FPTRAS
Theorem (Erdos and Szekeres)
Let k ∈ N. Then there exists R(k) < 22k such that any graph onn ≥ R(k) vertices contains either a clique or independent set on kvertices.
Corollary
Let G = (V ,E ) be an n-vertex graph, where n ≥ 22k . Then thenumber of k-vertex subsets U ⊂ V such that U induces either aclique or independent set in G is at least
(22k − k)!
(22k)!
n!
(n − k)!.
![Page 49: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/49.jpg)
Corollary
Let G = (V ,E ) be an n-vertex graph, where n ≥ 22k . Then thenumber of k-vertex subsets U ⊂ V such that U induces either aclique or independent set in G is at least
(22k − k)!
(22k)!
n!
(n − k)!.
If at least half of these “interesting” subsets are independentsets, we are done.
Thus we may assume from now on that G contains at least(22k−k)!2(22k )!
n!(n−k)! k-cliques.
![Page 50: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/50.jpg)
Even induced subgraphs: an FPTRAS
Definition
Let A ⊂ {1, . . . , k}. We say that a k-clique H in G isA-replaceable if there are subsets U ⊂ V (H) andW ⊂ V (G ) \ V (H), with |U| = |W | ∈ A, such thatG [(H \ U) ∪W ] has an even number of edges.
If every k-clique in G is {1, 2}-replaceable, we are done.
Thus we may assume from now on that there is at least onek-clique H in G that is not {1, 2}-replaceable.
We also assume G contains at least one even k-vertexsubgraph.
![Page 51: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/51.jpg)
Even induced subgraphs: an FPTRAS
k
![Page 52: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/52.jpg)
Even induced subgraphs: an FPTRAS
k r
![Page 53: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/53.jpg)
Even induced subgraphs: an FPTRAS
k r
k-1 r
k-1 r
![Page 54: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/54.jpg)
Even induced subgraphs: an FPTRAS
If(k2
)is odd, the following have an even number of edges:
k-2 k-2
If k ≡ 2 mod 4, this also has an even number of edges:
k-1
![Page 55: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/55.jpg)
Even induced subgraphs: an FPTRAS
If(k2
)is odd, the following have an even number of edges:
k-2 k-2
If k ≡ 2 mod 4, this also has an even number of edges:
k-1
![Page 56: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/56.jpg)
Even induced subgraphs: an FPTRAS
H
![Page 57: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/57.jpg)
Even induced subgraphs: an FPTRAS
H
![Page 58: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/58.jpg)
Even induced subgraphs: an FPTRAS
H
![Page 59: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/59.jpg)
Even induced subgraphs: an FPTRAS
H
a
b
x
y
![Page 60: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/60.jpg)
Even induced subgraphs: an FPTRAS
H
a x
yb
![Page 61: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/61.jpg)
Even induced subgraphs: an FPTRAS
H
a x
yb
![Page 62: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/62.jpg)
Even induced subgraphs: an FPTRAS
H
a x
yb
![Page 63: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/63.jpg)
Open problems
Can similar results be obtained for properties that only holdfor graphs H where
e(H) ≡ r mod p,
for p > 2?
What if we consider an arbitrary property that depends onlyon the number of edges?
![Page 64: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/64.jpg)
Open problems
Can similar results be obtained for properties that only holdfor graphs H where
e(H) ≡ r mod p,
for p > 2?
What if we consider an arbitrary property that depends onlyon the number of edges?
![Page 65: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/65.jpg)
Open problems
Can similar results be obtained for properties that only holdfor graphs H where
e(H) ≡ r mod p,
for p > 2?
What if we consider an arbitrary property that depends onlyon the number of edges?
![Page 66: Parameterised Subgraph Counting Problems - Glakitty/slides/strathclyde-may15.pdf · Parameterised Subgraph Counting Problems ... (with parameter ) to 0. An fpt Turing reduction from](https://reader031.fdocuments.net/reader031/viewer/2022022418/5a73b5107f8b9aa3618b5c04/html5/thumbnails/66.jpg)
THANK YOU