The Complexity of homomorphism and Constraint Satisfaction Problems - - PowerPoint PPT Presentation

The Complexity of homomorphism and Constraint Satisfaction Problems seen from the Other Side Martin Grohe, FOCS 2003 The Necessity of Bounded Treewidth for Efficient Inference in Bayesian Networks Kwisthout, Bodlaender, van der Gaag, ECAI 2010 Presented by Jenny Lam

Main theorem

Assume FPT = W[1].

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Graph homomorphism

Graph homomorphism

Graph homomorphism

Graph homomorphism

Graph homomorphism

Relational structures

A finite universe A finite set of relations

Relational structures

A finite universe A finite set of relations A R ⊆ Ar

Relational structures

A finite universe A finite set of relations A R ⊆ Ar arity

Relational structures

A finite universe A finite set of relations A R ⊆ Ar

Relational structures

A finite universe A finite set of relations A R ⊆ Ar Examples: graphs, CSPs, databases

Relational structures

size of structure = number of relations + size of universe + sum of relation sizes weighted by arity

Relational structures

Homomorphism A − → B h : A − → B for all relations RA a ∈ RA ⇐ ⇒ h(a) ∈ RB

Homomorphism problem

HOM(–,–) Instance: two relational structures A and B Problem: is there a homomorphism from A to B?

Homomorphism problem

HOM(C,D) Instance: two relational structures A ∈ C and B ∈ D, Problem: is there a homomorphism from A to B?

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

• f relational structures of bounded arity

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Homomorphic equivalence

Homomorphic equivalence

Homomorphic equivalence

Homomorphic equivalence

Homomorphic equivalence

≡

Homomorphic equivalence

Homomorphic equivalence

Homomorphic equivalence

≡

Homomorphic equivalence

Homomorphic equivalence

≡

Homomorphic equivalence

≡

Treewidth

Treewidth

Treewidth

Treewidth

4 3 2 2 3 width = 4 - 1 = 3

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C A ∈

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C tw ≤ A

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C tw ≤ A has bounded treewidth

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C A ∈

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C A ∈ ≡ B

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C A ∈ B tw ≤

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. bound k C A ∈ B tw ≤ has bounded treewidth mod hom equiv

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Parametrized complexity

Parametrized problem yes/no problem a parametrization of the problem

Parametrized complexity

Parametrized problem yes/no problem a parametrization of the problem p-Clique Instance: Graph G, k ∈ N Parameter: k Problem: is G has a clique of size k?

Parametrized complexity

Parametrized problem yes/no problem a parametrization of the problem

Parametrized complexity

Parametrized problem yes/no problem a parametrization of the problem p-HOM(C,D) Instance: relational structures A ∈ C, B ∈ D Parameter: |A| Problem: is there a homomorphism from A to B?

Fixed-parameter tractability

X The input: (Downey, Fellows, Parametrized Complexity) Classical complexity theory

Fixed-parameter tractability

The input: (Downey, Fellows, Parametrized Complexity) Parametrized complexity k X nα contribution to

• verall complexity

Fixed-parameter tractability

(P, k) is fixed-parameter tractable if it can be decided in time f (k) · |x|O(1)

Fixed-parameter tractability

(P, k) is fixed-parameter tractable if it can be decided in time f (k) · |x|O(1) polynomial in size of input

Fixed-parameter tractability

(P, k) is fixed-parameter tractable if it can be decided in time f (k) · |x|O(1) arbitrary (computable) in parameter

FPT-reduction

(P, k) is fpt-reducible to (P′, k′) if there is x ∈ P ⇐ ⇒ R(x) ∈ P′ R : { inputs to P} − → { inputs to P′} given x, problem R(x) can be solved in FPT-time k′(R(x)) ≤ g(k(x)) for some computable g

W-hierarchy

(P, k) ∈ W [1] if it is FPT-reducible to finding a weight k satisfying assignment to a boolean circuit of depth 1.

W-hierarchy

FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . .

W-hierarchy

FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p-Clique is W[1]-complete under FPT-reductions.

W-hierarchy

FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p-Clique is FPT-reducible to p-HOM(C,–) Does G have a k-clique? Is there a homomorphism Kk − → G?

W-hierarchy

FPT = W [0] ⊆ W [1] ⊆ W [2] ⊆ . . . p-HOM(C,–) is W[1]-hard if C contains all complete graphs

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. Dalmau et al, 2002. (w + 1) pebble-game

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. p-HOM(C,–) is not W[1]-hard. p-Clique to p-HOM(C,–) reduction.

Main theorem

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. p-HOM(C,–) is not W[1]-hard.

Constraint satisfaction problems

A CSP is a triplet (V , D, C) V : variables D: domain of the variables C: constraints

A 3-SAT formula as a CSP

(a ∨ b ∨ c) ∧ (b ∨ c ∨ d) ∧ (a ∨ d) V = {a, b, c, d} D = {T, F} C : a ∨ b ∨ c = {(T, T, T), (T, T, F), (T, F, T), . . .} b ∨ c ∨ d = {(T, T, T), (T, T, F), (T, F, T), . . .} a ∨ d = {(T, T), (T, F), (F, T)}

Constraint satisfaction problems

Constraint Satisfaction Instance: a CSP (V , D, C) Problem: is there an assignment V − → D satisfying all constraints in C?

Constraint satisfaction problems

I = (V , D, C) Hypergraph Universe: V Relations: scopes of relations in C

Constraint satisfaction problems

I = (V , D, C) Hypergraph Universe: V Relations: scopes of relations in C Universe: D Relations: actual relations in C Constraints

Constraint satisfaction problems

I = (V , D, C) Hypergraph Universe: V Relations: scopes of relations in C Universe: D Relations: actual relations in C Constraints ?

CSP satisfiability as homomorphism problem

(a ∨ b ∨ c) ∧ (b ∨ c ∨ d) ∧ (a ∨ d)

CSP satisfiability as homomorphism problem

(a ∨ b ∨ c) ∧ (b ∨ c ∨ d) ∧ (a ∨ d) A A = {a, b, c, d} RA = {(a, b, c)} ⊆ A3 SA = {(b, c, d)} ⊆ A3 T A = {(a, d)} ⊆ A2

CSP satisfiability as homomorphism problem

(a ∨ b ∨ c) ∧ (b ∨ c ∨ d) ∧ (a ∨ d) A A = {a, b, c, d} RA = {(a, b, c)} ⊆ A3 SA = {(b, c, d)} ⊆ A3 T A = {(a, d)} ⊆ A2 B B = {T, F} RB = {(T, T, T), . . .} SB = {(T, T, T), . . .} T B = {(T, T), . . .}

CSP result

Assume FPT = W[1]. For every recursively enumerable class C

• f relational structures of bounded arity

the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence.

CSP result

Assume FPT = W[1]. the following statements are equivalent

• 1. HOM(C,–) is in polynomial time.
• 2. p-HOM(C,–) is fixed-parameter tractable.
• 3. C has bounded treewidth modulo

homomorphic equivalence. For every recursively enumerable class C

• f graphs

CSP result

Assume FPT = W[1]. the following statements are equivalent

• 3. C has bounded treewidth modulo

homomorphic equivalence.

• 1. CSP(C) is in polynomial time.

For every recursively enumerable class C

• f graphs

A subexponentional lower bound

CSPs of bounded treewidth k nO(k) can be solved in time

A subexponentional lower bound

If there exists a recursively enumerable class G of graphs with unbounded treewidth, and a computable function f such that CSP(G) can be solved in time f (G) · |I|o(tw(G)/ log tw(G)) then ETH fails. Marx 2007, Can we beat treewidth?

Probabilistic Inference

Positive Inference Instance: a Bayesian network, evidence e, query variable x Problem: does Pr(x|e) > 0 hold?

The necessity of bounded treewidth

If there exists a computable function f such that Positive Inference can be decided in f (G) · |B|o(tw(G)/ log tw(G)) then ETH fails. G is the moralized graph of the Bayesian network B,

The necessity of bounded treewidth

Proof. Reduce Constraint Satisfaction preserving treewidth. to Positive Inference

TW-reducibility

A is tw-reducible to B if there exists a polynomial-time computable function g and a linear function l such that x ∈ A ⇐ ⇒ g(x) ∈ B tw(g(x)) = l(tw(x))

TW reduction of CSP to Bayesian network

X1 X3 X4 X2

TW reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2

TW reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 A

TW reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5 A1 A2 A3 A4 A5 A6

TW-reduction of CSP to Bayesian network

X1 X3 X4 X2 R1 R4 R3 R2 X1 X2 X4 X6 X3 X5 A1 A2 A3 A4 A5 A6

Summary

• 3. Small treewidth is necessary for
• 2. If CSPs with large treewidth can be solved

in sub-exponential time, ETH fails.

• 1. CSPs can be solved in polynomial time iff

they have bounded treewidth. effective Bayesian inference.