The complexity of general-valued CSPs seen from the other side - - PowerPoint PPT Presentation

The complexity of general-valued CSPs seen from the other side

Clement Carbonnel, Miguel Romero, Stanislav Zivny University of Oxford

IEEE Symposium on Foundations of Computer Science
 7 October 2018, Paris, France

General-valued Constraint Satisfaction Problem (VCSP)

Valued structure over :

A

σ

• (finite) universe
• interpretations for each

A

f A : Aar(f) → Q≥0 ∪ {∞}

f ∈ σ cost(h) = X

f∈σ, ¯ x∈Aar(f)

f A(¯ x)f B(h(¯ x))

For valued structures and over , the cost of is:

A

B

σ

h : A → B

Instance: Goal:

VCSP

Valued structures and over the same signature

A

B

σ

Compute

min

h:A→B cost(h)

• pt(A, B) =

A signature is a set of function symbols each of which has a fixed arity

σ

ar(f)

VCSP: particular cases

Instance: Goal:

VCSP

Valued structures and over the same signature

A

B

σ

Compute

min

h:A→B cost(h)

• pt(A, B) =

CSP = satisfy all constraints simultaneously
 = is there a homomorphism from to ? A B

• -valued structures

{0, ∞}

MinCSP = minimise unsatisfied constraints

• -valued structures

{0, 1}

Finite-valued CSP = -valued structures

Q≥0

• Finite valued (Thapper, Zivny STOC’13)
• CSP (Bulatov FOCS ’17; Zhuk FOCS’17)
• VCSP (Ochremiak, Kozic ICALP’15; Kolmogorov, Krokhin, Rolinek FOCS’15)

The complexity of VCSP

Instance: Goal:

VCSP

Valued structures and over the same signature

A

B

σ

Compute

min

h:A→B cost(h)

• pt(A, B) =

VCSP is NP-hard Tractable restrictions:

• Non-uniform restrictions: VCSP(−, {B})

VCSP(C, −)

• Structural restrictions:
• CSP, bounded arity (Dalmau, Kolaitis, Vardi CP’02; Grohe FOCS’03)

• CSP, unbounded arity: FPT classification (Marx STOC’10)

• CSP, bounded arity (Dalmau, Kolaitis, Vardi CP’02; Grohe FOCS’03)

• CSP, unbounded arity: FPT classification (Marx STOC’10)

Main Question:
 For which classes of bounded arity is tractable? VCSP(C, −)

C

• Finite valued (Thapper, Zivny STOC’13)
• CSP (Bulatov FOCS ’17; Zhuk FOCS’17)
• VCSP (Ochremiak, Kozic ICALP’15; Kolmogorov, Krokhin, Rolinek FOCS’15)

The complexity of VCSP

Instance: Goal:

VCSP

Valued structures and over the same signature

A

B

σ

Compute

min

h:A→B cost(h)

• pt(A, B) =

VCSP is NP-hard Tractable restrictions:

• Non-uniform restrictions: VCSP(−, {B})

VCSP(C, −)

• Structural restrictions:

Contributions

• Characterisation of the power of Sherali-Adams relaxations for VCSP
• Characterisation of the tractable structural restrictions for VCSP

(in the case of bounded arity)

Outline The case of CSP and bounded arity Tractable structural restrictions for VCSP Power of Sherali-Adams relaxations Open questions

Outline The case of CSP and bounded arity Tractable structural restrictions for VCSP Power of Sherali-Adams relaxations Open questions

Theorem (Freuder AAAI ’90):
 is in PTIME if the treewidth of is bounded


CSP(C, −)

C

Theorem (Dalmau, Kolaitis, Vardi CP’02):
 is in PTIME 
 if the treewidth modulo homomorphic equivalence of is bounded


CSP(C, −)

C

= treewidth of the core of

A

The case of CSP and bounded arity

A A

and are homomorphically equivalent 
 = there is homomorphism from to , and from to

A

A0 A0 A0

Treewidth modulo homomorphic equivalence of 
 = minimum treewidth over all homo. equiv. to

A A

A0

Theorem (Grohe FOCS ’03)
 Suppose is recursively enumerable and has bounded arity. If the treewidth modulo homo. equiv. of is unbounded, 
 then is W[1]-hard

C C

p-CSP(C, −) p-CSP(C, −): parameter |A|

Reduction from p-CLIQUE

The case of CSP and bounded arity

Complete classification: Theorem (Dalmau, Kolaitis, Vardi CP ’02; Grohe FOCS ’03)
 Assume FPT W[1]. 
 For every recursively enum. and of bounded arity, TFAE:

• 1. is in PTIME
• 2. is in FPT
• 3. The treewidth modulo homo. equiv. of is bounded

C C

p-CSP(C, −)

6=

CSP(C, −)

The case of CSP and bounded arity

Outline The case of CSP and bounded arity Tractable structural restrictions for VCSP Power of Sherali-Adams relaxations Open questions

VCSP and treewidth

Theorem (Folklore):
 is in PTIME if the treewidth of is bounded

C

VCSP(C, −)

Treewidth of a valued structure = treewidth of the positive part

A Pos(A)

VCSP: example beyond treewidth

1

2

1 σ = {φ(·, ·), µ(·)} 1 1 1 1

A2

B ≡

VCSP: example beyond treewidth

σ = {φ(·, ·), µ(·)} 1 1 1 1 1 1 1 1 1

A3

≡

1

2 3 2

1 C = {An | n ≥ 2}

The treewidth of is unbounded but is in PTIME

C

VCSP(C, −)

The tractability frontier for VCSP(C,—)?

Bounded treewidth modulo homomorphic equivalence
 (of the positive parts) Bounded treewidth

W[1]-hard PTIME

????

Classification for VCSP(C,—)

Theorem (Classification for VCSP(C,-))
 Assume FPT W[1]. 
 For every recursively enum. and of bounded arity, TFAE:

• 1. is in PTIME
• 2. is in FPT
• 3. The treewidth modulo valued equivalence of is bounded

C C

6=

VCSP(C, −) p-VCSP(C, −)

and over are valued equivalent if

A σ

(3) ⇒ (1) : (2) ⇒ (3) : (1) ⇒ (2) : trivial

Sherali-Adams relaxations Grohe’s reduction from p-CLIQUE + new tools

• Characterisation of valued equivalence in terms of certain type of homomorphisms

(inverse fractional homomorphisms)

• Notion of valued core of a valued structure

A0

for all valued structures over σ

• pt(A, B) = opt(A0, B)

B

Classification for VCSP(C,—)

Theorem (Classification for VCSP(C,-))
 Assume FPT W[1]. 
 For every recursively enum. and of bounded arity, TFAE:

• 1. is in PTIME
• 2. is in FPT
• 3. The treewidth modulo valued equivalence of is bounded

C C

6=

VCSP(C, −) p-VCSP(C, −)

• Classification for finite-valued structures
• Grohe’s classification: -valued structures

{0, ∞}

The tractability frontier for VCSP(C,—)

Bounded treewidth modulo homomorphic equivalence
 (of the positive parts) Bounded treewidth

W[1]-hard PTIME

Bounded treewidth modulo valued equivalence

Outline The case of CSP and bounded arity Tractable structural restrictions for VCSP Power of Sherali-Adams relaxations Open questions

Power of Sherali-Adams

Basic LP for an instance :

(A, B)

• variables for each and

λ(x, d) x ∈ A d ∈ B φ(τ)

• variables for each and scope

τ : S → B

S ⊆ A

k-th level of Sherali-Adams for :

(A, B)

φ(τ)

• variables for each and scope

τ : S → B

S ⊆ A

• variables for each where

λ(s)

|X| ≤ k

s : X → B

Theorem (Folklore): If the treewidth of is at most k-1, then the k-th level of Sherali-Adams is tight for

A A

For all , the k-th level is tight for

(A, B)

B

Power of Sherali-Adams

Theorem: If the treewidth modulo valued equivalence of is at most k-1, then the k-th level of Sherali-Adams is tight for

A A

treewidth of the valued core Theorem: Fix . Let be a valued structure and its valued core. Suppose that , where is the maximum arity of . TFAE:

• 1. The k-th level of Sherali Adams is tight for
• 2. The treewidth of is at most k-1

k ≥ 1

A A0 A

r ≤ k

r

A A0

Power of Sherali-Adams

A

Theorem (Power of Sherali-Adams): Fix . Let be a valued structure and its valued core. TFAE:

• 1. The k-th level of Sherali Adams is tight for
• 2. The treewidth modulo scopes of is at most k-1 and 


the overlap of is at most k k ≥ 1

A A0 A A

S S’ S’’ X X’

scope scope scope

≤ k − 1 ≤ k − 1 tw mod scopes ≤ k − 1

• verlap ≤ k : |S ∩ S′| ≤ k

Power of Sherali-Adams

A

Theorem (Power of Sherali-Adams): Fix . Let be a valued structure and its valued core. TFAE:

• 1. The k-th level of Sherali Adams is tight for
• 2. The treewidth modulo scopes of is at most k-1 and 


the overlap of is at most k k ≥ 1

A A0 A A

Results on the power of k-consistency for CSP +

(1) ⇒ (2) :

Inverse fractional homomorphism
 and valued cores

(Atserias, Bulatov, Dalmau ICALP’07)

Open problems

• Unbounded arity?
• MinCSP/MaxCSP ( -valued structures)

{0, 1}

• Classification for approximation of ?

VCSP(C, −)

Thank you!