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) ar ( f ) A signature is a set of function symbols each of which has a fixed arity σ Valued structure over : A σ A • (finite) universe f A : A ar ( f ) → Q ≥ 0 ∪ { ∞ } f ∈ σ • interpretations for each A For valued structures and over , the cost of is: B h : A → B σ X f A (¯ x ) f B ( h (¯ cost ( h ) = x )) x ∈ A ar ( f ) f ∈ σ , ¯ VCSP Instance: Valued structures and over the same signature A B σ opt ( A , B ) = h : A → B cost ( h ) min Goal: Compute

VCSP: particular cases VCSP Instance: Valued structures and over the same signature A B σ opt ( A , B ) = h : A → B cost ( h ) min Goal: Compute CSP = satisfy all constraints simultaneously 
 = is there a homomorphism from to ? A B { 0 , ∞ } • -valued structures MinCSP = minimise unsatisfied constraints { 0 , 1 } • -valued structures Finite-valued CSP = -valued structures Q ≥ 0

The complexity of VCSP VCSP Instance: Valued structures and over the same signature A B σ opt ( A , B ) = h : A → B cost ( h ) min Goal: Compute VCSP is NP-hard Tractable restrictions: •Non-uniform restrictions: VCSP( − , { B } ) - Finite valued (Thapper, Zivny STOC’13) - CSP (Bulatov FOCS ’17; Zhuk FOCS’17) - VCSP (Ochremiak, Kozic ICALP’15; Kolmogorov, Krokhin, Rolinek FOCS’15) •Structural restrictions: VCSP( C , − ) - CSP, bounded arity (Dalmau, Kolaitis, Vardi CP’02; Grohe FOCS’03) 
 - CSP, unbounded arity: FPT classification (Marx STOC’10)

The complexity of VCSP VCSP Instance: Valued structures and over the same signature A B σ opt ( A , B ) = h : A → B cost ( h ) min Goal: Compute VCSP is NP-hard Tractable restrictions: •Non-uniform restrictions: VCSP( − , { B } ) - Finite valued (Thapper, Zivny STOC’13) - CSP (Bulatov FOCS ’17; Zhuk FOCS’17) - VCSP (Ochremiak, Kozic ICALP’15; Kolmogorov, Krokhin, Rolinek FOCS’15) •Structural restrictions: VCSP( C , − ) - 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? C VCSP( C , − )

Contributions •Characterisation of the tractable structural restrictions for VCSP (in the case of bounded arity) •Characterisation of the power of Sherali-Adams relaxations for VCSP

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

The case of CSP and bounded arity Theorem (Freuder AAAI ’90): 
 CSP( C , − ) is in PTIME if the treewidth of is bounded 
 C A 0 and are homomorphically equivalent 
 A A 0 A 0 = there is homomorphism from to , and from to A A Treewidth modulo homomorphic equivalence of 
 A A 0 A = minimum treewidth over all homo. equiv. to = treewidth of the core of A Theorem (Dalmau, Kolaitis, Vardi CP’02): 
 CSP( C , − ) is in PTIME 
 if the treewidth modulo homomorphic equivalence of is bounded 
 C

The case of CSP and bounded arity Theorem (Grohe FOCS ’03) 
 Suppose is recursively enumerable and has bounded arity . C If the treewidth modulo homo. equiv. of is unbounded , 
 C p-CSP( C , − ) then is W[1]-hard 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]. 
 6 = For every recursively enum. and of bounded arity , TFAE: C CSP( C , − ) 1. is in PTIME p-CSP( C , − ) 2. is in FPT 3. The treewidth modulo homo. equiv. of is bounded C

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): 
 VCSP( C , − ) is in PTIME if the treewidth of is bounded C Pos( A ) A Treewidth of a valued structure = treewidth of the positive part

VCSP: example beyond treewidth σ = { φ ( · , · ) , µ ( · ) } A 2 1 1 ≡ 2 1 1 1 1 B

VCSP: example beyond treewidth σ = { φ ( · , · ) , µ ( · ) } A 3 1 1 1 ≡ 1 1 1 3 1 2 2 1 1 1 1 C = { A n | n ≥ 2 } The treewidth of is unbounded but is in PTIME C VCSP( C , − )

The tractability frontier for VCSP(C,—)? W[1]-hard Bounded treewidth modulo homomorphic equivalence 
 (of the positive parts) ???? PTIME Bounded treewidth

Classification for VCSP(C,—) A 0 A and over are valued equivalent if σ opt ( A , B ) = opt ( A 0 , B ) for all valued structures over σ B Theorem (Classification for VCSP(C,-)) 
 Assume FPT W[1]. 
 6 = For every recursively enum. and of bounded arity , TFAE: C 1. is in PTIME VCSP( C , − ) 2. is in FPT p-VCSP( C , − ) 3. The treewidth modulo valued equivalence of is bounded C (1) ⇒ (2) : trivial (3) ⇒ (1) : Sherali-Adams relaxations (2) ⇒ (3) : 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

Classification for VCSP(C,—) Theorem (Classification for VCSP(C,-)) 
 Assume FPT W[1]. 
 6 = For every recursively enum. and of bounded arity , TFAE: C 1. is in PTIME VCSP( C , − ) 2. is in FPT p-VCSP( C , − ) 3. The treewidth modulo valued equivalence of is bounded C { 0 , ∞ } - Grohe’s classification: -valued structures - Classification for finite-valued structures

The tractability frontier for VCSP(C,—) Bounded treewidth modulo homomorphic equivalence 
 (of the positive parts) W[1]-hard PTIME Bounded treewidth modulo valued equivalence Bounded treewidth

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

Power of Sherali-Adams ( A , B ) Basic LP for an instance : λ ( x, d ) x ∈ A d ∈ B - variables for each and φ ( τ ) τ : S → B - variables for each and scope S ⊆ A k-th level of Sherali-Adams for : ( A , B ) λ ( s ) | X | ≤ k - variables for each where s : X → B φ ( τ ) τ : S → B - variables for each and scope S ⊆ A Theorem (Folklore): If the treewidth of is at most k-1, then A the k-th level of Sherali-Adams is tight for A ( A , B ) For all , the k-th level is tight for B

Power of Sherali-Adams Theorem: If the treewidth modulo valued equivalence of is at most k-1, then A the k-th level of Sherali-Adams is tight for A treewidth of the valued core Theorem: A 0 A Fix . Let be a valued structure and its valued core. k ≥ 1 A Suppose that , where is the maximum arity of . TFAE: r ≤ k r 1. The k-th level of Sherali Adams is tight for A A 0 2. The treewidth of is at most k-1

Power of Sherali-Adams Theorem (Power of Sherali-Adams): A 0 A Fix . Let be a valued structure and its valued core. TFAE: k ≥ 1 1. The k-th level of Sherali Adams is tight for A A 2. The treewidth modulo scopes of is at most k-1 and 
 the overlap of is at most k A tw mod scopes ≤ k − 1 scope S overlap ≤ k : | S ∩ S ′ � | ≤ k ≤ k − 1 S’ scope X ≤ k − 1 S’’ X’ scope

Power of Sherali-Adams Theorem (Power of Sherali-Adams): A 0 A Fix . Let be a valued structure and its valued core. TFAE: k ≥ 1 1. The k-th level of Sherali Adams is tight for A A 2. The treewidth modulo scopes of is at most k-1 and 
 the overlap of is at most k A (1) ⇒ (2) : Inverse fractional homomorphism 
 Results on the power of k-consistency for CSP + 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!