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Schaefer proved a dichotomy theorem for boolean constraint satisfaction problems: he showed that for any finite set S of logical relations the satisfiability problem SAT(S) for S-formulas is either in P, or NP-complete

A dichotomy theorem for constraints on a three-element set

FOCS, pp.649-658, (2002)

Cited by: 187|Views168
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Abstract

The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on the possible form of constraints may affect the complexity, and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming...More

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Introduction
  • Contains only affine relations of at most two variables, the families (P) of polynomials associated to p-families of S-formulas are in VP.
  • There exists a p-family of S-formulas such that the corresponding polynomial family P is VNP-complete.
  • Together with Creignou and Hermann’s results, this suffices to establish the existence of a VNP-complete family for any set S containing non affine clauses.
Highlights
  • Schaefer [13] proved a dichotomy theorem for boolean constraint satisfaction problems: he showed that for any finite set S of logical relations the satisfiability problem SAT(S) for S-formulas is either in P, or NP-complete
  • An S-formula over a set of n variables is a conjunction of relations of S where the arguments of each relation are freely chosen among the n variables
  • Dichotomy theorems were obtained for counting problems, optimization problems and the decision problem of quantified boolean formulas
  • Constraint satisfaction problems were studied over non-boolean domains. This turned out to be a surprisingly difficult question, and it took a long time before a dichotomy theorem over domains of size 3 could be obtained [4]
  • For every set S containing an affine formula with at least three variables, there exists a VNP-complete family of polynomials associated to S-formulas
Results
  • By Valiant’s criterion (Proposition 2.20 in [2]), for any finite set S of logical relations and any p-family of S-formulas the polynomials (P) form a VNP family.
  • Since for a conjunction of such relations, the variables are either independent or completely bounded, a polynomial associated to a p-family of such formulas is factorizable.
  • For a set S of affine relations with at most two variables, every pfamily of polynomials associated to S-formulas is in VP.
  • To obtain the first property, the authors first establish a VNP-completeness result for the independent set polynomial IP(G).
  • The connection between independent sets and vertex covers does imply a relation between the polynomials IP(G) and VCP(G).
  • The authors first establish the existence of a VNP-complete family of polynomials associated to a p-family of affine formulas, and show how to reduce this family to each affine constraint with at least three variables.
  • Through c-reductions and p-projections, this suffices to establish the existence of VNP-complete families for affine formulas of at least three variables: Theorem 6.
  • 1. There exists a VNP-complete family of polynomials associated to {x ⊕ y ⊕ z = 0}-formulas.
  • 2. There exists a VNP-complete family of polynomials associated to {x⊕y ⊕z = 1}-formulas.
  • 3. For every set S containing an affine formula with at least three variables, there exists a VNP-complete family of polynomials associated to S-formulas.
  • The family (P) projects on a family of polynomials associated to S-formulas, which is VNP-complete.
Conclusion
  • In order to prove the #P-completeness of Problem 1, the authors first establish a many-one reduction from the #P-complete problem of computing the permanent of {0, 1}-matrices to the problem of computing the vertex cover polynomial of a weighted graph with weights in {0, 1, −1}.
  • By Theorem 5, the n×n partial permanent is equal to the independent set polynomial of the graph Gn; the reduction is obviously polynomial.
  • The partial permanent on entries in {0, 1, −1} reduces to the vertex cover polynomial on graphs with weights in {0, 1, −1}.
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