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# A dichotomy theorem for constraints on a three-element set

FOCS, pp.649-658, (2002)

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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}.

Reference

- P. Burgisser. On the structure of Valiant’s complexity classes. Discrete Mathematics and Theoretical Computer Science, 3:73–94, 1999.
- P. Burgisser. Completeness and Reduction in Algebraic Complexity Theory. Number 7 in Algorithms and Computation in Mathematics. Springer, 2000.
- I. Briquel and P. Koiran. A dichotomy theorem for polynomial evaluation. http://prunel.ccsd.cnrs.fr/ensl-00360974.
- A. Bulatov. A dichotomy theorem for constraint satisfaction problems on a 3element set. Journal of the ACM, 53(1):66–120, 2006.
- N. Creignou and M. Hermann. Complexity of generalized satisfiability counting problems. Information and Computation, 125:1–12, 1996.
- N. Creignou, S. Khanna, and M. Sudan. Complexity classification of boolean constraint satisfaction problems. SIAM monographs on discrete mathematics. 2001.
- F. M. Dong, M. D. Hendy, K. L. Teo, and C. H. C. Little. The vertex-cover polynomial of a graph. Discrete Mathematics, 250(1-3):71–78, 2002.
- M. Jerrum. Two-dimensional monomer-dimer systems are computationally intractable. Journal of Statistical Physics, 48:121–134, 1987.
- M. Jerrum. Counting, Sampling and Integrating: Algorithms and Complexity. Lectures in Mathematics - ETH Zurich. Birkhauser, Basel, 2003.
- N. Linial. Hard enumeration problems in geometry and combinatorics. SIAM Journal of Algebraic and Discrete Methods, 7(2):331–335, 1986.
- M. Lotz and J. A. Makowsky. On the algebraic complexity of some families of coloured Tutte polynomials. Advances in Applied Mathematics, 32(1):327–349, January 2004.
- J. S. Provan and M. O. Ball. The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. of Comp., 12(4):777–788, 1983.
- T. J. Schaefer. The complexity of satisfiability problems. In Conference Record of the 10th Symposium on Theory of Computing, pages 216–226, 1978.
- L. G. Valiant. Completeness classes in algebra. In Proc. 11th ACM Symposium on Theory of Computing, pages 249–261, 1979.
- L. G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8:189–201, 1979.
- L. G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal of Computing, 8(3):410–421, 1979.
- V. Zanko. #P-completeness via many-one reductions. International Journal of Foundations of Computer Science, 2(1):77–82, 1991.

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