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A New Algorithm for Optimal Constraint Satisfaction and Its Implications

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNCS,volume 3142)

Abstract

We present a novel method for exactly solving (in fact, counting solutions to) general constraint satisfaction optimization with at most two variables per constraint (e.g. MAX-2-CSP and MIN-2-CSP), which gives the first exponential improvement over the trivial algorithm. More precisely, the runtime bound has a constant factor improvement in the base of the exponent: the algorithm can count the number of optima in MAX-2-SAT and MAX-CUT instances in O(m 3 2ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1+ε)-approximation with roughly the same runtime, modulo polynomial factors. Our construction shows that improvement in the runtime exponent of either k-clique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2-CSP optimization. The overall approach also yields connections between the complexity of some (polynomial time) high dimensional search problems and some NP-hard problems. For example, if there are sufficiently faster algorithms for computing the diameter of n points in ℓ1, then there is an (2–ε)n algorithm for MAX-LIN. These results may be construed as either lower bounds on the high-dimensional problems, or hope that better algorithms exist for the corresponding hard problems.

Keywords

  • Matrix Multiplication
  • Constraint Satisfaction
  • Partial Assignment
  • Arbitrary Weight
  • Springer LNCS

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  • DOI: 10.1007/978-3-540-27836-8_101
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Williams, R. (2004). A New Algorithm for Optimal Constraint Satisfaction and Its Implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds) Automata, Languages and Programming. ICALP 2004. Lecture Notes in Computer Science, vol 3142. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27836-8_101

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  • DOI: https://doi.org/10.1007/978-3-540-27836-8_101

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22849-3

  • Online ISBN: 978-3-540-27836-8

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