A new algorithm for optimal 2-constraint satisfaction and its implications

  • Authors:

  • Ryan Williams

  • Affiliations:

  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA

  • Venue:
  • Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
  • Year:
  • 2005

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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 is 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(m32ωn/3) time, where ω k-clique solution (even when k = 3) or matrix multiplication over GF(2) would improve the runtime exponent for solving 2-CSP optimization.Our 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.