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

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(m 3 2 ωn/3) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1 +)-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. 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 1, then there is an (2 −) 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.

A Satisfiability Algorithm for Some Class of Dense Depth Two Threshold Circuits

“…The second one is MAX-k-SAT, the optimization version of k-CNF SAT. Even for MAX-3-SAT, no algorithms with constant savings over brute force search are known while such an algorithm is constructed for MAX-2-SAT in [19]. The third one is Integer Linear Programming (ILP) that is very useful in expressing combinatorial optimization problems both in theory and practice.…”

“…Their main subroutine is an algorithm for the Vector Domination Problem: given n vectors in R d , decide whether there is a pair of vectors such that the first vector is larger than the second vector in each coordinate. Relationship between this problem and satisfiability problem is studied in [19].…”

Short PCPs with Projection Queries

“…Do we believe that the satisfiability of (unrestricted) polynomial-size circuits can be solved faster than brute-force search? Even for seemingly simple problems such as MAX3SAT, no satisfiability algorithm better than brute-force search is known, despite attempts since a decade ago [Wil05]. Note that the MAX3SAT problem -given a 3CNF and an integer , is there an assignment that satisfies ≥ clauses?…”

New Plain-Exponential Time Classes for Graph Homomorphism

“…However, there is still a lot of room between these lower bounds and the best upper bound, which is O * n(H) cn(G) for a constant c < 1, using an algorithm by Williams [21]. In particular, as asked by Fomin, Heggernes, and Kratsch [10], can Hom(G, H) be solved in time OThis question is still open, but for a generalisation of graph homomorphism we know (again under a complexity theoretical assumption) that the answer is negative.…”