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

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(m2) time, where ω < 2.376 is the matrix product exponent over a ring. When constraints have arbitrary weights, there is a (1+ 2)-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− 2) 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.