On exact algorithms for the permutation CSP

Abstract

In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables V and a set of constraints C, in which the constraints are tuples of elements of V. The goal is to find a total ordering of the variables, π:V→[1,…,|V|], which satisfies as many constraints as possible. A constraint (v1,v2,…,vk) is satisfied by an ordering π when π(v1)<π(v2)<⋯<π(vk). An instance has arity k if all the constraints involve at most k elements.This problem expresses a variety of permutation problems including Feedback Arc Set and Betweenness problems. A naive algorithm, listing all the n! permutations, requires 2O(nlogn) time. Interestingly, Permutation CSP for arity 2 or 3 can be solved by Held–Karp-type algorithms in time O∗(2n), but no algorithm is known for arity at least 4 with running time significantly better than 2O(nlogn). In this paper we resolve the gap by showing that Arity 4 Permutation CSP cannot be solved in time 2o(nlogn) unless the exponential time hypothesis fails.