On Exact Algorithms for 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 constraints are tuples of elements of V. The goal is to find a total ordering of the variables, $\pi :V\to [1,...,|V|]$, which satisfies as many constraints as possible. A constraint $(v_1,v_2,...,v_k)$ is satisfied by an ordering $\pi$ when $\pi (v_1)<\pi (v_2)<...<\pi (v_k)$. An instance has arity $k$ if all the constraints involve at most $k$ elements. This problem expresses a variety of permutation problems including {\sc Feedback Arc Set} and {\sc Betweenness} problems. A naive algorithm, listing all the $n!$ permutations, requires $2^{O(n\log n)}$ time. Interestingly, {\sc Permutation CSP} for arity 2 or 3 can be solved by Held-Karp type algorithms in time $O^{\ast }(2^n)$, but no algorithm is known for arity at least 4 with running time significantly better than $2^{O(n\log n)}$. In this paper we resolve the gap by showing that {\sc Arity 4 Permutation CSP} cannot be solved in time $2^{o(n\log n)}$ unless ETH fails.