On exact algorithms for the permutation CSP

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, pi : V -> [1,...,vertical bar V vertical bar], 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 FEEDBACK ARC SET and BETWEENNESS problems. A naive algorithm, listing all the n! permutations, requires 2(O(n) (log n)) time. Interestingly, PERMUTATION CSP for arity 2 or 3 can be solved by Held-Karptype algorithms in time O*(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 ARITY 4 PERMUTATION CSP cannot be solved in time 2(O(n log n)) unless the exponential time hypothesis fails. (C) 2012 Elsevier B.V. All rights reserved.