On exact algorithms for the permutation CSP

• Published in: Theoretical computer science Vol. 511; pp. 109 - 116
• Main Authors: Kim, Eun Jung, Gonçalves, Daniel
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 04.11.2013; Elsevier
• Subjects: Algorithmic lower bound; Computer Science; Data Structures and Algorithms; Discrete Mathematics; Exact algorithm; Permutation constraint satisfaction problem; Exact algorithm; Algorithmic lower bound; Permutation constraint satisfaction problem
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2012.10.035

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.