On Exact Algorithms for Permutation CSP

TL;DR

This paper proves that solving the Arity 4 Permutation CSP problem cannot be done faster than $2^{o(n ext{log}n)}$ time unless ETH fails, closing a significant complexity gap.

Contribution

It establishes a tight lower bound for the complexity of Arity 4 Permutation CSP, showing no significantly faster algorithms exist under ETH.

Findings

Proves no $2^{o(n ext{log}n)}$ algorithm for Arity 4 Permutation CSP unless ETH fails.

Closes the complexity gap for Permutation CSP with arity 4 and above.

Highlights the computational difficulty of higher-arity permutation problems.

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, $\pi\ : V \rightarrow [1,...,|V|]$, 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 $O^*(2^n)$, 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.