Pattern Matching for k-Track Permutations

Abstract

Given permutations $\tau$ and $\pi$, the permutation pattern (PP) problem is to decide whether $\pi$ occurs in $\tau$ as an order-isomorphic subsequence. Although an FPT algorithm is known for PP parameterized by the size of the pattern $|\pi |$ [Guillemot and Marx 2014], the high complexity of this algorithm makes it impractical for most instances. In this paper we approach the PP problem from k-track permutations, i.e. those permutations that are the union of k increasing patterns or, equivalently, those permutation that avoid the decreasing pattern $(k+1) k \ldots 1$. Recently, k-track permutations have been shown to be central combinatorial objects in the study of the PP problem. Indeed, the PP problem is NP-complete when $\pi$ is 321-avoiding and $\tau$ is 4321-avoiding but is solvable in polynomial-time if both $\pi$ and $\tau$ avoid 321. We propose and implement an exact algorithm, FPT for parameters k and $|\pi |$, which allows to solve efficiently some large instances.