Pattern Matching for k-Track Permutations

TL;DR: This paper proposes and implements an exact algorithm, FPT for parameters k and \(|\pi |\), which allows to solve efficiently some large instances of the permutation pattern (PP) problem.

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.