Faster Exponential Algorithm for Permutation Pattern Matching

TL;DR

This paper introduces a significantly faster exponential algorithm for permutation pattern matching, improving the time complexity from approximately 1.6181^n to 1.415^n, and proves its optimality within a natural algorithm class.

Contribution

The paper presents a new exponential algorithm for permutation pattern matching with improved time complexity and establishes its optimality among similar algorithms.

Findings

New algorithm runs in O(1.415^n) time.

Proves the algorithm's optimality within a natural class.

Improves previous best exponential algorithms.

Abstract

The Permutation Pattern Matching problem asks, given two permutations $\sigma$ on $n$ elements and $\pi$, whether $\sigma$ admits a subsequence with the same relative order as $\pi$ (or, in the counting version, how many such subsequences are there). This natural problem was shown by Bose, Buss and Lubiw [IPL 1998] to be NP-complete, and hence we should seek exact exponential time solutions. The asymptotically fastest such solution up to date, by Berendsohn, Kozma and Marx [IPEC 2019], works in $\mathcal{O}(1.6181^n)$ time. We design a simple and faster $\mathcal{O}(1.415^{n})$ time algorithm for both the detection and the counting version. We also prove that this is optimal among a certain natural class of algorithms.