Long paths make pattern-counting hard, and deep trees make it harder

TL;DR

This paper investigates the computational complexity of a pattern counting problem in permutations, revealing how structural properties of permutation classes influence the difficulty of solving the problem under ETH assumptions.

Contribution

It identifies structural properties like the long path property and deep tree property that determine the complexity bounds of pattern counting in permutation classes.

Findings

LPP implies no $f(k)n^{o(\sqrt{k})}$ algorithms under ETH.

DTP implies no $f(k)n^{o(k/\log^2 k)}$ algorithms under ETH.

Certain monotone grid classes with LPP but not DTP allow $f(k)n^{O(\sqrt{k})}$ algorithms.

Abstract

We study the counting problem known as #PPM, whose input is a pair of permutations $\pi$ and $\tau$ (called pattern and text, respectively), and the task is to find the number of subsequences of $\tau$ that have the same relative order as $\pi$. A simple brute-force approach solves #PPM for a pattern of length $k$ and a text of length $n$ in time $O(n^{k+1})$, while Berendsohn, Kozma and Marx have recently shown that under the exponential time hypothesis (ETH), it cannot be solved in time $f(k) n^{o(k/\log k)}$ for any function $f$. In this paper, we consider the restriction of #PPM, known as $\mathcal{C}$-Pattern #PPM, where the pattern $\pi$ must belong to a hereditary permutation class $\mathcal{C}$. Our goal is to identify the structural properties of $\mathcal{C}$ that determine the complexity of $\mathcal{C}$-Pattern #PPM. We focus on two such structural properties, known as the long path property (LPP) and the deep tree property (DTP). Assuming ETH, we obtain these results: 1. If $C$ has the LPP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(\sqrt{k})}$ for any function $f$, and 2. if $C$ has the DTP, then $\mathcal{C}$-Pattern #PPM cannot be solved in time $f(k)n^{o(k/\log^2 k)}$ for any function $f$. Furthermore, when $\mathcal{C}$ is one of the so-called monotone grid classes, we show that if $\mathcal{C}$ has the LPP but not the DTP, then $\mathcal{C}$-Pattern #PPM can be solved in time $f(k)n^{O(\sqrt k)}$. In particular, the lower bounds above are tight up to the polylog terms in the exponents.