Kernelization lower bound for Permutation Pattern Matching

TL;DR

This paper proves that the Permutation Pattern Matching problem does not admit a polynomial kernel unless NP is contained in co-NP/poly, using a new polynomial reduction from the clique problem.

Contribution

It introduces a novel polynomial reduction from the clique problem to permutation pattern matching to establish kernelization lower bounds.

Findings

No polynomial kernel for permutation pattern matching unless NP ⊆ co-NP/poly

New polynomial reduction from clique to permutation pattern matching

Advances understanding of computational complexity in permutation problems

Abstract

A permutation $\pi$ contains a permutation $\sigma$ as a pattern if it contains a subsequence of length $|\sigma|$ whose elements are in the same relative order as in the permutation $\sigma$. This notion plays a major role in enumerative combinatorics. We prove that the problem does not have a polynomial kernel (under the widely believed complexity assumption $\mbox{NP} \not\subseteq \mbox{co-NP}/\mbox{poly}$) by introducing a new polynomial reduction from the clique problem to permutation pattern matching.