Chain minors are FPT

TL;DR

This paper proves that determining whether a finite poset P is a chain minor of another poset Q is fixed parameter tractable, providing an efficient algorithm for this problem.

Contribution

It introduces a fixed parameter tractable algorithm for deciding chain minor relations between finite posets, solving an open problem from prior research.

Findings

Algorithm runs in O(|Q| log |Q|) time for fixed P

Decides chain minor relation efficiently

Addresses an open problem in parameterized complexity

Abstract

Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain C_Q in Q with the property that f restricted to C_Q is an isomorphism of chains. We give an algorithm to decide whether a poset P is a chain minor of o poset Q that runs in time O(|Q| log |Q|) for every fixed poset P. This solves an open problem from the monograph by Downey and Fellows [Parameterized Complexity, 1999] who asked whether the problem was fixed parameter tractable.