A Probabilistic Approach to Problems Parameterized Above or Below Tight Bounds

TL;DR

This paper presents a novel probabilistic method for proving fixed-parameter tractability of problems parameterized above or below tight bounds, demonstrating its effectiveness on three previously studied problems.

Contribution

It introduces a new probabilistic approach for fixed-parameter tractability analysis and applies it to three problems of unknown complexity, showing some are fixed-parameter tractable.

Findings

A generalization of one problem is fixed-parameter tractable.

Non-trivial cases of two problems are fixed-parameter tractable.

The approach broadens tools for analyzing problems parameterized above or below bounds.

Abstract

We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by Mahajan, Raman and Sikdar (J. Comput. Syst. Sci. 75, 2009). We show that a generalization of one of the problems and non-trivial special cases of the other two are fixed-parameter tractable.