Breaking the 2^n-Barrier for Irredundance: A Parameterized Route to Solving Exact Puzzles

TL;DR

This paper introduces parameterized algorithms that break the exponential time barrier for computing irredundance numbers in graphs, providing faster solutions than naive enumeration and advancing exact exponential algorithms.

Contribution

It presents the first parameterized algorithms with sub-4^k running times for irredundance problems, surpassing the trivial exponential barrier and offering new insights into exponential time algorithmics.

Findings

Developed an algorithm with runtime O*(3.069^k) for IR(G)

First parameterized algorithms with exponential dependency on the parameter

Breaks the 2^n barrier for exact irredundance number computation

Abstract

The lower and the upper irredundance numbers of a graph $G$, denoted $ir(G)$ and $IR(G)$ respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph $G$ on $n$ vertices admits exact algorithms running in time less than the trivial $\Omega(2^n)$ enumeration barrier. We solve these open problems by devising parameterized algorithms for the dual of the natural parameterizations of the problems with running times faster than $O^*(4^{k})$. For example, we present an algorithm running in time $O^*(3.069^{k})$ for determining whether $IR(G)$ is at least $n-k$. Although the corresponding problem has been known to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Additionally, our work also appears to be the first example of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as an exact exponential-time algorithm fails.