A Survey on Graph Problems Parameterized Above and Below Guaranteed Values

TL;DR

This survey reviews algorithms and complexity results for graph problems parameterized above or below guaranteed solution values, highlighting recent advances, open problems, and the distinction from traditional parameterizations.

Contribution

It provides a comprehensive overview of the state-of-the-art in parameterized graph problems above and below guaranteed values, including algorithms, intractability results, and open challenges.

Findings

Many problems admit fixed-parameter algorithms

Certain problems are proven to be intractable in this parameterization

Open problems remain in developing efficient algorithms for some cases

Abstract

We survey the field of algorithms and complexity for graph problems parameterized above or below guaranteed values, a research area which was pioneered by Venkatesh Raman. Those problems seek, for a given graph $G$, a solution whose value is at least $g(G)+k$ or at most $g(G)-k$, where $g(G)$ is a guarantee on the value that any solution on $G$ takes. The goal is to design algorithms which find such solution in time whose complexity in $k$ is decoupled from that in the guarantee, or to rule out the existence of such algorithms by means of intractability results. We discuss a large number of algorithms and intractability results, and complement them by several open problems.