A Survey on Graph Problems Parameterized Above and Below Guaranteed Values

TL;DR

This survey studies graph problems parameterized above or below guaranteed values $g(G)$. It covers graph layout, vertex cover, cut, independent set, path and cycle problems, and TSP, emphasizing fixed-parameter tractability, kernelization, and reductions. The main contributions are compiling tight-bound based results, presenting explicit FPT algorithms and kernel bounds, and identifying open questions across problem classes. The findings reveal a rich spectrum of tractable and intractable cases depending on the guarantee used and the parameterization, with practical implications for kernelization and reduction rules.

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.

Theorems & Definitions (19)

• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8