Structural Graph Parameters, Fine-Grained Complexity, and Approximation
Résumé
In this work we summarize much of the research conducted by the author since his PhD defense in the area of structural parameterizations of intractable graph problems. The general objective of this research domain is the detailed investigation of the computational complexity of NP-hard graph problems using parameters that measure the structure of the given graph. This area has attracted much interest in recent years because most interesting graph problems are NP-hard, therefore there is a great need to better understand and deal with their complexity.
Our exposition begins with a summary of the scientific context and a survey of the main parameters that we use to measure graph structure, including treewidth, which is the most well-studied such parameter, and its variations. We then move on to present concrete results regarding
the complexity of well-known graph problems measured as a function of such structural parameters, separating our exposition into three parts:
1. In the first part, we use the tools of parameterized complexity to understand the trade-offs between different parameters. In particular, we are interested to know which problems which are fixed-parameter tractable (FPT) for one parameter are intractable when we use a more general parameter. Among the highlights of this chapter are the discovery of the first natural problem that separates treewidth from pathwidth (Grundy Coloring); the proof that SAT remains intractable in the very restricted case where we parameterize by neighborhood diversity; and the proof that natural generalizations of Dominating Set and Independent Set are already hard parameterized by vertex cover. 2. In the second part we take another look at the same problems using the tools of fine-grained complexity theory, notably the Exponential Time Hypothesis (ETH) and its Strong variant (SETH). This allows us to obtain much more detailed estimates of the complexity of various problems. Among the main results we present are the fact that ∃∀-SAT has a double-exponential complexity parameterized by treewidth under the ETH; and exact bounds on the complexity of k-Coloring and Dominating Set parameterized by clique-width, under the SETH ((2k − 2)cw and 4cw respectively). 3. In the last part we consider how the notion of approximation interacts with structural parameters and present a general technique for obtaining FPT approximation algorithms for problems which are intractable parameterized by treewidth, as well as several example applications.
We conclude the document with directions for further work as well as a list of concrete open problems.
Domaines
Informatique [cs]
Origine : Fichiers produits par l'(les) auteur(s)