Efficient parameterized approximation

TL;DR

This paper develops polynomial-time approximation algorithms for NP-hard problems leveraging structural parameters without requiring explicit structural input, achieving small additive errors and outperforming standard approximations for small parameters.

Contribution

It introduces a novel approach to approximation leveraging instance structure via modulators, providing algorithms with small additive errors without needing structural input.

Findings

Algorithms for Vertex Cover, Connected Vertex Cover, Chromatic Number, and Triangle Packing.

Most algorithms are tight under the Unique Games Conjecture.

Better approximation guarantees than standard methods for small modulator sizes.

Abstract

Many problems are NP-hard and, unless P = NP, do not admit polynomial-time exact algorithms. The fastest known exact algorithms exactly usually take time exponential in the input size. Much research effort has gone into obtaining faster exact algorithms for instances that are sufficiently well-structured, e.g., through parameterized algorithms with running time $f(k)\cdot n^{\mathcal{O}(1)}$ where n is the input size and k quantifies some structural property such as treewidth. When k is small, this is comparable to a polynomial-time exact algorithm and outperforms the fastest exact exponential-time algorithms for a large range of k. In this work, we are interested instead in leveraging instance structure for polynomial-time approximation algorithms. We aim for polynomial-time algorithms that produce a solution of value at most or at least (depending on minimization vs. maximization) $c\mathrm{OPT}\pm f(k)$ where c is a constant. Unlike for standard parameterized algorithms, we do not assume that structural information is provided with the input. Ideally, we can obtain algorithms with small additive error, i.e., $c=1$ and $f(k)$ is polynomial or even linear in $k$. For small k, this is similarly comparable to a polynomial-time exact algorithm and will beat general case approximation for a large range of k. We study Vertex Cover, Connected Vertex Cover, Chromatic Number, and Triangle Packing. The parameters we consider are the size of minimum modulators to graph classes on which the respective problem is tractable. For most problem-parameter combinations we give algorithms that compute a solution of size at least or at most $\mathrm{OPT}\pm k$. In the case of Vertex Cover, most of our algorithms are tight under the Unique Games Conjecture and provide better approximation guarantees than standard 2-approximations if the modulator is smaller than the optimum solution.