Efficient parameterized approximation

TL;DR

This work develops Efficient Parameterized Approximation (EPA), a framework for leveraging input structure via modulators to tractable graph classes to obtain polynomial-time approximations with additive error that depends on the parameter k. The authors instantiate EPA for four core problems—Vertex Cover, Connected Vertex Cover, Chromatic Number, and Triangle Packing—parameterized by modulators to classes such as clusters, co-clusters, forests, interval, chordal, cographs, split, and cocluster graphs. They present a spectrum of results: EPAs with additive bounds like OPT + c·k or OPT ± k, with many bounds tight under the Unique Games Conjecture; and several that outperform standard approximations (e.g., beating the classic 2-approx for Vertex Cover when k is small). The work connects EPA to existing frameworks (FPT, structural rounding) and demonstrates practical applicability by providing concrete, polynomial-time algorithms for multiple problems and graph classes, while outlining tight lower bounds and avenues for future research with broader classes and additive-error gains.

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.

Figures (4)

• Figure 1: A Hasse diagram of the graph classes and related modulator parameters used in this work: A line from one graph class to another means that the lower of the two classes is contained in the other. We will use abbreviations to refer to the vertex deletion problem to these graph classes. These abbreviations are indicated in parentheses.
• Figure 2: Small forbidden induced subgraphs for interval graphs.
• Figure 3: An instance showing that the analysis of the algorithm in \ref{['thm:col-cochordal']} is tight. The example is the complement of the pictured graph.
• Figure 4: The graph $P_3 + K_1$

Theorems & Definitions (67)

• Theorem 1
• proof
• Theorem 2: Demaine et al. Demaine2019
• proof
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Corollary 4