Efficient parameterized approximation

2025-01-24

AI Summary

This work addresses four NP-hard graph optimization problems—Vertex Cover, Connected Vertex Cover, Chromatic Number, and Triangle Packing—and introduces the first parameterized additive approximation framework parameterized by module size (k). Without prior structural knowledge, we design polynomial-time algorithms achieving tight additive error bounds of (mathrm{OPT} pm k) (with constant (c = 1)), overcoming inherent limitations of multiplicative approximation. Under the Unique Games Assumption (UGA), our result is optimal for Vertex Cover; moreover, when (k < mathrm{OPT}), it strictly improves upon the classic 2-approximation. Our approach integrates modular decomposition, identification of tractable graph classes, and combinatorial approximation analysis—systematically leveraging module structure to control additive error for the first time. This establishes a novel paradigm for structured graph optimization, bridging structural graph theory and approximation algorithms via additive guarantees.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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) $cmathrm{OPT}pm f(k)$ 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.

Problem

• Approximation Algorithms
• Graph Theory
• Computational Efficiency

Innovation

• Approximation Algorithm
• Complex Problem Solving
• Vertex Cover Optimization