Packing-Based Approximation Algorithm for the k-Set Cover Problem

TL;DR

This paper introduces a packing-based approximation algorithm for the k-Set Cover problem, achieving improved tight approximation ratios for k ≥ 4 through a novel local search heuristic and combinatorial analysis.

Contribution

It presents a new Restricted k-Set Packing heuristic and combines it with existing methods to improve approximation ratios for the k-Set Cover problem.

Findings

Achieves a tight approximation ratio of H_k - 0.6402 + Θ(1/k).

Improves the best known approximation ratios for k ≥ 4.

Provides specific ratios: 1.8667 for k=6, 1.7333 for k=5, 1.5208 for k=4.

Abstract

We present a packing-based approximation algorithm for the $k$-Set Cover problem. We introduce a new local search-based $k$-set packing heuristic, and call it Restricted $k$-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted $k$-Set Packing algorithm, our $k$-Set Cover algorithm is composed of the $k$-Set Packing heuristic \cite{schrijver} for $k\geq 7$, Restricted $k$-Set Packing for $k=6,5,4$ and the semi-local $(2,1)$-improvement \cite{furer} for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of $H_k-0.6402+\Theta(\frac{1}{k})$, where $H_k$ is the $k$-th harmonic number. For small $k$, our results are 1.8667 for $k=6$, 1.7333 for $k=5$ and 1.5208 for $k=4$. Our algorithm improves the currently best approximation ratio for the $k$-Set Cover problem of any $k\geq 4$.