An Improved Approximation for Maximum Weighted $k$-Set Packing

TL;DR

This paper presents an improved approximation algorithm for the weighted $k$-set packing problem, achieving better bounds for $k=3$ and extending techniques to larger $k$, with implications for related graph problems.

Contribution

The authors develop a novel local search algorithm based on squared weights, improving approximation factors for weighted $k$-set packing, especially for $k=3$, and extend these methods to larger $k$.

Findings

Achieves an approximation factor of 1.786 for weighted 3-set packing.

Provides a simple analysis for approximation factor 1.811 with specific exchange sizes.

Extends the approach to larger exchanges, improving bounds for all $k > 3$.

Abstract

We consider the weighted $k$-set packing problem, in which we are given a collection of weighted sets, each with at most $k$ elements and must return a collection of pairwise disjoint sets with maximum total weight. For $k = 3$, this problem generalizes the classical 3-dimensional matching problem listed as one of the Karp's original 21 NP-complete problems. We give an algorithm attaining an approximation factor of $1.786$ for weighted 3-set packing, improving on the recent best result of $2-\frac{1}{63,700,992}$ due to Neuwohner. Our algorithm is based on the local search procedure of Berman that attempts to improve the sum of squared weights rather than the problem's objective. When using exchanges of size at most $k$, this algorithm attains an approximation factor of $\frac{k+1}{2}$. Using exchanges of size $k^2(k-1) + k$, we provide a relatively simple analysis to obtain an approximation factor of 1.811 when $k = 3$. We then show that the tools we develop can be adapted to larger exchanges of size $2k^2(k-1) + k$ to give an approximation factor of 1.786. Although our primary focus is on the case $k = 3$, our approach in fact gives slightly stronger improvements on the factor $\frac{k+1}{2}$ for all $k > 3$. As in previous works, our guarantees hold also for the more general problem of finding a maximum weight independent set in a $(k+1)$-claw free graph.