An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs

TL;DR

This paper presents an improved approximation algorithm for the maximum weight independent set problem in d-claw free graphs, achieving a slightly better ratio than the longstanding best, with implications for the weighted k-Set Packing problem.

Contribution

The authors develop a broader class of local improvements that improve the approximation ratio for the problem, surpassing the previous 20-year-old best bound.

Findings

Achieves an approximation ratio of d/2 - 1/63700992 + ε.

Provides a polynomial-time d/2-approximation algorithm.

Improves approximation guarantees for weighted k-Set Packing problem.

Abstract

In this paper, we consider the task of computing an independent set of maximum weight in a given $d$-claw free graph $G=(V,E)$ equipped with a positive weight function $w:V\rightarrow\mathbb{R}^+$. In doing so, $d\geq 2$ is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman. It achieves a performance ratio of $\frac{d}{2}+\epsilon$ in time $\mathcal{O}(|V(G)|^{d+1}\cdot(|V(G)|+|E(G)|)\cdot (d-1)^2\cdot \left(\frac{d}{2\epsilon}+1\right)^2)$ for any $\epsilon>0$, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of $\frac{d}{2}-\frac{1}{63,700,992}+\epsilon$ for any $\epsilon>0$ at the cost of an additional factor of $\mathcal{O}(|V(G)|^{(d-1)^2})$ in the running time. In particular, our result implies a polynomial time $\frac{d}{2}$-approximation algorithm. Furthermore, the well-known reduction from the weighted $k$-Set Packing Problem to the Maximum Weight Independent Set Problem in $k+1$-claw free graphs provides a $\frac{k+1}{2}-\frac{1}{63,700,992}+\epsilon$-approximation algorithm for the weighted $k$-Set Packing Problem for any $\epsilon>0$. This improves on the previously best known approximation guarantee of $\frac{k+1}{2}+\epsilon$ originating from the result of Berman.