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

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→ℝ^+. In doing so, d≥ 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 d/2+ϵ in time 𝒪(|V(G)|^d+1·(|V(G)|+|E(G)|)· (d-1)^2·(d/2ϵ+1)^2) for any ϵ>0, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of d/2-1/63,700,992+ϵ for any ϵ>0 at the cost of an additional factor of 𝒪(|V(G)|^(d-1)^2) in the running time. In particular, our result implies a polynomial time 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 k+1/2-1/63,700,992+ϵ-approximation algorithm for the weighted k-Set Packing Problem for any ϵ>0. This improves on the previously best known approximation guarantee of k+1/2+ϵ originating from the result of Berman.

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