The Limits of Local Search for the Maximum Weight Independent Set Problem in d-Claw Free Graphs
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We consider the Maximum Weight Independent Set Problem (MWIS) in d-claw free graphs, i.e. 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_>0. For k≥ 1, the MWIS in k+1-claw free graphs generalizes the weighted k-Set Packing Problem. Given that for k≥ 3, this problem does not permit a polynomial time o(k/log k)-approximation unless P=NP, most previous algorithms for both weighted k-Set Packing and the MWIS in d-claw free graphs rely on local search. For the last twenty years, Berman's algorithm SquareImp, which yields a d/2+ϵ-approximation for the MWIS in d-claw free graphs, has remained unchallenged for both problems. Recently, it was improved by Neuwohner, obtaining an approximation guarantee slightly below d/2, and inevitably raising the question of how far one can get by using local search. In this paper, we finally answer this question asymptotically in the following sense: By considering local improvements of logarithmic size, we obtain approximation ratios of d-1+ϵ_d/2 for the MWIS in d-claw free graphs for d≥ 3 in quasi-polynomial time, where 0≤ϵ_d≤ 1 and lim_d→∞ϵ_d = 0. By employing the color coding technique, we can use the previous result to obtain a polynomial time k+ϵ_k+1/2-approximation for weighted k-Set Packing. On the other hand, we provide examples showing that no local improvement algorithm considering local improvements of size 𝒪(log(|𝒮|)) with respect to some power w^α of the weight function, where α∈ℝ is chosen arbitrarily, but fixed, can yield an approximation guarantee better than k/2 for the weighted k-Set Packing Problem with k≥ 3.
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