Simpler and Stronger Approaches for Non-Uniform Hypergraph Matching and the Füredi, Kahn, and Seymour Conjecture
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A well-known conjecture of Füredi, Kahn, and Seymour (1993) on non-uniform hypergraph matching states that for any hypergraph with edge weights w, there exists a matching M such that the inequality ∑_e∈ M g(e) w(e) ≥OPT_LP holds with g(e)=|e|-1+1/|e|, where OPT_LP denotes the optimal value of the canonical LP relaxation. While the conjecture remains open, the strongest result towards it was very recently obtained by Brubach, Sankararaman, Srinivasan, and Xu (2020)—building on and strengthening prior work by Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra (2012)—showing that the aforementioned inequality holds with g(e)=|e|+O(|e|(-|e|)). Actually, their method works in a more general sampling setting, where, given a point x of the canonical LP relaxation, the task is to efficiently sample a matching M containing each edge e with probability at least x(e)/g(e). We present simpler and easy-to-analyze procedures leading to improved results. More precisely, for any solution x to the canonical LP, we introduce a simple algorithm based on exponential clocks for Brubach et al.'s sampling setting achieving g(e)=|e|-(|e|-1)x(e). Apart from the slight improvement in g, our technique may open up new ways to attack the original conjecture. Moreover, we provide a short and arguably elegant analysis showing that a natural greedy approach for the original setting of the conjecture shows the inequality for the same g(e)=|e|-(|e|-1)x(e) even for the more general hypergraph b-matching problem.
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