There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colorssettling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan-and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that P = NP. Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

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There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colors—settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan—and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that $$\mathtt {P}\ne \mathtt {NP}$$ P ≠ NP . Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

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| exp(−|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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