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On hypergraphs withmhyperedges andnvertices, wherepdenotes the total size of the hyperedges, we provide the following results:We give an algorithm that runs in\(\widetilde{O}(mn^{2k-2})\)time for finding a minimumk-cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimumk-cut problem, fork> 2.We give an algorithm that runs in\(\widetilde{O}(n^{\max \lbrace r,2k-2\rbrace })\)time for finding a minimumk-cut in hypergraphs of constant rankr. This algorithm betters the previous best running times for both the minimum cut and minimumk-cut problems for dense hypergraphs.Both of our algorithms are Monte Carlo, i.e., they return a minimumk-cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a genericbranching randomized contractiontechnique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs. Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge-k-cut on hedgegraphs, which generalize hypergraphs. 

Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimizeregret, defined as the maximum difference between the solution’s cost and that of an optimal solution in hindsight once the input has been realized. For graph problems inP, such as shortest path and minimum spanning tree, robust polynomial-time algorithms that obtain a constant approximation on regret are known. In this paper, we study robust algorithms for minimizing regret inNP-hard graph optimization problems, and give constant approximations on regret for the classical traveling salesman and Steiner tree problems. 

This article presentsuniversalalgorithms for clustering problems, including the widely studiedk-median,k-means, andk-center objectives. The input is a metric space containing allpotentialclient locations. The algorithm must selectkcluster centers such that they are a good solution foranysubset of clients that actually realize. Specifically, we aim for lowregret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solutionSolfor a clustering problem is said to be an α , β-approximation if for all subsets of clientsC^′, it satisfiessol(C^′) ≤ α ċopt(C′) + β ċmr, whereopt(C′ is the cost of the optimal solution for clients (C′) andmris the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives ofk-median,k-means, andk-center that achieve (O(1),O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other ℓ_p-objectives and the setting where some subset of the clients arefixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1),O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.