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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.

 

We present an O(n³ g² log g + m) + Õ(n^{ω + 1}) time deterministic algorithm to find the minimum cycle basis of a directed graph embedded on an orientable surface of genus g. This result improves upon the previous fastest known running time of O(m³ n + m² n² log n) applicable to general directed graphs. While an O(n^ω + 2^{2g} n² + m) time deterministic algorithm was known for undirected graphs, the use of the underlying field ℚ in the directed case (as opposed to ℤ₂ for the undirected case) presents extra challenges. It turns out that some of our new observations are useful for both variants of the problem, so we present an O(n^ω + n² g² log g + m) time deterministic algorithm for undirected graphs as well. 

In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have a small dependence on n. Specifically, our Faulty k-center algorithms have only linear dependence on n, while for our algorithms for Faulty k-median and Faulty k-means the dependence is still only n^(1 + o(1)). 

We give an O(k³ Δ n log n min(k, log² n) log²(nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex capacities bounded by C, and k sources and sinks. This improves by a factor of max(k²,k log² n) over the fastest algorithm previously known for this problem [Wang, SODA 2019]. The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses O(k) invocations of an O(k³ n log³ n)-time algorithm for maximum flow algorithm in a planar graph with k apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al. Second, we show two alternatives for computing flows in the k-apex graphs that arise in our modification of Wang’s procedure faster than the algorithm of Borradaile et al. In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan [JACM 1988].