We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph @math with @math , @math and @math the best known algorithm to compute a global minimum cut in @math runs in time @math for the uncapacitated case and in @math time for the capacitated case. We show the following new results. 1. Given an uncapacitated hypergraph @math and an integer @math we describe an algorithm that runs in @math time to find a subhypergraph @math with sum of degrees @math that preserves all edge-connectivities up to @math (a @math -sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an @math time algorithm for computing a global minimum cut of @math where @math is the minimum cut value. 2. We generalize Matula's argument for graphs to hypergraphs and obtain a @math -approximation to the global minimum cut in a capacitated hypergraph in @math time. 3. We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in @math time and @math space. We utilize vertex ordering based ideas to obtain our results. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.
The k-cut problem is to find a partition of an edge weighted graph into k nonempty components, such that the total edge weight between components is minimum. This problem is NP-complete for an arbitrary k and its version involving fixing a vertex in each component is NP-hard even for k = 3. We present a polynomial algorithm for k fixed, that runs in Onki¾² 2-3k 2+4Tn, m steps, where Tn, m is the running time required to find the minimum s, t-cut on a graph with n vertices and m edges. Abstract The hypergraph k -cut problem is the problem of finding a minimum capacity set of hyperedges whose removal divides a given hypergraph into at least k connected components. We present an algorithm for this problem, that runs in strongly polynomial time if both k and the maximum size of the hyperedges are constants. Our algorithm extends the algorithm proposed by Thorup (2008) for computing minimum k -cuts of graphs from greedy packings of spanning trees. Two simple approximation algorithms for the minimum @math -cut problem are presented. Each algorithm finds a @math cut having weight within a factor of @math of the optimal. One of our algorithms is particularly efficient---it requires a total of only @math maximum flow computations for finding a set of near-optimal @math cuts, one for each value of @math between 2 and @math . We present a simple and fast deterministic algorithm for the minimum k-way cut problem in a capacitated graph, that is, finding a set of edges with minimum total capacity whose removal splits the graph into at least k components. The algorithm packs O(mk3 log n) trees. Each new tree is a minimal spanning tree with respect to the edge utilizations, and the utilization of an edge is the number of times it has been used in previous spanning trees divided by its capacity. We prove that each minimum k-way cut is crossed at most 2k-2 times by one of the trees. We can enumerate all such cuts in O(n2k) time, which is hence the running time of our algorithm producing all minimum k-way cuts. The previous fastest deterministic algorithm of [SICOMP'06] took O(n(4+o(1))k) time, so this is a near-quadratic improvement. Moreover, we essentially match the O(n(2-o(1))k) running time of the Monto Carlo (no correctness guarantee) randomized algorithm of Karger and Stein [JACM'96]. The minimum 3-way cut problem in an edge-weighted hypergraph is to find a partition of the vertices into 3 nonempty sets minimizing the total weight of hyperedges that have at least two endpoints in two different sets. In this paper we show that a minimum 3-way cut in hypergraphs can be found by using O(n^3) hypergraph minimum (s,t) cut computations, where n is the number of vertices in the hypergraph. Our simple algorithm is the first polynomial-time algorithm for finding minimum 3-way cuts in hypergraphs. 