Faster Exact and Approximate Algorithms for k-Cut

Abstract

In the k-Cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. The current best algorithms are an O(n (2−o(1))k ) randomized algorithm due to Karger and Stein, and anÕ(n 2k ) deterministic algorithm due to Thorup. Moreover, several 2-approximation algorithms are known for the problem (due to Saran and Vazirani, Naor and Rabani, and Ravi and Sinha).It has remained an open problem to (a) improve the runtime of exact algorithms, and (b) to get better approximation algorithms. In this paper we show an O(k O(k) n (2ω/3+o(1))k )-time algorithm for k-Cut. Moreover, we show an (1 + ε)-approximation algorithm that runs in time O((k/ε) O(k) n k+O(1) ), and a 1.81-approximation in fixed-parameter time O(2 OIn this paper we consider the k-Cut problem: given an edge-weighted graph G = (V, E, w) and an integer k, delete a minimum-weight set of edges so that G has at least k connected components. This problem is a natural generalization of the global min-cut problem, where the goal is to break the graph into k = 2 pieces. This problem has been actively studied in theory of both exact and approximation algorithms, where each result brought new insights and tools on graph cuts.It is not a priori clear how to obtain poly-time algorithms for any constant k, since guessing one vertex from each part only reduces the problem to the NP-hard Multiway Cut problem. Indeed, the first result along these lines was the work of Goldschmidt and Hochbaum [GH94] who gave an O(n (1/2−o(1))k 2 )-time exact algorithm for k-Cut. Since then, the exact exponent in terms of k has been actively studied. The current best runtime is achieved by an O(n 2(k−1) ) randomized algorithm due to Karger and Stein [KS96], which performs random edge contractions until the remaining graph has k nodes, and shows that the resulting cut is optimal with probability at least Ω(n −2(k−1) ). The asymptotic runtime ofÕ(n 2(k−1) ) was later matched by a deterministic algorithm of Thorup [Tho08]. His algorithm was based on tree-packing theorems; it showed how to efficiently find a tree for which the optimal k-cut crosses it 2k − 2 times. Enumerating over all possible 2k − 2 edges of this tree gives the algorithm.These elegant O(n 2k )-time algorithms are the state-of-the-art, and it has remained an open question to improve on them. An easy observation is that the problem is closely related to k-Clique, so we may not expect the exponent of n to go below (ω/3)k. Given the interest in fine-grained analysis of algorithms, where in the range [(ω/3)k, 2k − 2] does the correct answer lie?Our main results give faster deterministic and randomized algorithms for the problem.Theorem 1.1 (Faster Randomized Algorithm). Let W be a positive integer. There is a randomized algorithm for k-Cut on graphs with edge weights in [W ] with runtimethat succeeds with probability 1 − 1/poly(n).Theorem 1.2 (Even Faster Deterministic Algorithm). Let W be a positive integer. For any ε >...

Faster Minimum k-cut of a Simple Graph

“…If ℓ = k − 1, the minimum value possible, then we call it a tight T-tree. Theorem 2.3 (Thorup [24], rephrased in Corollary 2.3 of [9]). We can find a collection T ofÕ(k 3 m) trees such that there exists a (2k − 2)-T-tree in T .…”

“…We first state an easy claim from [9]. We then use it to bound the size of a nontrivial minimum k-cut by the minimum degree δ of the graph.…”

“…Lower bounds for the k-Cut problem have also been studied actively in the past decade. k-Cut on real-weighted graphs is at least as hard as minimum weighted (k − 1)-clique [9], the latter of which is conjectured to require n (1−o(1))k time for any constant k [26]. For k-Cut on unweighted graphs, the lower bound is weakened to n (ω/3−o(1))k , again from a reduction to (k − 1)-clique, where ω < 2.3727 is the matrix multiplication constant [9].…”

“…k-Cut on real-weighted graphs is at least as hard as minimum weighted (k − 1)-clique [9], the latter of which is conjectured to require n (1−o(1))k time for any constant k [26]. For k-Cut on unweighted graphs, the lower bound is weakened to n (ω/3−o(1))k , again from a reduction to (k − 1)-clique, where ω < 2.3727 is the matrix multiplication constant [9]. However, for "combinatorial" algorithms as described in [1,26], this lower bound is again n (1−o(1))k even for unweighted graphs, under the stronger hardness conjecture of k-clique for combinatorial algorithms [1,26].…”

“…Since n =Õ(n/λ k ), this running time becomes n (1+o(1))k , as needed.Tree Packing To solve k-Cut in n (1+o(1))k λ k k time on multi-graphs, 2 we begin with a tree packing result of Thorup [24], which says that we can compute a small collection of trees so that for one tree T , at most 2k − 2 edges of T have endpoints in different components of the minimum k-cut. we apply a reduction in [9] which, at a multiplicative cost of O(n k ) in the running time, produces a tree T such that exactly k − 1 edges of T have endpoints in different components of the minimum k-cut. (Note that k − 1 here is the smallest possible.)…”