Faster Minimum k-cut of a Simple Graph

Abstract

We consider the (exact, minimum) k-Cut problem: given a graph and an integer k, delete a minimumweight set of edges so that the remaining graph has at least k connected components. This problem is a natural generalization of the global minimum cut problem, where the goal is to break the graph into k = 2 pieces.Our main result is a (combinatorial) k-Cut algorithm on simple graphs that runs in n (1+o(1))k time for any constant k, improving upon the previously best n (2ω/3+o(1))k time algorithm of Gupta et al. [FOCS'18] and the previously best n (1.981+o(1))k time combinatorial algorithm of Gupta et al. [STOC'19]. For combinatorial algorithms, this algorithm is optimal up to o(1) factors assuming recent hardness conjectures: we show by a straightforward reduction that k-Cut on even a simple graph is as hard as (k − 1)-clique, establishing a lower bound of n (1−o(1))k for k-Cut. This settles, up to lower-order factors, the complexity of k-Cut on a simple graph for combinatorial algorithms. this algorithm is optimal up to o(1) factors assuming recent hardness conjectures: we show by a straightforward reduction that k-Cut on even a simple graph is as hard as (k − 1)-clique, establishing a lower bound of n (1−o(1))k for k-Cut. This settles, up to lower-order factors, the complexity of k-Cut on a simple graph for combinatorial algorithms. We remark that this is the first setting for k-Cut, except the restricted k = 2 case, where the running time has been determined up to o(1) factors.Theorem 1.1 (Main Result). For any parameter k, there is a (combinatorial, randomized) algorithm that computes the k-Cut of a simple graph in k O(k) n (1+o(1))k time.Theorem 1.2 (Lower Bound). Suppose we assume the conjecture that every combinatorial algorithm for k-clique requires n (1−o(1))k time for any constant k. Then, for any constant k, every combinatorial algorithm for k-Cut of a simple graph also requires n (1−o(1))k time. Our TechniquesOur k-Cut algorithm incorporates algorithmic techniques from a wide array of areas, from graph sparsification to fixed-parameter tractability to tree algorithms.Graph Sparsification Our first algorithmic ingredient is the Kawarabayashi-Thorup (KT) sparsification algorithm, which originated from the breakthrough paper of Kawarabayashi and Thorup on the deterministic minimum cut problem [16]. At a high level, given any simple graph G with minimum cut λ, the algorithm contracts G into a multi-graph ofÕ(m/λ) 1 edges so that any minimum cut of G that has at least two vertices on each side gets "preserved" in the contraction. That is, we never contract an edge in any such minimum cut. Kawarabayashi and Thorup used their contraction procedure to provide the firstÕ(m)-time deterministic algorithm for minimum cut of a simple graph. They first applied the contraction to G, obtaining a multi-graph G on m =Õ(m/λ) edges, and then ran theÕ(mλ)-time minimum cut algorithm of Gabow on G, which works for multi-graphs. This covers the case when the minimum cut of G has at least two vertices on each side; the other ca...

The Karger-Stein algorithm is optimal for k-cut

“…This improves the running time n (1.98+o(1))k for the general weighted case [GLL19] and even the running time n (1+o(1))k for the unweighted case [Li19], where the extra n o(k) term is still at least polynomial for fixed k. It is also almost tight under the hypothesis that Max-Weight (k − 1)-Cliqe requiresΩ(n (1−o(1))k ) time. while |V | > k do 3:…”

“…(2) For graphs with polynomial integer weights, we showed an algorithm to solve the problem in time approximately k O (k) n (2ω/3+o(1))k [GLL18]. And for unweighted graphs we showed how to get the k O (k ) n (1+o(1)k runtime [Li19]. Both these approaches were based on obtaining a spanning tree cut by a minimum k-cut in a small number of edges, and using involved dynamic programming methods on the tree to efficiently compute the edges and find the k-cut.…”

Breaking the $n^k$ Barrier for Minimum $k$-cut on Simple Graphs

“…Our high-level strategy mimics that of Li [Li19], in that we make use of the Kawarabayashi-Thorup graph sparsification technique on simple graphs, but our approach differs by exploiting matrix multiplicationbased methods as well. Below, we describe these two techniques and how we apply them.…”

“…Here, non-trivial means that the minimum cut does not have just a singleton vertex on one side. More recently, Li [Li19] has generalized the Kawarabayashi-Thorup sparsification to also preserve non-trivial minimum k-cuts (those without any singleton vertices as components), which led to an n (1+o(1))k -time minimum k-cut algorithm on simple graphs. The contracted graph has Õ(n/δ) vertices where δ is the minimum degree of the graph.…”

“…These algorithms remained the state of the art until a few years ago, when new progress was established on the problem [GLL18, GLL19,Li19], culminating in the Õ(n k ) time algorithm of Gupta, Harris, Lee, and Li [GHLL20] which is, surprisingly enough, just the original Karger-Stein recursive contraction algorithm with an improved analysis. The Õ(n k ) time algorithm also works for weighted graphs, and they show by a reduction to max-weight k-clique that their algorithm is asymptotically optimal, assuming the popular conjecture that max-weight k-clique cannot be solved faster than Ω(n k−O(1) ) time.…”