In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.
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Formal definition
Given an undirected graph G = (V, E) with an assignment of weights to the edges w: E → N and an integer k ∈ {2, 3, …, |V|}, partition V into k disjoint sets F = {C1, C2, …, Ck} while minimizing
For a fixed k, the problem is polynomial time solvable in O(|V|k2). However, the problem is NP-complete if k is part of the input. It is also NP-complete if we specify
Approximations
Several approximation algorithms exist with an approximation of 2 − 2/k. A simple greedy algorithm that achieves this approximation factor computes a minimum cut in each connected components and removes the lightest one. This algorithm requires a total of n − 1 max flow computations. Another algorithm achieving the same guarantee uses the Gomory–Hu tree representation of minimum cuts. Constructing the Gomory–Hu tree requires n − 1 max flow computations, but the algorithm requires an overall O(kn) max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm.
If we restrict the graph to a metric space, meaning a complete graph that satisfies the triangle inequality, and enforce that the output partitions are each of pre-specified sizes, the problem is approximable to within a factor of 3 for any fixed k. More recently, polynomial time approximation schemes (PTAS) were discovered for those problems.