Minimum k-cut
Introduction In mathematics, the '''minimum -cut''' is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least  connected components. These edges are referred to as '''-cut'''. The goal is to find the minimum-weight -cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.
Formal definition Given an undirected graph  with an assignment of weights to the edges  and an integer  partition  into  disjoint sets  while minimizing: For a fixed , the problem is polynomial time solvable in  However, the problem is NP-complete if  is part of the input. It is also NP-complete if we specify  vertices and ask for the minimum -cut which separates these vertices among each of the sets.
Approximations Several approximation algorithms exist with an approximation of  A simple greedy algorithm that achieves this approximation factor computes a minimum cut in each of the connected components and removes the lightest one. This algorithm requires a total of  max flow computations. Another algorithm achieving the same guarantee uses the Gomory–Hu tree representation of minimum cuts. Constructing the Gomory–Hu tree requires  max flow computations, but the algorithm requires an overall  max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm. Moreover, under the small set expansion hypothesis (a conjecture closely related to the unique games conjecture), the problem is NP-hard to approximate to within  factor for every constant , meaning that the aforementioned approximation algorithms are essentially tight for large .A variant of the problem asks for a minimum weight -cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed  if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality. More recently, polynomial time approximation schemes (PTAS) were discovered for those problems.While the minimum -cut problem is Parameterized complexity|W1-hard parameterized by , a parameterized approximation scheme can be obtained for this parameter.
See also * Maximum cut* Minimum cut
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