Minimum k-cut

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.

Contents

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

\sum_{i=1}^{k-1}\sum_{j=i%2B1}^k\sum_{\begin{smallmatrix} v_1 \in C_i \\ v_2 \in C_j \end{smallmatrix}} w ( \left \{ v_1, v_2 \right \} )

For a fixed k, the problem is polynomial time solvable in O(|V|k2).[1] However, the problem is NP-complete if k is part of the input.[2] It is also NP-complete if we specify k vertices and ask for the minimum k-cut which separates these vertices among each of the sets.[3]

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.[4][5]

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.[6]

See also

Notes

  1. ^ Goldschmidt & Hochbaum 1988.
  2. ^ Garey & Johnson 1979
  3. ^ [1], which cites [2]
  4. ^ Saran & Vazirani 1991.
  5. ^ Vazirani 2003, pp. 40–44.
  6. ^ Guttmann-Beck & Hassin 1999, pp. 198–207.

References