On minimum 3-cuts and approximating k-cuts using Cut Trees

Abstract

This paper describes two results on graph partitioning. Our first result is a non-crossing property of minimum 3-cuts. This property generalizes the results by Gomory-Hu on min-cuts (2-cute) in graphs. We also give an algorithm for finding minimum 3-cuts in O(n ³) Max-Flow computations. The second part of the paper describes a Performance Bounding technique based on Cut Trees for solving Partitioning Problems in weighted, undirected graphs. We show how to use this technique to derive approximation algorithms for two problems, the Minimum k-cut problem and the Multi-way cut problem.Our first illustration of the bounding technique is an algorithm for the Minimum k-cut which requires O(kn(m + n log n)) steps and gives an approximation of 2(1-1/k). We then generalise the Bounding Technique to achieve the approximation factor 2 — f(j, k) wheref(j, k) = j/k — (j — 2)/k ² + O(j/k ³), j ≥ 3. The algorithm presented for the Minimum k-cut problem is polynomial in n and k for fixed j. We also give an approximation algorithm for the planar Multi-way Cut problem.

Part of this work was done while the author was a visitor at the Max-Planck-Institute für Informatik, Saarbrücken, Germany.