Abstract
Given an edge-weighted graph G = ( V , E ) , a subset S ⊆ V , an integer k ⩾ 1 and a real b ⩾ 0 , the minimum subpartition problem asks to find a family of k nonempty disjoint subsets X 1 , X 2 , … , X k ⊆ S with d ( X i ) ⩽ b , 1 ⩽ i ⩽ k , so as to minimize ∑ 1 ⩽ i ⩽ k d ( X i ) , where d ( X ) denotes the total weight of edges between X and V − X . In this paper, we show that the minimum subpartition problem can be solved in O ( m n + n 2 log n ) time. The result is then applied to the minimum k-way cut problem and the graph strength problem to improve the previous best time bounds of 2-approximation algorithms for these problems to O ( m n + n 2 log n ) .
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