[Li, Angsheng] Chinese Acad Sci, Inst Software, State Key Lab Comp Sci, Beijing 100190, Peoples R China. [Zhang, Peng] Shandong Univ, Sch Comp Sci & Technol, Jinan 250101, Peoples R China.

Abstract:

We investigate the unbalanced cut problems. A cut (A, B) is called unbalanced if the size of its smaller side is at most k (called k-size) or exactly k (called Ek-size), where k is an input parameter. We consider two closely related unbalanced cut problems, in which the quality of a cut is measured with respect to the sparsity and the conductance, respectively. We show that even if the input graph is restricted to be a tree, the Ek-Sparsest Cut problem (to find an Ek-size cut with the minimum sparsity) is still NP-hard. We give a bicriteria approximation algorithm for the k-Sparsest Cut problem (to find a k-size cut with the minimum sparsity), which outputs a cut whose sparsity is at most O(log n) times the optimum and whose smaller side has size at most O(log n) k. As a consequence, this leads to a (O(log n), O(log n))-bicriteria approximation algorithm for the Min k-Conductance problem (to find a k-size cut with the minimum conductance).

English Abstract:

We investigate the unbalanced cut problems. A cut (A, B) is called unbalanced if the size of its smaller side is at most k (called k-size) or exactly k (called Ek-size), where k is an input parameter. We consider two closely related unbalanced cut problems, in which the quality of a cut is measured with respect to the sparsity and the conductance, respectively. We show that even if the input graph is restricted to be a tree, the Ek-Sparsest Cut problem (to find an Ek-size cut with the minimum sparsity) is still NP-hard. We give a bicriteria approximation algorithm for the k-Sparsest Cut problem (to find a k-size cut with the minimum sparsity), which outputs a cut whose sparsity is at most O(log n) times the optimum and whose smaller side has size at most O(log n) k. As a consequence, this leads to a (O(log n), O(log n))-bicriteria approximation algorithm for the Min k-Conductance problem (to find a k-size cut with the minimum conductance).