Approximation Algorithm for Sparsest k-Partitioning

Given a graph G, the sparsest-cut problem asks to find the set of vertices S which has the least expansion defined as φG(S) def = w(E(S, S̄)) min{w(S), w(S̄)} , where w is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer k, compute a k-partition {P1, . . . , Pk} of the vertex set so as to minimize φG({P1, . . . , Pk}) def = max i φG(Pi). Our main result is a polynomial time bi-criteria approximation algorithm which outputs a (1− ε)k-partition of the vertex set such that each piece has expansion at most Oε( √ log n log k) times OPT . We also study balanced versions of this problem. ∗Supported in part by NSF awards CCF-0915903 and CCF-1217793. 1 ar X iv :1 30 6. 43 84 v2 [ cs .D S] 8 O ct 2 01 3