We study algorithms for the SUBMODULAR MUL-TIWAY PARTITION problem (SUB-MP). An instance of SUB-MP consists of a finite ground set V , a subset S = {s1, s2, . . . , s k } ⊆ V of k elements called terminals, and a non-negative submodular set function f : 2 V → R+ on V provided as a value oracle. The goal is to partition V into k sets A1, . . . , A k to minimize k i=1 f (Ai) such that for 1 ≤ i ≤ k, si ∈ Ai. SUB-MP generalizes some well-known problems such as the MULTIWAY CUT problem in graphs and hypergraphs, and the NODE-WEIGHED MULTIWAY CUT problem in graphs. SUB-MP for arbitrary submodular functions (instead of just symmetric functions) was considered by Zhao, Nagamochi and Ibaraki [29] . Previous algorithms were based on greedy splitting and divide and conquer strategies. In recent work [5] we proposed a convex-programming relaxation for SUB-MP based on the Lovász-extension of a submodular function and showed its applicability for some special cases. In this paper we obtain the following results for arbitrary submodular functions via this relaxation. • A 2-approximation for SUB-MP. This improves the (k − 1)approximation from [29]. • A (1.5− 1 k )-approximation for SUB-MP when f is symmetric. This improves the 2(1 − 1 k )-approximation from [23], [29].
Chekuri and Ene REF obtained a (1.5 − 1/k)-approximation for SYM-SUB-MP and 2-approximation for SUB-MP.
Approximation Algorithms for Submodular Multiway Partition
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Mathematics Computer Science