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].
These problems were considered by Zhao, Nagamochi and Ibaraki [31] who analyzed greedy-splitting algorithms, and more recently by Chekuri and Ene REF who used a Lovász-extension based convex relaxation.
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