Approximation Algorithms for Submodular Multiway Partition

We study algorithms for the SUBMODULAR Multiway PARTITION problem (SUB-MP). An instance of SUB-MP consists of a finite ground set V, a subset S = {s
₁
, S
₂
, ..., 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 A
₁
,...,A
_k
to minimize Σ
_i=1
^k
f(A
_i
) such that for 1 ≤ i ≤ k, s
_i
∈ A
_i
. 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 sub- modular 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 Lovasz-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. (1) A 2-approximation for SUB-MP. This improves the (k - 1)-approximation from [29]. (2) A (1.5 - 1/k)-approximation for SUB-MP when f is symmetric. This improves the 2(1 - 1/k)-approximation from [23], [29].