Approximation Algorithms for Submodular Multiway Partition

We study algorithms for the {\sc Sub modular Multiway Partition}problem (\SubMP). An instance of \SubMP consists of a finite ground set $V$, a subset $S = \{s_1,s_2,\ldots,s_k\} \subseteq V$ of $k$elements called terminals, and a non-negative sub modular set function$f:2^V\rightarrow \mathbb{R}_+$ on $V$ provided as a value oracle. The goal is to partition $V$ into $k$ sets $A_1,\ldots,A_k$ to minimize $\sum_{i=1}^kf(A_i)$ such that for $1 \le i \le k$, $s_i \inA_i$. \SubMP generalizes some well-known problems such as the {\scMultiway Cut} problem in graphs and hyper graphs, and the {\scNode-weighed Multiway Cut} problem in graphs. \SubMP for arbitrary sub modular functions (instead of just symmetric functions) was considered by Zhao, Nagamochi and Ibaraki \cite{ZhaoNI05}. Previous algorithms were based on greedy splitting and divide and conquer strategies. In recent work \cite{ChekuriE11} we proposed a convex-programming relaxation for \SubMP based on the Lov\'asz-extension of a sub modular function and showed its applicability for some special cases. In this paper we obtain the following results for arbitrary sub modular functions via this relaxation. \begin{itemize} \item A $2$-approximation for \SubMP. This improves the $(k-1)$-approximation from \cite{ZhaoNI05}. \item A $(1.5-\frac{1}{k})$-approximation for \SubMP when $f$ is {\em symmetric}. This improves the $2(1-\frac{1}{k})$-approximation from \cite{Queyranne99, ZhaoNI05}.\end{itemize}