Approximation Algorithms for Submodular Multiway Partition

We study algorithms for the Submodular Multiway Partition problem (SubMP). An instance of SubMP consists of a finite ground set $V$, a subset of $k$ elements $S = {s_1,s_2,…,s_k}$ called terminals, and a non-negative submodular set function $f:2^V\rightarrow \mathbb{R}+$ on $V$ provided as a value oracle. The goal is to partition $V$ into $k$ sets $A_1,…,A_k$ such that for $1 \le i \le k$, $s_i \in A_i$ and $\sum{i=1}^k f(A_i)$ is minimized. SubMP generalizes some well-known problems such as the Multiway Cut problem in graphs and hypergraphs, and the Node-weighed Multiway Cut problem in graphs. SubMP for arbitrarysubmodular 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 very recent work \cite{ChekuriE11} we proposed a convex-programming relaxation for SubMP based on the Lov'asz-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. (i) A 2-approximation for SubMP. This improves the $(k-1)$-approximation from \cite{ZhaoNI05} and (ii) A $(1.5-1/k)$-approximation for SubMP when $f$ is symmetric. This improves the $2(1-1/k)$-approximation from \cite{Queyranne99,ZhaoNI05}.

💡 Research Summary

The paper studies the Submodular Multiway Partition (SubMP) problem, which asks for a partition of a ground set (V) into (k) parts (A_1,\dots,A_k) such that each prescribed terminal (s_i) belongs to (A_i) and the sum (\sum_{i=1}^k f(A_i)) is minimized, where (f) is a non‑negative submodular set function given via a value oracle. SubMP captures many classic cut problems: the graph multiway cut, hypergraph multiway cut, and node‑weighted multiway cut are all special cases. Prior work achieved a ((k-1))-approximation for the general (possibly asymmetric) case and a (2(1-1/k))-approximation for the symmetric case, using greedy splitting or divide‑and‑conquer techniques.

The authors introduce a convex programming relaxation based on the Lovász extension (\hat f) of the submodular function. They formulate the relaxation (SUB‑MP‑REL) as
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