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• W2119003693 abstract "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 <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> , S <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sub> , ..., s <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> } ⊆ V of k elements called terminals, and a non-negative submodular set function f : 2 <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>V</sup> → ℝ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>+</sub> on V provided as a value oracle. The goal is to partition V into k sets A <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> ,...,A <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> to minimize Σ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i=1</sub> <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sup> f(A <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> ) such that for 1 ≤ i ≤ k, s <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> ∈ A <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>i</sub> . 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]." @default.
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• W2119003693 date "2011-10-01" @default.
• W2119003693 modified "2023-10-16" @default.
• W2119003693 title "Approximation Algorithms for Submodular Multiway Partition" @default.
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• W2119003693 doi "https://doi.org/10.1109/focs.2011.34" @default.
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