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Submodular Minimization via Pathwidth

  • Hiroshi Nagamochi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7287)

Abstract

In this paper, we present a submodular minimization algorithm based on a new relationship between minimizers of a submodular set function and pathwidth defined on submodular set functions. Given a submodular set function f on a finite set V with n ≥ 2 elements and an ordered pair s,t ∈ V, let λ s,t denote the minimum f(X) over all sets X with s ∈ X ⊆ V − {t}. The pathwidth Λ(σ) of a sequence σ of all n elements in V is defined to be the maximum f(V(σ′)) over all nonempty and proper prefixes σ′ of σ, where V(σ′) denotes the set of elements occurred in σ′. The pathwidth Λ s,t of f from s to t is defined to be the minimum pathwidth Λ(σ) over all sequences σ of V which start with element s and end up with t. Given a real k ≥ f({s}), our algorithm checks whether Λ s,t  ≤ k or not and computes λ s,t (when Λ s,t  ≤ k) in O(n Δ(k) + 1) oracle-time, where Δ(k) is the number of distinct values of f(X) with f(X) ≤ k overall sets X with s ∈ X ⊆ V − {t}.

Keywords

Search Tree Distinct Element Polynomial Time Algorithm Submodular Function Proper Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hiroshi Nagamochi
    • 1
  1. 1.Graduate School of InformaticsKyoto UniversityJapan

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