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}.
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Nagamochi, H. (2012). Submodular Minimization via Pathwidth. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_54
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DOI: https://doi.org/10.1007/978-3-642-29952-0_54
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