Submodular minimization via pathwidth

• Authors: Hiroshi Nagamochi
• Venue: TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
• Year: 2012

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 Λs ,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 Λs ,t ≤k or not and computes λ s ,t (when Λs ,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 }.