Minimization of Locally Defined Submodular Functions by Optimal Soft Arc Consistency

Abstract

Submodular function minimization is a polynomially solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have highorder polynomial time complexity. In many applications the objective function is locally defined in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every valued constraint satisfaction problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally defined submodular functions. From a theoretical point of view, this brings to light a previously unknown connection between submodularity and soft arc consistency.

The Complexity of Valued Constraint Satisfaction Problems

“…An important and well-studied subproblem of SFM is the minimisation of submodular functions of bounded arity (SFM b ), also known as locally-defined submodular functions [73], or submodular functions with succinct representation [114]. In this scenario the submodular function to be minimised is defined by the sum of a collection of functions which each depend only on a bounded number of variables.…”

“…7) [188]. An alternative approach to solving VCSP instances with submodular constraints, based on linear programming, has been proposed in [73]. Recently, Thapper and Živný have shown that linear programming can be used to solve VCSPs with submodular constraints with respect to any lattice.…”

“…Other approaches for solving instances from VCSP(Γ sub ) include linear programming [73,75] and submodular flows [186].…”

Supervised hypergraph labeling

“…It has been shown in the literature that for a submodular clique potential function, there are algorithms to produce a globally optimal solution in polynomial time [5,14] for two class problems. This section analyzes the proposed clique function to identify what properties make it submodular and therefore can be solved efficiently and accurately with existing algorithms.…”

“…Problems with nonlinear g c (·) can be solved using a linear programming (LP) formulation suggested in [5,14]. See [11,22] and references therein to learn more about various inference algorithms available in the literature.…”