Exact Algorithms for Combinatorial Optimization Problems with Submodular Objective Functions

Many combinatorial optimization problems have natural formulations as submodular minimization problems over well-studied combinatorial structures. A standard approach to these problems is to linearize the objective function by introducing new variables and constraints, yielding an extended formulation. We propose two new approaches for constrained submodular minimization problems. The first is a linearization approach that requires only a small number of additional variables. We exploit a tight polyhedral description of this new model and an efficient separation algorithm. The second approach uses Lagrangean decomposition to create two subproblems which are solved with polynomial time algorithms; the first subproblem corresponds to the objective function while the second consists of the constraints. The bounds obtained from both approaches are then used in branch and bound-algorithms. We apply our general results to problems from wireless network design and mean-risk optimization. Our experimental results show that both approaches compare favorably to the standard techniques.