A branch‐and‐cut algorithm for the single‐commodity, uncapacitated, fixed‐charge network flow problem

Abstract

We present a branch-and-cut algorithm to solve the single-commodity, uncapacitated, fixed-charge network flow problem, which includes the Steiner tree problem, uncapacitated lot-sizing problems, and the fixed-charge transportation problem as special cases. The cuts used are simple dicut inequalities and their variants. A crucial problem when separating these inequalities is to find the right cut set on which to generate the inequalities. The prototype branch-and-cut system, bc-nd, includes a separation heuristic for the dicut inequalities and problem-specific primal heuristics, branching, and pruning rules. Computational results show that bc-nd is competitive compared to a variety of special purpose algorithms for problems with explicit flow costs. We also examine how general purpose MIP systems perform on such problems when provided with formulations that have been tightened a priori with dicut inequalities.

Single-commodity robust network design problem: Complexity, instances and heuristic solutions

“…Ortega and Wolsey [24] report on the performance of general MIP solvers on various network design problems and develop an exact algorithm for the single-commodity fixed-charge network design problem (all arcs may be bought at a fixed-charge and then be used at full capacity).…”

The Mcf-separator: detecting and exploiting multi-commodity flow structures in MIPs

“…It is well known that cutting planes defined on network cuts are among the most effective when used within branch-and-cut frameworks to solve network design problems, see [5,13,14,24,30,38,45,49] for computational studies. Cut-based inequalities define facets of the corresponding polyhedra under very mild conditions and they are usually sparse.…”

Networks, uncertainty, applications and a crusade for optimality

“…Ortega and Wolsey [Ortega and Wolsey, 2003] report on the performance of general MIP solvers on various network design problems and develop an exact algorithm for the single-commodity fixed-charge network design problem (all arcs may be bought at a fixed-charge and then be used at full capacity).…”