The Steiner tree problem (STP) is a classical NP-hard combinatorial optimization problem with applications in computational biology and network wiring. The objective of this problem is to find a minimum cost subgraph of a given undirected graph G with edge costs, that spans a subset of vertices called terminals. We present currently used linear programming formulations of the problem based on two different approaches-the bidirected cut relaxation (BCR) and the hypergraphic formulations (HYP), the former offering better computational performance, and the latter better bounds on the integrality gap. As our contribution, we propose a new hierarchy of path-based extended formulations for STP. We show that this hierarchy provides better integrality gaps on graph instances used to define the worst-case lower bounds on the integrality gap for both BCR and HYP. We prove that each consecutive level of our hierarchy is at least as strong as the previous one. Additionally, we also present numerical results showing that several difficult STP instances can be solved to integer optimality by using this hierarchy. Our approach can be adapted to variants of STP or applied to hypergraphic formulations for further potential improvement on the integrality gap bounds, in exchange for additional computational effort. KEYWORDS extended formulation, flow-based formulation, integrality gap, linear programming relaxation, mixed-integer linear programming, Steiner tree Networks. 2020;75:3-17.wileyonlinelibrary.com/journal/net
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.