Stronger path‐based extended formulation for the Steiner tree problem

Abstract

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

Solving Steiner trees: Recent advances, challenges, and perspectives

“…In the remainder of this section we review four MIP formulations for the STP: (1) the bidirected cut formulation (DCUT), which is part of the most successful exact algorithmic frameworks; (2) hypergraphic formulations (HYP), which are used to derive the state‐of‐the‐art approximation algorithms; and the two strong compact MIP formulations that are known to dominate DCUT or HYP: (3) the common flow formulation (CF) of Polzin and Vahdati Daneshmand [182] and (4) the brand new path‐based formulation (MCF‐ λ ) of Filipecki and Van Vyve [92] published in a recent Networks 2020 issue.…”

“…Very recently, in their Networks article, Filipecki and Van Vyve [92] proposed new MCF‐based formulations with aggregated‐path variables. The authors generalized the MCF formulation by considering flows over paths, instead of just flows over edges.…”

“…Finally, to make the model independent on the choice of the root node, similarly as for the common flow formulation, an additional root index is introduced. In the following we simplify the original notation and only introduce the formulation MCF‐2, whereas the more general model can be found in Filipecki and Van Vyve [92].…”

“…In the generalization of this model given by Filipecki and Van Vyve [92] paths of length up to λ (where λ ≥ 2 is a given input parameter) are used to route the flow and flow‐preservation constraints are imposed for all subpaths of length λ − 1. The sets are comprised of paths of length λ along with all paths of length 1 to λ − 1 that start at r .…”

“…Finally, Filipecki and Van Vyve [92] did not compare their new hierarchy of formulations with the hierarchy of common flow models studied in Polzin and Vahdati Daneshmand [187]; see Section 3.3. This comparison remains to be done in order to better understand the relationship between these flow‐based models, as well as their strengths and weaknesses.…”