A vertex-separator-based integer linear programming formulation for the partitioned Steiner tree problem

Given an undirected graph G with a cost function on vertices, a collection of subgraphs of G such that in each subgraph, there are some distinguished vertices called terminals, the Partitioned Steiner Tree Problem (PSTP) asks for a minimum cost vertex set such that, in each of the given subgraph Gi, the graph induced by the vertex set spans the terminal set in Gi. The PSTP generalizes the well-known Steiner tree problem and has important applications in computational sustainability, network design, and social network analysis. However, for solving the PSTP, conventional integer programming approaches based on single-commodity flow, multi-commodity flow and subtour elimination integer linear programs, suffer from low computational efficiency due to a substantial number of variables. In this paper, we propose a compact vertex-separator-based integer linear programming formulation with much fewer variables. Enhancing inequalities are also studied for tightening the formulation. We further investigate a branch-and-cut algorithm, a local-branching heuristic algorithm, and a hybrid algorithm combining them. In experiments where both public real-world and synthetic graphs are used, our hybrid algorithm outperforms all conventional approaches, especially for large graphs with more than ten thousand vertices. Further tests also validate the effectiveness of the proposed formulation and enhancing inequalities.