The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP . The focus of this article is on the role of the SCIP Optimization Suite in supporting research. SCIP ’s main design principles are discussed, followed by a presentation of the latest performance improvements and developments in version 8.0, which serve both as examples of SCIP ’s application as a research tool and as a platform for further developments. Furthermore, this article gives an overview of interfaces to other programming and modeling languages, new features that expand the possibilities for user interaction with the framework, and the latest developments in several extensions built upon SCIP .

The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with a large number of practical applications. It attracted particular interest during the 11th DIMACS Challenge in 2014, and since then, several PCSTP solvers have been introduced in the literature. Although these new solvers further, and often drastically, improved on the results of the DIMACS Challenge, many PCSTP benchmark instances have remained unsolved. The following article describes further advances in the state of the art in exact PCSTP solving. It introduces new techniques and algorithms for PCSTP, involving various new transformations (or reductions) of PCSTP instances to equivalent problems, for example, to decrease the problem size or to obtain a better integer programming formulation. Several of the new techniques and algorithms provably dominate previous approaches. Further theoretical properties of the new components, such as their complexity, are discussed. Also, new complexity results for the exact solution of PCSTP and related problems are described, which form the base of the algorithm design. Finally, the new developments also translate into a strong computational performance: the resulting exact PCSTP solver outperforms all previous approaches, both in terms of runtime and solvability. In particular, it solves several formerly intractable benchmark instances from the 11th DIMACS Challenge to optimality. Moreover, several recently introduced large-scale instances with up to 10 million edges, previously considered to be too large for any exact approach, can now be solved to optimality in less than two hours. Summary of Contribution: The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with many practical applications. The article introduces and analyses new techniques and algorithms for PCSTP that ultimately aim for improved (practical) exact solution. The algorithmic developments are underpinned by results on theoretical aspects, such as fixed-parameter tractability of PCSTP. Computationally, we considerably push the limits of tractibility, being able to solve PCSTP instances with up to 10 million edges. The new solver, which also considerably outperforms the state of the art on smaller instances, will be made publicly available as part of the SCIP Optimization Suite.

Abstract Connectivity is a central concept in combinatorial optimization, graph theory, and operations research. In many applications, one is interested in finding an optimal subset of vertices with the essential requirement that the vertices are connected, but not how they are connected. In other words, it is not relevant which edges are selected to obtain connectivity. This article is concerned with the exact solution of such problems via integer programming. We analyze and compare (mixed) integer programming formulations with respect to the strength of their linear programming relaxations. Along the way, we also provide a tighter (compact) description of the connected subgraph polytope—the convex hull of subsets of vertices that induce a connected subgraph. Furthermore, we give a (compact) complete description of the connected subgraph polytope for graphs with no four independent vertices.

Relay node placement, which aims to connect pre-deployed sensor nodes to base stations, is essential in minimizing the costs of wireless sensor networks. In this paper, we formulate the new Node-Weighted Partial Terminal Steiner Tree Problem (NWPTSTP) for minimum-cost relay node placement in two-tiered wireless sensor networks. The objective is to minimize the sum of heterogeneous production and placement costs of relay nodes and the sum of outage probabilities of transmission routes in a routing tree simultaneously. This extends the previous work that considers the costs of relay nodes to be homogeneous. After formulating NWPTSTP for this purpose, we prove that it can be transformed to the existing node-weighted Steiner tree problem. Subsequently, we conduct some theoretical analyses on the emerging Physarum-inspired algorithms to reveal their potential of computing Steiner trees. Based on these analyses, we propose a new Physarum-inspired algorithm for solving NWPTSTP. We conduct computational trials to show that: 1) in comparison to a state-of-the-art approximation algorithm for solving the node-weighted Steiner tree problem, our Physarum-inspired algorithm can produce better solutions in a smaller amount of time; and 2) in comparison to two state-of-the-art relay node placement algorithms, our Physarum-inspired algorithm can design wireless sensor networks with 25% lower relay cost and similar quality of service (specifically, 5% shorter network lifetime, 2% longer delay, and 0% loss of goodput). This indicates the usefulness of our Physarum-inspired algorithm for minimum-cost relay node placement in budget-limited scenarios.

Abstract The concept of reduction has frequently distinguished itself as a pivotal ingredient of exact solving approaches for the Steiner tree problem in graphs. In this article we broaden the focus and consider reduction techniques for three Steiner problem variants that have been extensively discussed in the literature and entail various practical applications: The prize‐collecting Steiner tree problem, the rooted prize‐collecting Steiner tree problem and the maximum‐weight connected subgraph problem. By introducing and subsequently deploying numerous new reduction methods, we are able to drastically decrease the size of a large number of benchmark instances, already solving more than 90% of them to optimality. Furthermore, we demonstrate the impact of these techniques on exact solving, using the example of the state‐of‐the‐art Steiner problem solver SCIP‐Jack .