Reduction techniques for the prize collecting Steiner tree problem and the maximum‐weight connected subgraph problem

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.Proof. First, note that both of the minima attained in (7) and (8) correspond to a path. Next, let t (1)min ) is minimal. The corresponding path between t i and t (1) min contains an edge {v j , v k }∈ (N(t i )) and since the

Combining NP-Hard Reduction Techniques and Strong Heuristics in an Exact Algorithm for the Maximum-Weight Connected Subgraph Problem

“…For the MWCSP the first ground was broken in the course of the 11th DIMACS Challenge, with two articles [4,15] containing reduction techniques as part of an exact solving approach. Two years later a significantly more comprehensive reduction package for the MWCSP was introduced in [29] and a dual-ascent based branch-and-bound algorithm with strong reduction properties was described in [21]. The reductions techniques introduced in the following continue the work started in [29] by introducing both bound-based and alternative-based reduction methods.…”

“…Two years later a significantly more comprehensive reduction package for the MWCSP was introduced in [29] and a dual-ascent based branch-and-bound algorithm with strong reduction properties was described in [21]. The reductions techniques introduced in the following continue the work started in [29] by introducing both bound-based and alternative-based reduction methods. To render proof techniques more perspicuous, throughout this section it will without loss of generality be assumed that each solution to P M W is given as a tree (and not as an arbitrary connected subgraph).…”

“…The term bound-based reductions describes preprocessing methods that identify edges and vertices for elimination by examining whether they induce an upper bound that is lower than a given lower bound (or vice versa) [26,29]. In the following we will introduce a new reduction technique that could also be used to generalize the bound-based Voronoi reduction concept for the Steiner tree problem in graphs [26] and the prize-collecting Steiner tree problem [29]-in this way the proof technique for the followings three propositions is similar to the Voronoi reduction proofs in [26] and [29] and is therefore presented in the appendix only.…”

“…Several articles have contributed to a solid theoretical understanding of the problem [8,33] while others have addressed exact solving approaches [7,16,17]. Recently, also the scope of preprocessing techniques has been considerably enhanced [29].…”

The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks

“…Pre-processing techniques and exact algorithms are often combined together to produce optimal solutions in small instances (e.g. the combination in [14]), where pre-processing techniques reduce instance sizes and exact algorithms produce optimal solutions in the reduced instances (note that, the state-of-the-art pre-processing techniques, such as those in [15], are only fast enough to reduce small instances with thousands of vertices, while we deal with much larger instances with millions of vertices in this paper). On the other hand, heuristic algorithms and post-processing techniques are often combined together to produce fast suboptimal solutions in large instances (e.g.…”