Improved algorithms for the Steiner problem in networks

Abstract

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors.

Primal-Dual Approaches to the Steiner Problem

“…There are many theoretical and empirical investigations which indicate that the directed relaxation is (at least usually) a much stronger relaxation than the undirected one (see for example [3,4]). In [19]' we could achieve impressive empirical results (including extremely tight lower and upper bounds) using this relaxation.…”

“…'In the original work of Wong [22] An empirically more successful variant uses a size. criterion: at each iteration, an active component of minimum size is chosen (see [6,19]). Note that such a component is always a root component.…”

“…This problem is particularly relevant if an instance is tobe solved to optimality: although the lower bounds generated by DAc usually come elose to v(LPc), the remaining gap is sometimes larger than desiredin the context of exact algorithms. In [19]~we used methods like Lagrangean relaxation and row generation to further improve the bounds provided by dual-ascent. But these methods tend to be slow for very large instances.…”

“…The main emphasis will be on lower bounds, with upper bounds mainly usedin a primal-dual context to prove a performance guarantee for the lower bounds. Despite the fact that calculating tight lower bounds efficiently is highly desirable (for example in the context of exact algorithms or reduction tests [19,6,16]), this issue has found 'much less attention in the literature. For re cent developments concerning upper bounds, see [19].…”

Decomposition methods for the two-stage stochastic Steiner tree problem

“…Each scenario k ∈ K has probability p k ∈ (0, 1], k∈K p k = 1, as well as second-stage edge costs c k : E → R ≥0 and terminals T k ⊆ V, r ∈ T k . The objective is to select first-stage edges E 0 S ⊆ E and second-stage edges E k S ⊆ E for each k ∈ K such that the subgraph induced by E 0 S ∪ E k S , G[E 0 S ∪ E k S ], connects T k and the expected cost Our contribution For the deterministic STP a wealth of theoretical results [6,11,20,29] and empirically successful computational techniques are known [8,10,31]. However, as noted in [2,27], the generalization of results from the STP to the SSTP is not straightforward.…”