The regenerator location problem

Abstract

In this paper we introduce the regenerator location problem (RLP), which deals with a constraint on the geographical extent of transmission in optical networks. Specifically, an optical signal can only travel a maximum distance of d max before its quality deteriorates to the point that it must be regenerated by installing regenerators at nodes of the network. As the cost of a regenerator is high we wish to deploy as few regenerators as possible in the network, while ensuring all nodes can communicate with each other. We show that the RLP is NP-Complete. We then devise three heuristics for the RLP. We show how to represent the RLP as a max leaf spanning tree problem (MLSTP) on a transformed graph. Using this fact we model the RLP as a Steiner arborescence problem (SAP) with a unit degree constraint on the root node. We also devise a branch-and-cut procedure to the directed cut formulation for the SAP problem. In our computational results over 740 test instances, the heuristic procedures obtained the optimal solution in 454 instances, while the branch-and-cut procedure obtained the optimal solution in 536 instances. These results indicate the quality of the heuristic solutions are quite good, and the branch-and-cut approach is viable for the optimal solution of problems with up to 100 nodes. Our approaches are also directly applicable to the MLSTP indicating that both the heuristics and branch-and-cut approach are viable options for the MLSTP.

Regenerator Location Problem and survivable extensions: A hub covering location perspective

“…We assume without loss of generality that w e 6 d max for every e 2 E since any edge violating this condition can simply be deleted from G. Let A ¼ fði; jÞ [ ðj; iÞ : fi; jg 2 Eg be the arc set induced by the edges in G. We assume that w ij ¼ w ji ¼ w e for every e ¼ fi; jg 2 E. Let d ij be the length of the shortest path from i to j in G. After solving an all pairs shortest path problem in G, we create the unweighted closure graph ðV; E c Þ, where the edges correspond to pairs of nodes that can communicate with no need for regeneration, i.e., E c ¼ ffi; jg : i; j 2 V; i -j; d ij 6 d max g. Let A c ¼ fði; jÞ [ ðj; iÞ : fi; jg 2 E c g be the set of arcs induced by the edges in the closure graph. A similar construction, under the name of ''communication graph'', is introduced to the literature in Chen et al (2009). Throughout the text the notation ½G c will correspond to the operation of ''taking the closure'' of an input graph G ¼ ðV; EÞ.…”

“…Then, the problem is finding the minimum number of regenerators (and their locations) to facilitate transportation of the commodity between any two nodes. In telecommunications literature applications, this version of the hub covering problem is known as the Regenerator Location Problem (RLP) (Chen et al, 2009). In this study, we introduce two new dimensions to it: an arbitrary hub node fails.…”

“…Taking the geographical aspect of the RLP into account, Chen et al (2009) introduce it to the operations research literature, proving its NP-completeness and showing that it can be modeled as a special Steiner arborescence problem (SAP). Using this fact, they formulate a MIP model that can provide optimal solutions for relatively small instances of the problem.…”

“…Similar to this study, we formulate the RLP as a MIP and solve it with a branch and cut procedure. However, Chen et al (2009) first transform the RLP to a SAP and then apply branch and cut. This transformation necessitates duplicating the nodes in the original input graph and hence prolongs the solution times.…”

Optimizing busy time on parallel machines

“…Our study is also motivated by problems in optical network design (see, e.g., [8], [10], [11]). Optical wavelength-division multiplexing (WDM) is the leading technology that enables us to deal with the enormous growth of traffic in communication networks, like the Internet.…”