A branch-and-cut algorithm for the Multiple Steiner TSP with Order constraints (original) (raw)
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The Multiple Steiner TSP with order constraints: complexity and optimization algorithms
Soft Computing
We consider a variant of the Travelling Salesman Problem (TSP), the Multiple Steiner TSP with Order constraints (MSTSPO). Consider a weighted undirected graph and a set of salesmen, and each salesman is associated with a set of compulsory vertices to visit, called terminals. The MSTSPO consists in finding a minimum-cost subgraph containing for each salesman a tour going in a specified order through its terminals. Along with its importance from a theoretical point of view, the problem is also challenging in practice since it has applications in telecommunication networks. We show that the problem is NP-hard even for a single salesman and propose integer programming formulations. We then devise both Branch-and-Cut and Branch-and-Price algorithms to solve the problem. The extensive computational results are presented, showing the efficiency of our algorithms. Keywords Steiner TSP • Order constraints • Integer linear programming • Branch-and-Price algorithm • Branch-and-Cut algorithm Communicated by V. Loia.
Telecommunication networks can be seen as the stacking of several layers like, for instance, IP-over-Optical networks. This infrastructure has to be sufficiently survivable to restore the traffic in the event of a failure. Moreover, it should have adequate capacities so that the demands can be routed between the origin-destinations. In this paper we consider the Multilayer Capacitated Survivable IP Network Design problem. We study two variants of this problem with simple and multiple capacities. We give two multicommodity flow formulations for each variant of this problem and describe some valid inequalities. In particular, we characterize valid inequalities obtained using Chvatal-Gomory procedure from the well known Cutset inequalities. We show that some of these inequalities are facet defining. We discuss separation routines for all the valid inequalities. Using these results, we develop a Branch-and-Cut algorithm and a Branch-and-Cut-and-Price algorithm for each variant and present extensive computational results.
A Cutting Plane Algorithm for Multicommodity Survivable Network Design Problems
INFORMS Journal on Computing, 1998
We present a cutting plane algorithm for solving the following telecommunications network design problem: given point-to-point traffic demands in a network, specified survivability requirements and a discrete cost/capacity function for each link, find minimum cost capacity expansions satisfying the given demands. This algorithm is based on the polyhedral study described in . In this paper we describe the underlying problem, the model and the main ingredients in our algorithm. This includes: initial formulation, feasibility test, separation for strong cutting planes and primal heuristics. Computational results for a set of real-world problems are reported.
A Branch-and-Cut algorithm for the Capacitated Multi-Failure Survivable Network Design problem
Computers & Industrial Engineering, 2018
Telecommunication networks can be seen as the stacking of several layers like, for instance, IP-over-Optical networks. This infrastructure should have sufficient capacities to route some demands between their origindestination nodes. In this paper we consider the Capacitated Multi-Failure Survivable Network Design problem. We study two variants of this problem with simple and multiple capacities. We give two multicommodity flow formulations for each variant of this problem and describe some valid inequalities. In particular, we characterize valid inequalities obtained using Chvatal-Gomory procedure from the well known Cutset inequalities. We show that some of these inequalities are facet defining. We discuss separation routines for all the valid inequalities. Using these results, we develop a Branch-and-Cut algorithm and a Branch-and-Cut-and-Price algorithm for each variant and present extensive computational results.
Single-Layer Cuts for Multi-Layer Network Design Problems
Operations Research/Computer Science Interfaces, 2008
We study a planning problem arising in SDH/WDM multi-layer telecommunication network design. The goal is to find a minimum cost installation of link and node hardware of both network layers such that traffic demands can be realized via grooming and a survivable routing. We present a mixed-integer programming formulation that takes many practical side constraints into account, including node hardware, several bitrates, and survivability against single physical node or link failures. This model is solved using a branch-and-cut approach with problem-specific preprocessing and cutting planes based on either of the two layers. On several realistic two-layer planning scenarios, we show that these cutting planes are still useful in the multi-layer context, helping to increase the dual bound and to reduce the optimality gaps.
Survivable network design using polyhedral approaches
2011
We consider the problem of designing a survivable telecommunication network using facilities of a fixed capacity. Given a graph G = (V,E), the traffic demand among the nodes, and the cost of installing facilities on the edges of G, we wish to design the minimum cost network, so that under any single edge failure, the network permits the flow of all traffic using the remaining capacity. The problem is modeled as a mixed integer program, which can be converted into a pure integer program by applying the well-known Japanese Theorem on multi-commodity flows. Using a key theorem that characterizes the facet inequalities of this integer program, we derive several families of 3-and 4-partition facets, which help to achieve extremely tight lower bounds on the problem. Using these bounds, problems of up to 20 nodes and 40 edges have been solved optimally in a pervious work. Using heuristic approaches based on this framework, we solve problems of up to 40 nodes and 80 edges to obtain solutions that are approximately within 5% of optimal solutions.
Strong Formulations for 2-Node-Connected Steiner Network Problems
Lecture Notes in Computer Science, 2008
We consider a survivable network design problem known as the 2-Node-Connected Steiner Network Problem (2NCON): we are given a weighted undirected graph with a node partition into two sets of customer nodes and one set of Steiner nodes. We ask for the minimum weight connected subgraph containing all customer nodes, in which the nodes of the second customer set are nodewise 2-connected. This problem class has received lively attention in the past, especially with regard to exact ILP formulations and their polyhedral properties. In this paper, we present a transformation of this problem into a related problem considering directed graphs and use this to establish two novel ILP formulations to solve 2NCON, based on multi-commodity flow and on directed cuts, respectively. We prove the advantages of our formulations and compare both approaches theoretically as well as experimentally. Thereby we solve instances with up to 1600 nodes to provable optimality.
Mathematical Programming, 2011
The hop-constrained minimum spanning tree problem (HMSTP) is an NP-hard problem arising in the design of centralized telecommunication networks with quality of service constraints. We show that the HMSTP is equivalent to a Steiner tree problem (STP) in an appropriate layered graph. We prove that the directed cut model for the STP defined in the layered graph, dominates the best previously known models for the HMSTP. We also show that the Steiner directed cuts in the extended layered graph space can be viewed as being a stronger version of some previously known HMSTP cuts in the original design space. Moreover, we show that these strengthened cuts can be combined and projected into new families of cuts, including facet defining ones, in the original design space. We also adapt the proposed approach to the diameter-constrained minimum spanning tree problem (DMSTP). Computational results with a branch-and-cut algorithm show that the proposed method is significantly better than previously known methods on both problems.
Strong Lower Bounds for a Survivable Network Design Problem
Electronic Notes in Discrete Mathematics, 2010
We consider a generalization of the Prize Collecting Steiner Tree Problem on a graph with special redundancy requirements on a subset of the customer nodes suitable to model a real world problem occurring in the extension of fiber optic communication networks. We strengthen an existing connection-based mixed integer programming formulation involving exponentially many variables using a recent result with respect
Directed Steiner problems with connectivity constraints
Discrete Applied Mathematics, 1993
We present a generalization of the Steiner problem in a directed graph. Given nonnegative weights on the arcs, the problem is to find a minimum weight subset F of the arc set such that the subgraph induced by F contains a given number of arc-disjoint directed paths from a certain root node to each given terminal node. Some applications of the problem are discussed and properties of associated polyhedra are studied. Results from a cutting plane algorithm are reported.