A Factor 2 Approximation Algorithm for the Generalized Steiner Network Problem

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. Received March 6, 1998     

A primal-dual approximation algorithm for generalized steiner network problems

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.Research supported by an NSF Graduate Fellowship, DARPA contracts N00014-91-J-1698 and N00014-92-J-1799, and AT&T Bell Laboratories.Research supported in part by Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.Part of this work was done while the author was visiting AT&T Bell Laboratories and Bellcore.     

On the minimum number of edges of two-connected graphs with given diameter

We answer the following question: what is the minimum number of edges of a 2-connected graph with a given diameter? This problem stems from survivable telecommunication network design with grade-of-service constraints. In this paper, we prove tight bounds for 2-connected graphs and for 2-edge-connected graphs. The bound for 2-connected graphs was a conjecture of B. Bollobás (AMH 75–80) [3].     

Two-level evolutionary approach to the survivable mesh-based transport network topological design

The complete topology design problem of survivable mesh-based transport networks is to address simultaneously design of network topology, working path routing, and spare capacity allocation based on span-restoration. Each constituent problem in the complete design problem could be formulated as an Integer Programming (IP) and is proved to be NP\mathcal{NP} -hard. Due to a large amount of decision variables and constraints involved in the IP formulation, to solve the problem directly by exact algorithms (e.g. branch-and-bound) would be impractical if not impossible. In this paper, we present a two-level evolutionary approach to address the complete topology design problem. In the low-level, two parameterized greedy heuristics are developed to jointly construct feasible solutions (i.e., closed graph topologies satisfying all the mesh-based network survivable constraints) of the complete problem. Unlike existing “zoom-in”-based heuristics in which subsets of the constraints are considered, the proposed heuristics take all constraints into account. An estimation of distribution algorithm works on the top of the heuristics to tune the control parameters. As a result, optimal solution to the considered problem is more likely to be constructed from the heuristics with the optimal control parameters. The proposed algorithm is evaluated experimentally in comparison with the latest heuristics based on the IP software CPLEX, and the “zoom-in”-based approach on 28 test networks problems. The experimental results demonstrate that the proposed algorithm is more effective in finding high-quality topologies than the IP-based heuristic algorithm in 21 out of 28 test instances with much less computational costs, and performs significantly better than the “zoom-in”-based approach in 19 instances with the same computational costs.     

A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

     

Design of survivable IP-over-optical networks

In the past years, telecommunications networks have seen an important evolution with the advances in optical technologies and the explosive growth of the Internet. Several optical systems allow a very large transport capacity, and data traffic has dramatically increased. Telecommunications networks are now moving towards a model of high-speed routers interconnected by intelligent optical core networks. Moreover, there is a general consensus that the control plan of the optical networks should utilize IP-based protocols for dynamic provisioning and restoration of lightpaths. The interaction of the IP routers with the optical core networks permits to achieve end-to-end connections, and the lightpaths of the optical networks define the topology of the IP network. This new infrastructure has to be sufficiently survivable, so that network services can be restored in the event of a catastrophic failure. In this paper we consider a multilayer survivable network design problem that may be of practical interest for IP-over-optical neworks. We give an integer programming formulation for this problem and discuss the associated polytope. We describe some valid inequalities and study when these are facet defining. We discuss separation algorithms for these inequalities and introduce some reduction operations. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.     

The convex hull of two core capacitated network design problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost.This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.Research of this author was supported in part by a Faculty Grant from the Katz Graduate School of Business, University of Pittsburgh.     

On routing in VLSI design and communication networks

In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. First we propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APX-hard. We present the integer linear programming (LP) formulation of both problems and solve the LP relaxations by the fast approximation algorithms for min-max resource-sharing problems in [K. Jansen, H. Zhang, Approximation algorithms for general packing problems and their application to the multicast congestion problem, Math. Programming, to appear, doi:10.1007/s10107-007-0106-8] (which is a generalization of the approximation algorithm proposed by Grigoriadis and Khachiyan [Coordination complexity of parallel price-directive decomposition, Math. Oper. Res. 2 (1996) 321-340]). For the global routing problem, we investigate the particular property of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally, we develop asymptotic approximation algorithms for both problems with ratios depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.     

Algorithms for the design of network topologies with balanced disjoint rings

We define the Balanced Disjoint Rings (BDR) problem as developing a method to partition of the nodes in a network to form a given number of disjoint rings of minimum total link length in such a way that there is almost the same number of nodes in each ring. The BDR problem has potential applications in the design of survivable network structures in telecommunications as well as in the identification of local distribution/collection routes in logistics. The BDR problem can also be considered a generalization of the traveling salesman problem since we are interested in multiple tours instead of a single tour. We develop an efficient heuristic solution methodology that involves various GRASP-based randomized solution construction routines that allow a multi-start framework and a n effective combination of cyclic-exchange and single-move neighborhoods in a local search improvement procedure. The algorithms perform very well in our numerical studies, providing encouraging optimality and lower bound gaps with very reasonable runtimes.     

Creating Advanced Bases For Large Scale Linear Programs Exploiting Embedded Network Structure

In this paper, we investigate how an embedded pure network structure arising in many linear programming (LP) problems can be exploited to create improved sparse simplex solution algorithms. The original coefficient matrix is partitioned into network and non-network parts. For this partitioning, a decomposition technique can be applied. The embedded network flow problem can be solved to optimality using a fast network flow algorithm. We investigate two alternative decompositions namely, Lagrangean and Benders. In the Lagrangean approach, the optimal solution of a network flow problem and in Benders the combined solution of the master and the subproblem are used to compute good (near optimal and near feasible) solutions for a given LP problem. In both cases, we terminate the decomposition algorithms after a preset number of passes and active variables identified by this procedure are then used to create an advanced basis for the original LP problem. We present comparisons with unit basis and a well established crash procedure. We find that the computational results of applying these techniques to a selection of Netlib models are promising enough to encourage further research in this area.     

Stochastic Survivable Network Design Problems

In this paper we introduce survivable network design problems under a two-stage stochastic model with fixed recourse and finitely many scenarios. We propose a new cut-based formulation based on orientation properties which is stronger than the undirected cut-based model. We use a two-stage branch&cut algorithm for solving the decomposed model to provable optimality. In order to accelerate the computations, we suggest a new cut strengthening technique for the decomposed L-shaped optimality cuts that is computationally fast and easy to implement.     

Survivable network design with shared-protection routing

In this paper we study the problem of designing a survivable telecommunication network with shared-protection routing. We develop a heuristic algorithm to solve this problem. Recent results in the area of global re-routing have been used to obtain very tight lower bounds for the problem. Our results indicate that in a majority of problem instances, the average gap between the heuristic solutions and the lower bounds is within 5%. Computational experience is reported on randomly generated problem instances with up to 35 nodes, 80 edges and 595 demand pairs and also on the instances available in SNDlib database.     

LP-Based Heuristic Algorithms for Interconnecting Token Rings via Source Routing Bridges

We develop a method to determine the topology of a network that interconnects a number of token rings using source routing bridges. The purpose is to compute a topology that provides low response delays for network users at a minimal cost of bridge installations. We formulate this network design problem as a mixed binary integer linear program. We develop effective heuristic algorithms. The algorithms exploit the topology and routing solutions of the linear programming relaxation in a sophisticated manner which we believe is new in the literature. The model incorporates performance issues, such as network stability, bridge overflow, back pressure effect and broadcast storm, that are specific to the underlying communication technology. By formally incorporating these performance issues, we tighten the model formulation and improve the quality of the LP bound considerably. Computational results are reported for problems with up to 20 token rings and 190 potential bridge locations.     

Solving singularly constrained generalized network problems

The singularly constrained generalized network problem represents a large class of capacitated linear programming (LP) problems. This class includes any LP problem whose coefficient matrix, ignoring single upper bound constraints, containsm + 1 rows which may be ordered such that each column has at most two non-zero entries in the firstm rows. The paper describes efficient procedures for solving such problems and presents computational results which indicate that, on large problems, these procedures are at least twenty-five times more efficient than the state of the art LP systemapex-iii.This research was partly supported by ONR Contract N00014-76-C-0383 with Decision Analysis and Research Institute and by Project NR047-021, ONR Contracts N00014-75-C-0616 and N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

A branch-and-cut algorithm for the Multiple Steiner TSP with Order constraints

The paper deals with a problem motivated by survivability issues in multilayer IP-over-WDM telecommunication networks. Given a set of traffic demands for which we know a survivable routing in the IP layer, our purpose is to look for the corresponding survivable topology in the WDM layer. The problem amounts to Multiple Steiner TSPs with order constraints. We propose an integer linear programming formulation for the problem and investigate the associated polytope. We also present new valid inequalities and discuss their facial aspect. Based on this, we devise a Branch-and-cut algorithm and present preliminary computational results.     

Survivable network design with demand uncertainty

The objective in designing a communications network is to find the most cost efficient network design that specifies hardware devices to be installed, the type of transmission links to be installed, and the routing strategy to be followed. In this paper algorithmic ideas are presented for improving tractability in solving the survivable network design problem by taking into account uncertainty in the traffic requirements. Strategies for improving separation of metric inequalities are presented and an iterative approach for obtaining solutions, that significantly reduces computing times, is introduced. Computational results are provided based on data collected from an operational network.     

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.