Network Optimization Problems: Algorithms, Applications and Complexity

The connected facility location polytope

Chapter 1 Applications of network optimization

We present the first constant-factor approximation algorithms for the following problem. Given a metric space ( V , c ), a finite set D ⊆ V of terminals/customers with demands d : D → R + , a facility opening cost f ∈ R + and a capacity u ∈R + , find a partition D = D 1 ⊍…⊍ D k and Steiner trees T i for D i ( i = 1, …, k ) with c ( E ( T i )) + d ( D i ) ≤ u for i = 1,…, k such that Σ i = 1 k c ( E ( T i )) + kf is minimum. This problem arises in VLSI design. It generalizes the bin-packing problem and the Steiner tree problem. In contrast to other network design and facility location problems, it has the additional feature of upper bounds on the service cost that each facility can handle. Among other results, we obtain a 4.1-approximation in polynomial time, a 4.5-approximation in cubic time, and a 5-approximation as fast as computing a minimum spanning tree on ( D , c ).

We consider the problem of finding a shortest rectilinear Steiner tree for a given set of points in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if we intersect the Steiner tree with some obstacle, then no connected component of the induced subtree must be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles. We show that the length-restricted Steiner tree problem can be approximated with a performance guarantee of 2 in O(n logn) time, where n denotes the size of the associated Hanan grid. Optimal length-restricted Steiner trees can be characterized to have a special structure. In particular, we prove that a certain graph, which is a variant of the Hanan grid, always contains an optimal solution. Based on this structural result, we can improve the performance guarantee of approximation algorithms for the special case that all obstacles are of rectangular shape or of constant complexity, i.e. they are represented by at most a constant number of edges. For such a scenario, we give a $\frac{5}{4}\alpha$ -approximation and a $\frac{2k}{2k-1}\alpha$ -approximation for any integral k ≥ 4, where α denotes the performance guarantee for the ordinary Steiner tree problem in graphs.

In the past few decades, the architecture of optical networks has undergone significant evolution, from the earliest wavelength-division multiplexing (WDM) optical networks to elastic optical networks (EONs) and later to space-division multiplexing (SDM) EONs, to address the continuous growth of Internet traffic. By 2024, Pbps-level optical networks are expected, far exceeding the capacity limit of single-mode fibers. The massive SDM era is on the horizon. In this context, a newly designed architecture for optical networks called the spatial channel network (SCN) architecture, which achieves high cost efficiency by means of practical hierarchical optical cross-connects, has recently been proposed. However, the evolution of optical network architectures will simultaneously present challenges related to network optimization. For instance, with the evolution from WDM optical networks to EONs, the kernel network optimization problem was transformed from the routing and wavelength assignment (RWA) problem into the routing and spectrum assignment (RSA) problem due to the additionally introduced constraint of spectrum contiguity. Similarly, specially designed algorithms are also expected to be essential for addressing the network optimization problem in SCNs. In this paper, we define this new problem as the routing, spatial channel, and spectrum assignment (RSCSA) problem. We propose an integer linear programming (ILP) model and a heuristic algorithm to solve the RSCSA problem. We examine the performance of the proposed approaches via simulation experiments. The results show that both proposed approaches are effective in finding the optimal solutions or solutions close to the lower bounds. To the best of our knowledge, this is the first work to focus on the network optimization problem in SCNs.

This paper investigates the application of a minimum loss path finding algorithm to determine the maximum flow in generalized networks that are characterized by arc losses or gains. In these generalized network flow problems, each arc has not only a defined capacity but also a loss or gain factor, which must be taken into consideration when calculating the maximum achievable flow. This extension of the traditional maximum flow problem requires a more comprehensive approach, where the maximum amount of flow is determined by accounting for additional factors such as costs, varying arc capacities, and the specific loss or gain associated with each arc. This paper extends the classic Ford–Fulkerson algorithm, adapting it to iteratively identify source-to-sink (s − t) residual directed paths with minimum cumulative loss and generalized augmenting paths (GAPs), thus enabling the efficient computation of maximum flow in such complex networks. Moreover, to enhance the computational performance of the proposed algorithm, we conducted extensive studies on parallelization techniques using graphics processing units (GPUs). Significant improvements in the algorithm’s efficiency and scalability were achieved. The results demonstrate the potential of GPU-accelerated computations in handling real-world applications where generalized network flows with arc losses and gains are prevalent, such as in telecommunications, transportation, or logistics networks.

This paper, of which a preliminary version appeared in ISTCS'92, is concerned with generalized network flow problems. In a generalized network, each edgee = (u, v) has a positive ‘flow multiplier’a e associated with it. The interpretation is that if a flow ofx e enters the edge at nodeu, then a flow ofa e x e exits the edge atv. The uncapacitated generalized transshipment problem (UGT) is defined on a generalized network where demands and supplies (real numbers) are associated with the vertices and costs (real numbers) are associated with the edges. The goal is to find a flow such that the excess or deficit at each vertex equals the desired value of the supply or demand, and the sum over the edges of the product of the cost and the flow is minimized. Adler and Cosares [Operations Research 39 (1991) 955–960] reduced the restricted uncapacitated generalized transshipment problem, where only demand nodes are present, to a system of linear inequalities with two variables per inequality. The algorithms presented by the authors in [SIAM Journal on Computing, to appear result in a faster algorithm for restricted UGT. Generalized circulation is defined on a generalized network with demands at the nodes and capacity constraints on the edges (i.e., upper bounds on the amount of flow). The goal is to find a flow such that the excesses at the nodes are proportional to the demands and maximized. We present a new algorithm that solves the capacitated generalized flow problem by iteratively solving instances of UGT. The algorithm can be used to find an optimal flow or an approximation thereof. When used to find a constant factor approximation, the algorithm is not only more efficient than previous algorithms but also strongly polynomial. It is believed to be the first strongly polynomial approximation algorithm for generalized circulation. The existence of such an approximation algorithm is interesting since it is not known whether the exact problem has a strongly polynomial algorithm.

We consider the Minimum Elmore Delay Steiner Tree Problem, which is a key problem in VLSI design: We are given a set of pins which have to be connected by a Steiner tree. One of the pins is the source. Challenging timing constraints impose tight bounds on the delay of propagating a signal from the source to the other pins. The commonly used measure is Elmore delay (Elmore in J Appl Phys 19:55---63, 1948). We consider two variants: minimizing the maximum Elmore delay or a weighted sum of Elmore delays. Both variants are strongly NP-hard even for very restricted special cases. Although it is a central problem in VLSI design (Kahng and Robins in On optimal interconnections for VLSI. Kluwer, Boston, 1995; Korte and Vygen in Building bridges--between mathematics and computer science. Springer, Berlin, pp 333---368, 2008), no approximation algorithms were known so far. In this work, we give the first constant-factor approximation algorithm. It works for both variants. The algorithm achieves an approximation ratio of 3.39 in the rectilinear plane and 4.11 in general metric spaces. We can show that our algorithm is best possible in a certain sense. We also demonstrate that our algorithm leads to improvements on real world VLSI instances compared to the currently used standard method of computing short Steiner trees.

This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes—primary and secondary—and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4/3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem.

: Computational procedures for solving three general network flow problems are presented, together with proofs establishing their validity. Two of the problems are concerned with the determination of feasible flows (i.e., flows that lie between prescribed bounds in every arc of the network) whose costs are minimum, where arc costs are proportional to the magnitude of arc flow. The third problem involves the determination of whether or not a given feasible flow has minimum cost. The utility of the three procedures is illustrated in the context of a general transportation application. The material presented constitutes an adaptation and unified discussion of various known concepts and results in network flow theory.

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. We first show that the problem is NP-hard and cannot be approximated within unless P=NP, and provide a fast polynomial time deterministic approximation algorithm with performance ratio .

An overview of the key topics discussed at the 13th International Teletraffic Congress (ITC13) held in Copenhagen, Denmark, in June 1991, particularly the sessions on network design and planning, is given. Nineteen papers were presented at these sessions. Four issues that are of particular importance to ISDN design and planning are emphasized: design of networks with digital information transfer capabilities; design of networks with advanced network/traffic control techniques; use of GOS and reliability objectives for network design and planning; and design of intelligent networks. >

The optimization problems in communication networks have received the attention of many researchers in such related fields as network designer, network analysis, and network administration. The use of computer communication networks has been increasing rapidly in order to share expensive hardware/software resources and provide access to main systems from distant locations. These network problems have many applications in telecommunications, computer networking, and related domains in electric, gas, and sewer networks. In computer networking, LANs (local area networks) are commonly used as the communication infrastructure that meets the demands of users in the local environment. These networks typically consist of several LAN segments connected together via bridges. The use of these transparent bridges requires.loop-free paths between LAN segments. Therefore, only spanning tree topologies can be used as active LAN configurations. Recently, genetic algorithms have greatly advanced in related research fields such as network optimization problems, combinatorial optimization, multiobjective optimization, and so on. Genetic algorithms have also received a great deal of attention because of their ability as optimization techniques for many real-world problems. In this paper, we attempt to solve the LAN topology design problem with bicriteria which minimize the cost and average message delay using genetic algorithms, and propose a method of searching the Pareto solutions. We also employ the Prufer number in order to represent the chromosomes, because the interconnection between the network service centers must yield spanning tree configurations. Finally, we conduct experiments to certify the quality of the networks designs obtained by using genetic algorithms.

Designing and constructing networks under uncertainty in the construction stage: Definition and exact algorithmic approach

Given a set of nodes in the plane and a constant t ≥ 1, a Euclidean t-spanner is a network in which, for any pair of nodes, the ratio of the network distance and the Euclidean distance of the two nodes is at most t. These networks have applications in transportation or communication network design and have been studied extensively. In this paper we study 1-spanners under the Manhattan (or L1-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of nodes can be seen as a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of nodes. It is not known whether it is NP-hard to compute minimum Manhattan networks, i.e. Manhattan networks of minimum total length. In this paper we present a factor-3 approximation algorithm for this problem. Given a set of n nodes, our algorithm takes O(n log n) time and linear space.