Heuristics, LPs, and Trees on Trees: Network Design Analyses

Abstract

We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a LP relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and LP relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, both the heuristic and the LP relaxation of the combined overlay optimization model have performance guarantees of 4/3. We extend this analysis to multiple overlays on the same base solution, producing the first known worst-case bounds (approximately proportional to the square root of the number of commodities) for the uncapacitated multicommodity network design problem. In a companion paper, we develop heuristic performance guarantees for various new multi-tier. survivable network design models that incorporate both multiple facility types or technologies and differential node connectivity levels. IntroductionThis paper considers a general class of models, which we call overlay optimization problems, that combines two sets of decisions: the choice of activity levels to provide a basic level of service to all customers, and decisions regarding which of these activities to enhance to meet more stringent service requirements for subsets of important customers. The activity levels might represent facility installation and sizing decisions, with the basic and enhanced activities representing two levels of technology that differ in speed, capacity. or functionality. We treat the installation of higher grade facilities as "overlaying" or upgrading the base facilities at extra cost. Overlay optimization has potential applications in logistics and infrastructure planning including the design of telecommunications, transportation, electric power, and pipeline networks. For instance, in transportation planning, customers correspond to cities and towns. Basic service represents providing access to every town via a paved road, but certain important cities must be interconnected by, say, all-weather highways. Likewise, in telecommunications planning, the base decisions model the installation of switching and transmission facilities that can accommodate basic voice and data services (e.g., DS 1 services), while overlay variables represent upgrading certain facilities to high capacity, broadband (e.g., fiber optic) system,, between selected locations.We analyze the worst-case performance of a generic heuristic strategy for solving o...

Composite Variable Formulations for Express Shipment Service Network Design

“…The DP is embedded within a Lagrangian relaxation scheme and the method is shown to provide good lower and upper bounds. Balakrishnan et al [7] present worst-case bounds for heuristics and LP relaxations of the overlay optimization problem and demonstrate worst-case bounds for the uncapacitated multicommodity network design problem. Balakrishnan et al [8] introduce a multi-tier survivable network design problem for which they derive a solution procedure that solves the single-tier subproblems as matroids.…”

Improved Approximation Algorithms for the Quality of Service Multicast Tree Problem

“…An interesting open question is to extend this analysis to the case of three non-zero rates. The best known approximation factor for this case is α(5 + 4 √ 2)/7 < 2.359 [2], [14].…”

“…Colbourn and Xue [6] presented an O(r 3 n) time algorithm for solving the problem on a series-parallel graph, where n is the number of nodes and r is the number of grades of service (distinct rates). Some results for the case of few rates were obtained by Balakrishnan et al in [1] and [2]. Specifically, in [2] (see also [14]) they suggested an algorithm for the case of two non-zero rates with an approximation ratio of 4 3 α < 2.066, where α < 1.550 is the best approximation ratio of an algorithm for the Steiner tree problem.…”

“…Some results for the case of few rates were obtained by Balakrishnan et al in [1] and [2]. Specifically, in [2] (see also [14]) they suggested an algorithm for the case of two non-zero rates with an approximation ratio of 4 3 α < 2.066, where α < 1.550 is the best approximation ratio of an algorithm for the Steiner tree problem. Recently, Charikar et al [5] gave the first constant-factor approximation algorithm for an unbounded number of rates.…”

“…Runtime and approximation ratios of previously known algorithms and of the algorithms given in this paper. * Steiner tree algorithm LCA [12] LCA + RNS [11] BR [15], [3] MST [ [2], [14] <2.066+ ε Improved ratio -1.960 + ε 2.237 2.414 * In the runtime, n and m denote the number of nodes and edges in the original graph G = (V, E), respectively. Approximation ratios associated with polynomial-time approximation schemes are accompanied by a +ε to indicate that they approach the quoted value from above and do not reach this value in polynomial time.…”