Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

Abstract

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. The nodes of a multi-level network have varying levels of importance; more critical or higher level nodes require more expensive higher grade interconnections. Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the multilevel network design problem seeks a connected design that minimizes total cost while spanning all the nodes, and connecting nodes at each level via facilities of the corresponding or higher type. The TLND problem is a special case of multi-level network design with L = 2. This 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, and analyze the worst-case performance for a composite TLND heuristic based upon Steiner and spanning tree computations. When the ratio of higher to lower grade facility costs is the same for all edges, the worstcase performance ratio of the TLND heuristic is 4/3 if we can solve an embedded Steiner network problem optimally. For other cases, we express the TLND heuristic worst-case ratio in terms of the performance ratio of the Steiner solution method. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of the multi-level network design problem

Designing Hierarchical Survivable Networks

“…Therefore, Theorem 1 implies that the worst-case ratio of the composite heuristic for TLND problems with 10-proportional costs must not exceed 2. These worst-case bounds for the composite heuristic for HND and TLND problems are tight (Balakrishnan et al [1992a]). …”

“…Another recent stream of research in the network design literature attempts to provide reliable designs by using multiple interconnection technologies. Examples are the Hierarchical Network Design Problem (Current, Revelle, and Cohon [1986]) and the more general Multi-Level Network Design Problem (Balakrishnan, Magnanti, and Mirchandani [1992a]); this "serviceability" approach to network design provides higher grade (more reliable and more costly) service between certain "important" pairs of nodes, and lower grade service between other nodes. This approach does not incorporate multiple paths.…”

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

“…Since PB = 1, Theorem 1 implies that the worst-case ratio of the composite heuristic does not exceed 2. Balakrishnan et al [1992] have previously described worst-case examples for the HND and TLND problems showing that these bounds are tight. By using a more sophisticated heuristic for the overlay subproblem, e can improve the worst-case bound.…”

“…For the HND problem, the BU heuristic finds an MST of graph G using primary costs cj, and installs primary facilities on all edges of this tree. In our subsequent -5-analysis, we ignore the possibility of further reducing the cost of the BU heuristic solution by locally downgrading some primary facilities into secondary facilities whenever possible (for the HND problem, Balakrishnan et al [1992] have shown that local improvement does not reduce the heuristic's worst-case performance ratio).…”

“…Balakrishnan et al [1992] provide HND and TLND examples with unrelated costs to show that the worst-case bound (2.20) for the OC heuristic is tight. Let us now examine the LP gap.…”