The Constrained Bottleneck Problem in Networks

Abstract

We consider problems on networks that are captured by two performance measures. One performance measure is any general cost function of a solution; the other is a bottleneck measure that describes the worst (maximum cost) component of the solution. The paper contains algorithms to solve three problems. In one problem, we minimize the bottleneck subject to a constraint on the generalized cost. In the second problem, we minimize the generalized cost subject to a constraint on the bottleneck. In the third problem, we consider the two criteria simultaneously and find all the Pareto optimum solutions. The major result is that the introduction of the bottleneck measure changes the complexity of the original (general cost) problem by a factor which is at most linear in the number of links.

An efficient heuristic algorithm for the bottleneck traveling salesman problem

“…We observe that a heuristic solution to CBSTP can be obtained by a simple modification of our BST-heuristic, taking insights from [23]. Again, note that the possible objective function values of CBTSP are the distinct edge costs of E and wez perform binary search over this set.…”

“…Subject to H ∈ F, ∑ e∈H W e ≤B CBSTP can be solved using the general purpose algorithm of Berman et al [23], designed for general constrained bottleneck problems. We observe that a heuristic solution to CBSTP can be obtained by a simple modification of our BST-heuristic, taking insights from [23].…”

Generalized multiple objective bottleneck problems

“…Polynomial algorithms considering a sum and a bottleneck objective are presented in [11][12][13][14]. More recently, shortest path problems involving two bottleneck and a sum objective were analyzed in [15] and [16].…”

The weighted sum of split and diameter clustering

“…In location theory, center and median are two measures that reflect two different interests. Various ways of combining two objectives have been studied (see Berman, Einav and Handler 1990;Hansen 1979; and for a comprehensive text on multiple criteria decision making, Zeleny, 1982).…”