An Exact Algorithm for Robust Network Design

Abstract

Abstract. Modern life heavily relies on communication networks that operate efficiently. A crucial issue for the design of communication networks is robustness with respect to traffic fluctuations, since they often lead to congestion and traffic bottlenecks. In this paper, we address an NP-hard single commodity robust network design problem, where the traffic demands change over time. For k different times of the day, we are given for each node the amount of single-commodity flow it wants to send or to receive. The task is to determine the minimum-cost edge capacities such that the flow can be routed integrally through the net at all times. We present an exact branch-and-cut algorithm, based on a decomposition into biconnected network components, a clever primal heuristic for generating feasible solutions from the linear-programming relaxation, and a general cutting-plane separation routine that is based on projection and lifting. By presenting extensive experimental results on realistic instances from the literature, we show that a suitable combination of these algorithmic components can solve most of these instances to optimality. Furthermore, cutting-plane separation considerably improves the algorithmic performance.

Single-commodity robust network design problem: Complexity, instances and heuristic solutions

“…According to the results that we present in the following subsections, we are able to get a better understanding of empirical results in [9,2], and we propose a family of randomly generated instances that are challenging for the natural flow formulation already for |V | = 50 and K = 10.…”

“…The problem was first studied by Buchheim, Liers and Sanità [9]. They gave an exact branch-and-cut algorithm that solves a flowmodel MIP through sophisticated general-purpose cutting planes.…”

“…Motivated by the computational investigations in [9,2], in the following, we analyze some special cases:…”

“…Contribution of the paper. Preliminary computational investigations have been performed on classical graphs from the literature with random balances [9] and on special hypercubes with {−1, 0, 1} balances [2]. The results in both papers have shown that the former instances are surprisingly easy for a general-purpose Mixed-Integer Programming (MIP) solver on the natural flow-formulation (1)- (5), while the latter instances are structurally difficult.…”

“…Integral flows are enforced through integrality of the capacity variables, as all balances are integral [17]. As described in [9], an example of a practical application of the considered problem is the following: some clients wish to download some program stored on several servers. For a client, it is not important which server he or she is downloading from, as long as the demand is satisfied.…”