Complexity and Approximation of the Continuous Network Design Problem

We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. We are given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed. The goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly NP-complete and APX-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a closed form expression of its approximation guarantee for arbitrary sets S of allowed latency functions. Second, we propose a different approximation algorithm and show that it has the same approximation guarantee. As our final -- and arguably most interesting -- result regarding approximation, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we give a closed form expression. For affine latencies, e.g., this algorithm achieves a 1.195-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment.