Abstract
We revisit a classical problem in transportation, known as the (bilevel) continuous network design problem, CNDP for short. 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 costs of the induced Wardrop equilibrium and the investment costs for installing the edge's capacities. While this problem is considered to be challenging in the literature, its complexity status was still unknown. We close this gap, showing that CNDP is strongly $\mathsf{NP}$-hard and $\mathsf{APX}$-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 [P. Marcotte, Math. Prog., 34 (1986), pp. 142--162]. We derive a closed form expression of its approximation guarantee for arbitrary sets of latency functions. We then propose a different approximation algorithm and show that it has the same approximation guarantee. Then, we prove that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we derive a closed form expression. For affine latencies, for example, this best-of-two approach achieves an approximation factor of $49/41\approx 1.195$, which improves on the factor of $5/4$ that has been shown before by Marcotte.
Highlights
The continuous network design problem (CNDP) introduced by Dafermos [7], Dantzig et al [9], and Abdulaal et al [1] is one of the most classical network design problems in transport
Given a graph in which the latency of each edge depends on the ratio of the edge flow and the capacity installed at that edge, 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 for installing the capacity
While all our hardness proofs rely on instances with an arbitrary number of commodities and respective sinks, we show that for instances in which all commodities share a common sink, CNDP can be solved to optimality in polynomial time
Summary
The continuous network design problem (CNDP) introduced by Dafermos [7], Dantzig et al [9], and Abdulaal et al [1] is one of the most classical network design problems in transport. An exception is the work of Marcotte [18] who considered several algorithms based on solutions of associated convex optimization problems which can be solved in polynomial time [11]. He derives worst-case bounds for his heuristics and, in particular, for affine latency functions he devises an approximation algorithm with an approximation factor of 5/4. We propose a new algorithm which we call ScaleUniformly This algorithm first computes an optimal solution of the relaxation (as before) and uniformly scales the capacities with a certain parameter λ(S) that depends on the class of allowable latency functions S. Some proofs missing in this extended abstract can be found in the full version
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