New Complexity Results and Algorithms for the Minimum Tollbooth Problem

TL;DR

This paper investigates the computational complexity of the Minimum Tollbooth problem, proving NP-hardness for certain cases, and presents the first polynomial-time exact algorithm for series-parallel graphs.

Contribution

It establishes NP-hardness results for the MINTB problem and introduces the first polynomial-time exact solution for series-parallel graphs.

Findings

MINTB is NP-hard to approximate within 1.1377 for single commodity networks.

A new NP-hardness result for a variant allowing edge removal.

First polynomial-time exact algorithm for MINTB in series-parallel graphs.

Abstract

The inefficiency of the Wardrop equilibrium of nonatomic routing games can be eliminated by placing tolls on the edges of a network so that the socially optimal flow is induced as an equilibrium flow. A solution where the minimum number of edges are tolled may be preferable over others due to its ease of implementation in real networks. In this paper we consider the minimum tollbooth (MINTB) problem, which seeks social optimum inducing tolls with minimum support. We prove for single commodity networks with linear latencies that the problem is NP-hard to approximate within a factor of $1.1377$ through a reduction from the minimum vertex cover problem. Insights from network design motivate us to formulate a new variation of the problem where, in addition to placing tolls, it is allowed to remove unused edges by the social optimum. We prove that this new problem remains NP-hard even for single commodity networks with linear latencies, using a reduction from the partition problem. On the positive side, we give the first exact polynomial solution to the MINTB problem in an important class of graphs---series-parallel graphs. Our algorithm solves MINTB by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally.