New Complexity Results and Algorithms for the Minimum Tollbooth Problem

Abstract

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 (M IN T B) 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 M IN T B problem in an important class of graphs-series-parallel graphs. Our algorithm solves M IN T B by first tabulating the candidate solutions for subgraphs of the series-parallel network and then combining them optimally.

The Minimum Tollbooth Problem in Atomic Network Congestion Games with Unsplittable Flows

“…The algorithm gives tolls, such that a given state is a pure Nash equilibrium of the tolled game while using the minimum number of toll booths. This algorithm is based on a similar method presented by Basu, Lianeas, and Nikolova in [4], who consider non-atomic games, exploiting the recursive structure of series-parallel graphs.…”

“…This section presents an algorithm that optimally implements a state S in an atomic network congestion game based on a series-parallel graph in polynomial time. We base the procedure on a similar approach from Basu, Lianeas, and Nikolova, who show the same result in [4] for non-atomic games. Starting at the simple base case of parallel-link networks, we inductively decide on the number of tolled edges for larger components, based on the optimality of the smaller ones.…”

“…We also show that for a non-trivial graph class, it is possible to efficiently calculate a solution with the smallest possible number of toll booths. Based on the algorithm by Basu, Lianeas, and Nikolova from [4], we construct a polynomial time algorithm on games based on series-parallel graphs that turns any given state of the untolled game into a pure Nash equilibrium of the tolled game with the minimum number of tolled edges.…”

Negative Prices in Network Pricing Games

“…Hearn and Ramana [25] and later Harks et al [21] considered the problem of characterizing the set of all optimal tolls and used this polyhedron to propose secondary optimization problems. Recently, Basu et al [4] improved some of the complexity results for one of these problems called the minimum tollbooth problem.…”

The Minimum Tollbooth Problem in Atomic Network Congestion Games with Unsplittable Flows

“…Work on this topic is mainly restricted to non-atomic games, with several heuristics as a result (e.g., [1,2,3,21,28]). The NP-hardness of the non-atomic case with multiple commodities was shown by Bai, Hearn, and Lawphongpanich in [1] and later for the single commodity case by Basu, Lianeas, and Nikolova in [4]. On the positive side, Basu, Lianeas, and Nikolova give a polynomial time algorithm, if the network is a series-parallel graph [4].…”