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

## TL;DR

This paper studies the problem of placing the minimum number of tolls in atomic congestion games to achieve social optima as Nash equilibria, proving NP-hardness and providing efficient solutions for series-parallel graphs.

## Contribution

It introduces the minimum tollbooth problem for atomic games, proves its computational hardness, and offers a polynomial-time algorithm for series-parallel networks.

## Key findings

- The problem is NP-hard and W[2]-hard when parameterized by tolled edges.
- A polynomial-time algorithm exists for series-parallel graphs.
- Tolls can be optimally placed to align Nash equilibria with social optima in certain network classes.

## Abstract

This work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it as not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness when parameterizing the problem with the number of tolled edges. On the positive side, we present a polynomial time algorithm for networks 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.