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# Packing Cars into Narrow Roads: PTASs for Limited Supply Highway

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LIPIcs.ESA.2019.54.pdf
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## Cite As

Fabrizio Grandoni and Andreas Wiese. Packing Cars into Narrow Roads: PTASs for Limited Supply Highway. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ESA.2019.54

## Abstract

In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers’s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA'11, SICOMP'16]. In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway. As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Packing and covering problems
##### Keywords
• approximation algorithms
• pricing problems
• highway problem
• unsplittable flow on a path

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## References

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