On the complexity of the highway problem

Khaled Elbassioni, Rajiv Raman, Saurabh Ray, René Sitters

Research output: Contribution to journalArticle

Abstract

In the highway problem, we are given a path, and a set of buyers interested in buying sub-paths of this path; each buyer declares a non-negative budget, which is the maximum amount of money she is willing to pay for that sub-path. The problem is to assign non-negative prices to the edges of the path such that we maximize the profit obtained by selling the edges to the buyers who can afford to buy their sub-paths, where a buyer can afford to buy her sub-path if the sum of prices in the sub-path is at most her budget. In this paper, we show that the highway problem is strongly NP-hard; this settles the complexity of the problem in view of the existence of a polynomial-time approximation scheme, as was recently shown in Grandoni and Rothvoß (2011) [15]. We also consider the coupon model, where we allow some items to be priced below zero to improve the overall profit. We show that allowing negative prices makes the problem APX-hard. As a corollary, we show that the bipartite vertex pricing problem is APX-hard with budgets in 1,2,3, both in the cases with negative and non-negative prices.

Original languageEnglish (US)
Pages (from-to)70-77
Number of pages8
JournalTheoretical Computer Science
Volume460
DOIs
StatePublished - Nov 16 2012

Fingerprint

Profitability
Path
Sales
Polynomials
Non-negative
Profit
Costs
Polynomial Time Approximation Scheme
Pricing
Assign
Corollary
NP-complete problem
Maximise
Zero
Vertex of a graph

Keywords

  • Approximation algorithms
  • Complexity
  • Hardness of approximation
  • Interval graphs
  • NP-hardness
  • Pricing

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

On the complexity of the highway problem. / Elbassioni, Khaled; Raman, Rajiv; Ray, Saurabh; Sitters, René.

In: Theoretical Computer Science, Vol. 460, 16.11.2012, p. 70-77.

Research output: Contribution to journalArticle

Elbassioni, Khaled ; Raman, Rajiv ; Ray, Saurabh ; Sitters, René. / On the complexity of the highway problem. In: Theoretical Computer Science. 2012 ; Vol. 460. pp. 70-77.
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