On the complexity of the highway problem

R.A. Sitters, K.M. Elbassioni, R Raman, S. Ray

Research output: Contribution to JournalArticleAcademicpeer-review

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. © 2012 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)70-77
JournalTheoretical Computer Science
Volume460
DOIs
Publication statusPublished - 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

Cite this

Sitters, R.A. ; Elbassioni, K.M. ; Raman, R ; Ray, S. / On the complexity of the highway problem. In: Theoretical Computer Science. 2012 ; Vol. 460. pp. 70-77.
@article{71a0a5c81a4042fe804acb0bfa22a527,
title = "On the complexity of the highway problem",
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 Rothvo{\ss} (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. {\circledC} 2012 Elsevier B.V. All rights reserved.",
author = "R.A. Sitters and K.M. Elbassioni and R Raman and S. Ray",
year = "2012",
doi = "10.1016/j.tcs.2012.07.028",
language = "English",
volume = "460",
pages = "70--77",
journal = "Theoretical Computer Science",
issn = "0304-3975",
publisher = "Elsevier",

}

On the complexity of the highway problem. / Sitters, R.A.; Elbassioni, K.M.; Raman, R; Ray, S.

In: Theoretical Computer Science, Vol. 460, 2012, p. 70-77.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - On the complexity of the highway problem

AU - Sitters, R.A.

AU - Elbassioni, K.M.

AU - Raman, R

AU - Ray, S.

PY - 2012

Y1 - 2012

N2 - 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. © 2012 Elsevier B.V. All rights reserved.

AB - 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. © 2012 Elsevier B.V. All rights reserved.

U2 - 10.1016/j.tcs.2012.07.028

DO - 10.1016/j.tcs.2012.07.028

M3 - Article

VL - 460

SP - 70

EP - 77

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -