Abstract
An instance of the tollbooth problem consists of an undirected network and a collection of single-minded customers, each of which is interested in purchasing a fixed path subject to an individual budget constraint. The objective is to assign a per-unit price to each edge in a way that maximizes the collective revenue obtained from all customers. The revenue generated by any customer is equal to the overall price of the edges in her desired path, when this cost falls within her budget; otherwise, that customer will not purchase any edge.
Our main result is a deterministic algorithm for the tollbooth problem on trees whose approximation ratio is O(logm / loglogm), where m denotes the number of edges in the underlying graph. This finding improves on the currently best performance guarantees for trees, due to Elbassioni et al. (SAGT ’09), as well as for paths (commonly known as the highway problem), due to Balcan and Blum (EC ’06). An additional interesting consequence is a computational separation between tollbooth pricing on trees and the original prototype problem of single-minded unlimited supply pricing, under a plausible hardness hypothesis due to Demaine et al. (SODA ’06).
Due to space limitations, some details are omitted from this extended abstract. All missing details are provided in the full version of this paper [11].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, New York (2000)
Balcan, M.F., Blum, A.: Approximation algorithms and online mechanisms for item pricing. Theory of Computing 3(1), 179–195 (2007)
Balcan, M.F., Blum, A., Mansour, Y.: Item pricing for revenue maximization. In: EC, pp. 50–59 (2008)
Briest, P., Krysta, P.: Single-minded unlimited supply pricing on sparse instances. In: SODA, pp. 1093–1102 (2006)
Cheung, M., Swamy, C.: Approximation algorithms for single-minded envy-free profit-maximization problems with limited supply. In: FOCS, pp. 35–44 (2008)
Demaine, E.D., Feige, U., Hajiaghayi, M., Salavatipour, M.R.: Combination can be hard: Approximability of the unique coverage problem. SIAM Journal of Computing 38(4), 1464–1483 (2008)
Elbassioni, K.M., Raman, R., Ray, S., Sitters, R.: On profit-maximizing pricing for the highway and tollbooth problems. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 275–286. Springer, Heidelberg (2009)
Elbassioni, K.M., Sitters, R., Zhang, Y.: A quasi-PTAS for profit-maximizing pricing on line graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 451–462. Springer, Heidelberg (2007)
Even, G., Garg, N., Könemann, J., Ravi, R., Sinha, A.: Covering graphs using trees and stars. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 24–35. Springer, Heidelberg (2003)
Frederickson, G.N., Johnson, D.B.: Generating and searching sets induced by networks. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 221–233. Springer, Heidelberg (1980)
Gamzu, I., Segev, D.: A sublogarithmic approximation for highway and tollbooth pricing, http://arxiv.org/abs/1002.2084
Gamzu, I., Segev, D.: An improved approximation algoritm for maximum graph orientation (2010) (in preparation)
Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On profit-maximizing envy-free pricing. In: SODA, pp. 1164–1173 (2005)
Hartline, J.D., Koltun, V.: Near-optimal pricing in near-linear time. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 422–431. Springer, Heidelberg (2005)
Hassin, R., Segev, D.: Robust subgraphs for trees and paths. ACM Transactions on Algorithms 2(2), 263–281 (2006)
Khandekar, R., Kimbrel, T., Makarychev, K., Sviridenko, M.: On hardness of pricing items for single-minded bidders. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 202–216. Springer, Heidelberg (2009)
Krauthgamer, R., Mehta, A., Rudra, A.: Pricing commodities, or how to sell when buyers have restricted valuations. In: Kaklamanis, C., Skutella, M. (eds.) WAOA 2007. LNCS, vol. 4927, pp. 1–14. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gamzu, I., Segev, D. (2010). A Sublogarithmic Approximation for Highway and Tollbooth Pricing. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_49
Download citation
DOI: https://doi.org/10.1007/978-3-642-14165-2_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14164-5
Online ISBN: 978-3-642-14165-2
eBook Packages: Computer ScienceComputer Science (R0)