Abstract
We consider the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph Gā=ā(V,E) and k request pairs R 1, ā¦, R k , where each R i consists of a source-destination pair (s i ,t i ), a demand d i and a weight w i . The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(logn) approximation for UFP on trees when all weights are identical; this yields an O(log2 n) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results.
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
Andrews, M., Chuzhoy, J., Khanna, S., Zhang, L.: Hardness of the undirected edge-disjoint paths problem with congestion. In: Proc. of IEEE FOCS, pp. 226ā241 (2005)
Azar, Y., Regev, O.: Combinatorial algorithms for the unsplittable flow problem. AlgorithmicaĀ 441(1), 49ā66 (2006)
Bansal, N., Friggstad, Z., Khandekar, R., Salavatipour, M.R.: A logarithmic approximation for unsplittable flow on line graphs. In: Proc. of ACM-SIAM SODA, pp. 702ā709 (2009)
Bansal, N., Chakrabarti, A., Epstein, A., Schieber, B.: A Quasi-PTAS for unsplittable flow on line graphs. In: Proc. of ACM STOC, pp. 721ā729 (2006)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. JACMĀ 48(5), 1069ā1090 (2001)
Bar-Noy, A., Guha, S., Naor, J., Schieber, B.: Approximating the Throughput of Multiple Machines in Real-Time Scheduling. SICOMPĀ 31(2), 331ā352 (2001)
Baveja, A., Srinivasan, A.: Approximation algorithms for disjoint paths and related routing and packing problems. Math. Oper. Res.Ā 25(2), 255ā280 (2000)
Bienstock, D.: Approximate formulations for 0-1 knapsack sets. Oper. Res. Lett.Ā 36(3), 317ā320 (2008)
Calinescu, G., Chakrabarti, A., Karloff, H., Rabani, Y.: Improved approximation algorithms for resource allocation. In: Proc. of IPCO, pp. 439ā456 (2001)
Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proc. of ACM-SIAM SODA, pp. 106ā115 (2000)
Carr, R., Vempala, S.: Randomized meta-rounding. Random Structures and AlgorithmsĀ 20(3), 343ā352 (2002)
Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation Algorithms for the Unsplittable Flow Problem. AlgorithmicaĀ 47(1), 53ā78 (2007)
Chekuri, C., Khanna, S., Shepherd, F.B.: An \(O(\sqrt{n})\) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow. Theory of ComputingĀ 2, 137ā146 (2006)
Chekuri, C., Khanna, S., Shepherd, F.B.: Edge-Disjoint Paths in Planar Graphs with Constant Congestion. In: Proc. of ACM STOC, pp. 757ā766 (2006)
Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity Demand Flow in a Tree and Packing Integer Programs. ACM Trans. on AlgorithmsĀ 3(3) (2007)
Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K.: Hardness of Routing with Congestion in Directed Graphs. In: Proc. of ACM STOC, pp. 165ā178 (2007)
Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. AlgorithmicaĀ 18(1), 3ā20 (1997)
Guruswami, V., Khanna, S., Shepherd, B., Rajaraman, R., Yannakakis, M.: Near-Optimal Hardness Results and Approximation Algorithms for Edge-Disjoint Paths and Related Problems. J. of Computer and System SciencesĀ 67(3), 473ā496 (2003)
HĆ„stad, J.: Clique is Hard to Approximate within n 1Īµ. Acta MathematicaĀ 182, 105ā142 (1999)
Kleinberg, J.M.: Approximation algorithms for disjoint paths problems. PhD thesis, MIT EECS (1996)
Kolliopoulos, S.G.: Edge-disjoint Paths and Unsplittable Flow. In: Gonzalez, T.F. (ed.) Handbook of Approximation Algorithms and Metaheuristics. Chapman & Hall/CRC, Boca Raton (2007)
Kolliopoulos, S.G., Stein, C.: Approximating disjoint-path problems using greedy algorithms and Packing Integer Programs. Math. Prog. AĀ (99), 63ā87 (2004)
Kolman, P.: A Note on the Greedy Algorithm for the Unsplittable Flow Problem. Information Processing LettersĀ 88(3), 101ā105 (2003)
Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. J. of AlgorithmsĀ 61(1), 20ā44 (2006)
Pritchard, D.: Approximability of Sparse Integer Programs. In: Proc. of ESA (to appear, 2009); arXiv.org preprint, arXiv:0904.0859v1, http://arxiv.org/abs/0904.0859
Raghavan, P.: Probabilistic Construction of Deterministic Algorithms: Approximating Packing Integer Programs. JCSSĀ 37(2), 130ā143 (1988)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Shepherd, B., Vetta, A.: The Demand Matching Problem. Math. of Operations ResearchĀ 32(3), 563ā578 (2007)
Sherali, H., Adams, W.P.: A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-Integer Zero-One Programming Problems. Discrete Applied Mathematics and Combinatorial Operations Research and Computer ScienceĀ 52 (1994)
Srinivasan, A.: Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems. In: Proc. of IEEE FOCS, pp. 416ā425 (1997)
Srinivasan, A.: New Approaches to Covering and Packing Problems. In: Proc. of ACM-SIAM SODA, pp. 567ā576 (2001)
Van Vyve, M., Wolsey, L.A.: Approximate Extended Formulations. Math. Prog.Ā 105, 501ā522 (2006)
Varadarajan, K., Venkataraman, G.: Graph Decomposition and a Greedy Algorithm for Edge-disjoint Paths. In: Proc. of ACM-SIAM SODA, pp. 379ā380 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chekuri, C., Ene, A., Korula, N. (2009). Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-03685-9_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03684-2
Online ISBN: 978-3-642-03685-9
eBook Packages: Computer ScienceComputer Science (R0)