Integer plane multiflow maximisation: one-quarter-approximation and gaps
Pages 403 - 419
Abstract
In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of value at least half the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing at most half the value thus providing a 1/4-approximation algorithm and integrality gap for maximum integer multiflows in the plane.
References
[1]
Appel K and Haken W A proof of the four color theorem Discret. Math. 1976 16 179-180
[2]
Baker BS Approximation algorithms for np-complete problems on planar graphs J. ACM 1994 41 1 153-180
[3]
Cheriyan J, Karloff H, Khandekar R, and Könemann J On the integrality ratio for tree augmentation Oper. Res. Lett. 2008 36 4 399-401
[4]
Edmonds J and Giles R A min-max relation for submodular functions in graphs Ann. Discret. Math. 1977 I 185-204
[5]
Fiorini S, Hardy N, Reed B, and Vetta A Approximate min-max relations for odd cycles in planar graphs Math. Program. 1995 110 1 71-91
[6]
Frank, A.: Connections in Combinatorial Optimization. Oxford University Press (2011)
[7]
Frank A and Szigeti Z A note on packing paths in planar graphs Math. Program. 1995 70 201-209
[8]
Garg, N., Kumar, N.: Dual half-integrality for uncrossable cut cover and its application to maximum half-integral flow. In: European Symposium on Algorithms (2020)
[9]
Garg, N., Kumar, N., Sebő, A.: Integer plane multiflow maximisation: flow-cut gap and one-quarter-approximation. In: International Conference on Integer Programming and Combinatorial Optimization, pp. 144–157. Springer (2020)
[10]
Garg N, Vazirani VV, and Yannakakis M Approximate max-flow min-(multi) cut theorems and their applications SIAM J. Comput. 1996 25 2 235-251
[11]
Garg N, Vazirani VV, and Yannakakis M Primal-dual approximation algorithms for integral flow and multicut in trees Algorithmica 1997 18 1 3-20
[12]
Hoffman, A.J., Kruskal J.B.: Integral boundary points of convex polyhedra. Linear Inequal. Relat. Syst., 223–246 (1956)
[13]
Huang, C.C., Mari, M., Mathieu, C., Schewior, K., Vygen, J.: An approximation algorithm for fully planar edge-disjoint paths. arXiv preprint arXiv:2001.01715 (2020)
[14]
Huang, C.C., Mari, M., Mathieu, C., Vygen, J.: Approximating maximum integral multiflows on bounded genus graphs. arXiv preprint arXiv:2005.00575 (2020)
[15]
Klein, P., Plotkin, S.A., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of the Twenty-fifth Annual ACM Symposium on Theory of Computing, pp. 682–690. ACM (1993)
[16]
Klein, P.N., Mathieu, C., Zhou, H.: Correlation clustering and two-edge-connected augmentation for planar graphs. In: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)
[17]
Korach E and Penn M Tight integral duality gap in the Chinese postman problem Math. Program. 1992 55 183-191
[18]
Korte B and Vygen J Combinatorial Optimization: Theory and Algorithms 2012 Berlin and Heidelberg, collection Algorithms and Combinatorics, fifth edition Springer
[19]
Král D and Voss H Edge-disjoint odd cycles in planar graphs J. Combinat. Theory Ser. B 2004 90 107-120
[20]
Matsumoto K, Nishizeki T, and Saito N Planar multicommodity fows, maximum matchings and negative cycles SIAM J. Comput. 1986 15 2 495-510
[21]
Middendorf M and Pfeiffer F On the complexity of the disjoint paths problem Combinatorica 1993 13 1 97-107
[22]
Okamura H and Seymour PD Multicommodity flows in planar graphs J. Combinat. Theory Ser. B 1981 31 1 75-81
[23]
Robertson, N., Sanders, D.P., Seymour, P., Thomas, R.: Efficiently four-coloring planar graphs. In: Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, pp. 571–575 (1996)
[24]
Schrijver, A.: Theory of Linear and Integer Programming. Wiley and Chichester (1986)
[25]
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer (2003)
[26]
Seguin-Charbonneau, L., Shepherd, F.B.: Maximum edge-disjoint paths in planar graphs with congestion 2. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 200–209. IEEE (2011)
[27]
Seymour PD On odd cuts and plane multicommodity flows Proc. Lond. Math. Soc. 1981 3 1 178-192
[28]
Tardos, É., Vazirani, V.V.: Improved bounds for the max-flow min-multicut ratio for planar and k\_r, r-free graphs. Inf. Process. Lett. 47(2), 77–80 (1993).
[29]
Tutte, W.T.: Lectures on matroids. J. Res. Nat. Bureau Stand. B(69) (1965)
[30]
Williamson DP, Goemans MX, Mihail M, and Vazirani VV A primal-dual approximation algorithm for generalized Steiner network problems Combinatorica 1995 15 3 435-454
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© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2021.
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Publication History
Published: 01 September 2022
Accepted: 29 July 2021
Received: 19 September 2020
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