LP Relaxation and Tree Packing for Minimum k-cuts
Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008].
In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008].
k-cut
LP relaxation
tree packing
7:1-7:18
Regular Paper
Chandra
Chekuri
Chandra Chekuri
Kent
Quanrud
Kent Quanrud
Chao
Xu
Chao Xu
10.4230/OASIcs.SOSA.2019.7
Ajit Agrawal, Philip Klein, and R Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 24(3):440-456, 1995.
Francisco Barahona. On the k-cut problem. Operations Research Letters, 26(3):99-105, 2000.
Chandra Chekuri, Sudipto Guha, and Joseph Naor. The Steiner k-cut problem. SIAM Journal on Discrete Mathematics, 20(1):261-271, 2006.
Chandra Chekuri and Kent Quanrud. Near-linear time approximation schemes for some implicit fractional packing problems. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 801-820. SIAM, 2017.
William H Cunningham. Optimal attack and reinforcement of a network. Journal of the ACM (JACM), 32(3):549-561, 1985.
Rodney G. Downey, Vladimir Estivill-Castro, Michael R. Fellows, Elena Prieto-Rodriguez, and Frances A. Rosamond. Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems. Electr. Notes Theor. Comput. Sci., 78:209-222, 2003.
Satoru Fujishige. Theory of principal partitions revisited. In Research Trends in Combinatorial Optimization, pages 127-162. Springer, 2009.
Harold N. Gabow and K. S. Manu. Packing algorithms for arborescences (and spanning trees) in capacitated graphs. Mathematical Programming, 82(1):83-109, June 1998.
Michel X Goemans and David P Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24(2):296-317, 1995.
O. Goldschmidt and D.S. Hochbaum. A polynomial algorithm for the k-cut problem for fixed k. Mathematics of Operations Research, pages 24-37, 1994.
Anupam Gupta, Euiwoong Lee, and Jason Li. Faster Exact and Approximate Algorithms for k-Cut. In Proceedings of IEEE FOCS, 2018.
Monika Henzinger, Satish Rao, and Di Wang. Local Flow Partitioning for Faster Edge Connectivity. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 1919-1938. SIAM, 2017.
David R Karger. Random Sampling in Graph Optimization Problems. PhD thesis, Stanford University, February 1995.
David R. Karger. Minimum Cuts in Near-linear Time. J. ACM, 47(1):46-76, January 2000. URL: http://dx.doi.org/10.1145/331605.331608.
http://dx.doi.org/10.1145/331605.331608
David R Karger and Clifford Stein. A new approach to the minimum cut problem. Journal of the ACM (JACM), 43(4):601-640, 1996.
Ken-ichi Kawarabayashi and Mikkel Thorup. Deterministic Global Minimum Cut of a Simple Graph in Near-Linear Time. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, pages 665-674. ACM, 2015.
Vladimir Kolmogorov. A faster algorithm for computing the principal sequence of partitions of a graph. Algorithmica, 56(4):394-412, 2010.
Pasin Manurangsi. Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis. In Proc. of ICALP, volume 80 of LIPIcs, pages 79:1-79:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017.
Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(1-6):583-596, 1992. URL: http://dx.doi.org/10.1007/BF01758778.
http://dx.doi.org/10.1007/BF01758778
Hiroshi Nagamochi and Yoko Kamidoi. Minimum cost subpartitions in graphs. Information Processing Letters, 102(2):79-84, 2007. URL: http://dx.doi.org/10.1016/j.ipl.2006.11.011.
http://dx.doi.org/10.1016/j.ipl.2006.11.011
J Naor and Yuval Rabani. Tree Packing and Approximating k-Cuts. In Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, volume 103, page 26. SIAM, 2001.
H. Narayanan. The principal lattice of partitions of a submodular function. Linear Algebra and its Applications, 144:179-216, 1991.
Kent Quanrud. Fast and Deterministic Approximations for k-Cut. CoRR, abs/1807.07143, 2018. URL: http://arxiv.org/abs/1807.07143.
http://arxiv.org/abs/1807.07143
R Ravi and Amitabh Sinha. Approximating k-cuts using network strength as a lagrangean relaxation. European Journal of Operational Research, 186(1):77-90, 2008.
Huzur Saran and Vijay V. Vazirani. Finding k-Cuts Within Twice the Optimal. SIAM J. Comput., 24(1):101-108, February 1995.
Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science &Business Media, 2003.
Mechthild Stoer and Frank Wagner. A simple min-cut algorithm. Journal of the ACM, 44(4):585-591, July 1997. URL: http://dx.doi.org/10.1145/263867.263872.
http://dx.doi.org/10.1145/263867.263872
Mikkel Thorup. Fully-dynamic min-cut. Combinatorica, 27(1):91-127, 2007.
Mikkel Thorup. Minimum k-way cuts via deterministic greedy tree packing. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pages 159-166. ACM, 2008.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode