Abstract
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfactory approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is unavoidable. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a general reduction which identifies some sufficient conditions on the considered graph class in order to prove the hardness of the problem. We focus on two combinatorially simple but very different classes, namely trees and solid grid graphs. The latter are finite connected subgraphs of the infinite two-dimensional grid without holes. We apply the reduction to show that for solid grid graphs it is NP-hard to approximate the optimum number of cut edges within any satisfactory ratio. We also consider solutions in which the sets may deviate from being equal-sized. Our reduction is applied to grids and trees to prove that no fully polynomial time algorithm exists that computes solutions in which the sets are arbitrarily close to equal-sized. This is true even if the number of edges cut is allowed to increase when the limit on the set sizes decreases. These are the first bicriteria inapproximability results for the k-BALANCED PARTITIONING problem.
The author is supported by the Swiss National Science Foundation under grant 200021_125201/1.
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References
Andreev, K., Räcke, H.: Balanced graph partitioning. Theory of Computing Systems 39(6), 929–939 (2006)
Arbenz, P., van Lenthe, G., Mennel, U., Müller, R., Sala, M.: Multi-level μ-finite element analysis for human bone structures. In: Proceedings of the 8th Workshop on State-of-the-art in Scientific and Parallel Computing (PARA), pp. 240–250 (2007)
Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. In: Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC), pp. 222–231 (2004)
Bhatt, S., Leighton, F.T.: A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences 28(2), 300–343 (1984)
Chevalier, C., Pellegrini, F.: PT-Scotch: A tool for efficient parallel graph ordering. Parallel Computing 34(68), 318–331 (2008)
Delling, D., Goldberg, A., Pajor, T., Werneck, R.: Customizable route planning. Experimental Algorithms, 376–387 (2011)
Díaz, J., Serna, M.J., Torán, J.: Parallel approximation schemes for problems on planar graphs. Acta Informatica 33(4), 387–408 (1996)
Diestel, R., Jensen, T.R., Gorbunov, K.Y., Thomassen, C.: Highly connected sets and the excluded grid theorem. Journal of Combinatorial Theory, Series B 75(1), 61–73 (1999)
Diks, K., Djidjev, H.N., Sykora, O., Vrto, I.: Edge separators of planar and outerplanar graphs with applications. Journal of Algorithms 14(2), 258–279 (1993)
Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, USA (2005)
Even, G., Naor, J., Rao, S., Schieber, B.: Fast approximate graph partitioning algorithms. SIAM Journal on Computing 28(6), 2187–2214 (1999)
Feldmann, A.E.: Balanced Partitioning of Grids and Related Graphs: A Theoretical Study of Data Distribution in Parallel Finite Element Model Simulations. PhD thesis, ETH Zurich, Diss.-Nr. ETH: 20371 (April 2012)
Feldmann, A.E., Das, S., Widmayer, P.: Restricted cuts for bisections in solid grids: A proof via polygons. In: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science (WG), pp. 143–154 (2011)
Feldmann, A.E., Foschini, L.: Balanced partitions of trees and applications. In: 29th International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 100–111 (2012)
Feldmann, A.E., Widmayer, P.: An O(n 4) time algorithm to compute the bisection width of solid grid graphs. In: Proceedings of the 19th Annual European Symposium on Algorithms (ESA), pp. 143–154 (2011)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co. (1979)
Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoretical Computer Science 1(3), 237–267 (1976)
Karypis, G., Kumar, V.: METIS-unstructured graph partitioning and sparse matrix ordering system, version 2.0. Technical report, University of Minnesota (1995)
Khot, S.A., Vishnoi, N.K.: The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 53–62 (2005)
Klein, P., Plotkin, S., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pp. 682–690 (1993)
Krauthgamer, R., Naor, J., Schwartz, R.: Partitioning graphs into balanced components. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 942–949 (2009)
Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)
Lipton, R., Tarjan, R.: Applications of a planar separator theorem. SIAM Journal on Computing 9, 615–627 (1980)
MacGregor, R.M.: On Partitioning a Graph: a Theoretical and Empirical Study. PhD thesis, University of California, Berkeley (1978)
Papadimitriou, C., Sideri, M.: The bisection width of grid graphs. Theory of Computing Systems 29, 97–110 (1996)
Park, J.K., Phillips, C.A.: Finding minimum-quotient cuts in planar graphs. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (STOC), pp. 766–775 (1993)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC) (2008)
Simon, H.D., Teng, S.H.: How good is recursive bisection? SIAM Journal on Scientific Computing 18(5), 1436–1445 (1997)
Wu, Z., Leahy, R.: An optimal graph theoretic approach to data clustering: Theory and its application to image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(11), 1101–1113 (1993)
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Feldmann, A.E. (2012). Fast Balanced Partitioning Is Hard Even on Grids and Trees. In: Rovan, B., Sassone, V., Widmayer, P. (eds) Mathematical Foundations of Computer Science 2012. MFCS 2012. Lecture Notes in Computer Science, vol 7464. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32589-2_34
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