Abstract
Given a tree T on n vertices, and \(k, b, s_1, \ldots , s_b \in \mathbb {N}\), the Tree Partitioning problem asks if at most k edges can be removed from T so that the resulting components can be grouped into b groups such that the number of vertices in group i is \(s_i\), for \(i =1, \ldots , b\). The case where \(s_1=\cdots =s_b =n/b\), referred to as the Balanced Tree Partitioning problem, was shown to be \({\mathcal {NP}}\)-complete for trees of maximum degree at most 5, and the complexity of the problem for trees of maximum degree 4 and 3 was posed as an open question. The parameterized complexity of Balanced Tree Partitioning was also posed as an open question in another work. In this paper, we answer both open questions negatively. We show that Balanced Tree Partitioning (and hence, Tree Partitioning) is \({\mathcal {NP}}\)-complete for trees of maximum degree 3, thus closing the door on the complexity of Balanced Tree Partitioning, as the simple case when T is a path is in \({\mathcal {P}}\). In terms of the parameterized complexity of the problems, we show that both Balanced Tree Partitioning and Tree Partitioning are W[1]-complete parameterized by k. Using a compact representation of the solution space for an instance of the problem, we present a dynamic programming algorithm for the weighted version of Tree Partitioning (and hence for that of Balanced Tree Partitioning) that runs in subexponential-time \(2^{O(\sqrt{n})}\), adding a natural problem to the list of problems that can be solved in subexponential time. Finally, we extend this subexponential-time algorithm to the Weighted Graph Partitioning problem on graphs of treewidth \(o(n/\lg {n})\), and we also show an application of this subexponential-time algorithm for approximating the Weighted Graph Partitioning problem.
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Notes
In a variant of the Balanced Tree Partitioning problem, the group sizes in the solution are allowed to differ by 1. All the results in this paper still hold for this variant of the problem.
We assume that all weights can be efficiently encoded.
References
Andreev, K., Räcke, H.: Balanced graph partitioning. Theory Comput. Syst. 39(6), 929–939 (2006)
Arbenz, P., van Lenthe, G., Mennel, U., Müller, R., Sala, M.: Multi-level \(\mu\)-finite element analysis for human bone structures. In: Proceedings of the 8th International Workshop on Applied Parallel Computing, Volume 4699 of Lecture Notes in Computer Science, pp. 240–250. Springer, Berlin (2007)
Bhatt, S., Leighton, F.: A framework for solving VLSI graph layout problems. J. Comput. Syst. Sci. 28(2), 300–343 (1984)
Bodlaender, H., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)
Bodlaender, H., Drange, P., Dregi, M., Fomin, F., Lokshtanov, D., Pilipczuk, M.: A \(c^k n\) 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)
Boscznay, Á.: On the lower estimation of non-averaging sets. Acta Math. Hung. 53(1–1), 155–157 (1989)
Chen, J., Kanj, I., Perkovic, L., Sedgwick, E., Xia, G.: Genus characterizes the complexity of certain graph problems: some tight results. J. Comput. Syst. Sci. 73(6), 892–907 (2007)
Chen, Y., Flum, J., Grohe, M.: Machine-based methods in parameterized complexity theory. Theor. Comput. Sci. 339(2–3), 167–199 (2005)
Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Cygan, M., Fomin, F., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms, 1st edn. Springer, Berlin (2015)
Delling, D., Goldberg, A., Pajor, T., Werneck, R.: Customizable route planning. In: Proceedings of the 10th International Symposium on Experimental Algorithms, Volume 6630 of Lecture Notes in Computer Science, pp. 376–387. Springer, Berlin (2011)
Demaine, E., Fomin, F., Hajiaghayi, M., Thilikos, D.: Subexponential parameterized algorithms on bounded-genus graphs and \(H\)-minor-free graphs. J. ACM 52, 866–893 (2005)
Downey, R., Fellows, M.: Fundamentals of Parameterized Complexity. Springer, New York (2013)
Feldmann, A.: Balanced partitions of grids and related graphs. Ph.D. thesis, ETH, Zurich, Switzerland (2012)
Feldmann, A., Foschini, L.: Balanced partitions of trees and applications. Algorithmica 71(2), 354–376 (2015)
Feldmann, A., Widmayer, P.: An \(O(n^4)\) time algorithm to compute the bisection width of solid grid graphs. Algorithmica 71(1), 181–200 (2015)
Fellows, M., Hermelin, D., Rosamond, F., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2010)
Fomin, F., Kolay, S., Lokshtanov, D., Panolan, F., Saurabh, S.: Subexponential algorithms for rectilinear Steiner tree and arborescence problems. In: Proceedings of the 32nd International Symposium on Computational Geometry, Volume 51 of LIPIcs, pp. 39:1–39:15 (2016)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)
Hardy, G., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 17(2), 75–115 (1918)
Jansen, K., Kratsch, S., Marx, D., Schlotter, I.: Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci. 79(1), 39–49 (2013)
Jones, C.: Generalized hockey stick identities and \(N\)-dimensional blockwalking. Fibonacci Q. 34(3), 280–288 (1996)
Klein, P., Marx, D.: A subexponential parameterized algorithm for subset TSP on planar graphs. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1812–1830 (2014)
Kloks, T.: Treewidth, Computations and Approximations. Vol. 842 of Lecture Notes in Computer Science. Springer, Berlin (1994)
MacGregor, R.: On partitioning a graph: a theoretical and empirical study. Ph.D. thesis, University of California at Berkeley, California, USA (1978)
Madry, A.: Fast approximation algorithms for cut-based problems in undirected graphs. In: Proceedings of the 51st Annual Symposium on Foundations of Computer Science, pp. 245–254 (2010)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci. 67(4), 757–771 (2003)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 255–264 (2008)
Räcke, H., Stotz, R.: Improved approximation algorithms for balanced partitioning problems. In: Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science, Volume 47 of LIPIcs, pp. 58:1–58:14 (2016)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)
van Bevern, R., Feldmann, A., Sorge, M., Suchý, O.: On the parameterized complexity of computing balanced partitions in graphs. Theory Comput. Syst. 57(1), 1–35 (2015)
Acknowledgements
Q. Feng: Supported by the National Natural Science Foundation of China under Grants 61872450, 61672536, 61828205, and 71631008.
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A preliminary version of the paper appeared in Proceedings of the 15th International Symposium on Algorithms and Data Structures (WADS), volume 10389 of Lecture Notes in Computer Science. Springer, 2017.
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An, Z., Feng, Q., Kanj, I. et al. The Complexity of Tree Partitioning. Algorithmica 82, 2606–2643 (2020). https://doi.org/10.1007/s00453-020-00701-x
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DOI: https://doi.org/10.1007/s00453-020-00701-x