Abstract
The graph bisection problem asks to partition the n vertices of a graph into two sets of equal size so that the number of edges across the cut is minimum. We study finite, connected subgraphs of the infinite two-dimensional grid that do not have holes. Since bisection is an intricate problem, our interest is in the tradeoff between runtime and solution quality that we get by limiting ourselves to a special type of cut, namely cuts with at most one bend each (corner cuts). We prove that optimum corner cuts get us arbitrarily close to equal sized parts, and that this limitation makes us lose only a constant factor in the quality of the solution. We obtain our result by a thorough study of cuts in polygons and the effect of limiting these to corner cuts.
We gratefully acknowledge discussions with Peter Arbenz who introduced the human bone simulation problem to us, and the support of this work through the Swiss National Science Foundation under Grant No. 200021_125201/1.
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References
Arbenz, P., Müller, R.: Microstructural finite element analysis of human bone structures. ERCIM News 74, 31–32 (2008)
Díaz, J., Serna, M.J., Torán, J.: Parallel approximation schemes for problems on planar graphs. Acta Informatica 33(4), 387–408 (1996)
Feldmann, A.E., Das, S., Widmayer, P.: Simple Cuts are Fast and Good: Optimum Right-Angled Cuts in Solid Grids. In: Wu, W., Daescu, O. (eds.) COCOA 2010, Part I. LNCS, vol. 6508, pp. 11–20. Springer, Heidelberg (2010)
Feldmann, A.E., Das, S., Widmayer, P.: Restricted cuts for bisections in solid grids: A proof via polygons. Technical Report 731, Institute of Theoretical Computer Science, ETH Zürich (July 2011)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-completeness. W.H. Freeman and Co., San Fransisco (1979)
Goldberg, M., Miller, Z.: A parallel algorithm for bisection width in trees. Computers & Mathematics with Applications 15(4), 259–266 (1988)
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)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing (2008)
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Feldmann, A.E., Das, S., Widmayer, P. (2011). Restricted Cuts for Bisections in Solid Grids: A Proof via Polygons. In: Kolman, P., Kratochvíl, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes in Computer Science, vol 6986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25870-1_14
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DOI: https://doi.org/10.1007/978-3-642-25870-1_14
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