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An \(O(n^4)\) Time Algorithm to Compute the Bisection Width of Solid Grid Graphs

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Abstract

The bisection problem asks for a partition of the \(n\) vertices of a graph into two sets of size at most \(\lceil n/2\rceil \), so that the number of edges connecting the sets is minimised. A grid graph is a finite connected subgraph of the infinite two-dimensional grid. It is called solid if it has no holes. Papadimitriou and Sideri (Theory Comput Syst 29:97–110, 1996) gave an \(O(n^5)\) time algorithm to solve the bisection problem on solid grid graphs. We propose a novel approach that exploits structural properties of optimal cuts within a dynamic program. We show that our new technique leads to an \(O(n^4)\) time algorithm.

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Acknowledgments

We gratefully acknowledge discussions with Peter Arbenz, and the support of this work through the Swiss National Science Foundation under Grant No. 200021_125201/1. A preliminary version [8] appeared in the Proceedings of the 19th Annual European Symposium on Algorithms (ESA).

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

    Andreas Emil Feldmann

  2. Institute of Theoretical Computer Science, ETH Zurich, Zurich, Switzerland

    Peter Widmayer

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  1. Andreas Emil Feldmann
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  2. Peter Widmayer
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Correspondence to Andreas Emil Feldmann.

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Feldmann, A.E., Widmayer, P. An \(O(n^4)\) Time Algorithm to Compute the Bisection Width of Solid Grid Graphs. Algorithmica 71, 181–200 (2015). https://doi.org/10.1007/s00453-014-9928-y

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  • DOI: https://doi.org/10.1007/s00453-014-9928-y

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