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Balanced Partitions of Trees and Applications

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Abstract

We study the problem of finding the minimum number of edges that, when cut, form a partition of the vertices into k sets of equal size. This is called the k-BALANCED PARTITIONING problem. The problem is known to be inapproximable within any finite factor on general graphs, while little is known about restricted graph classes.

We show that the k-BALANCED PARTITIONING problem remains APX-hard even when restricted to unweighted tree instances with constant maximum degree. If instead the diameter of the tree is constant we prove that the problem is NP-hard to approximate within n c, for any constant c<1.

If vertex sets are allowed to deviate from being equal-sized by a factor of at most 1+ε, we show that solutions can be computed on weighted trees with cut cost no worse than the minimum attainable when requiring equal-sized sets. This result is then extended to general graphs via decompositions into trees and improves the previously best approximation ratio from O(log1.5(n)/ε 2) [Andreev and Räcke in Theory Comput. Syst. 39(6):929–939, 2006] to O(logn). This also settles the open problem of whether an algorithm exists for which the number of edges cut is independent of ε.

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Notes

  1. We thank an anonymous reviewer for pointing out this folklore result.

  2. We thank Nikhil Bansal for pointing out the connection between the reductions in [18] and the results by Petrank [29].

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zürich, Zürich, Switzerland

    Andreas Emil Feldmann

  2. Department of Computer Science, U.C. Santa Barbara, Santa Barbara, USA

    Luca Foschini

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  1. Andreas Emil Feldmann
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  2. Luca Foschini
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Correspondence to Andreas Emil Feldmann.

Additional information

The first author is supported by the Swiss National Science Foundation under grant 200021_125201/1. The second author is supported by the National Science Foundation grant IIS 0904501. An extended abstract [13] of this article appeared at the 29th International Symposium on Theoretical Aspects of Computer Science (STACS).

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Feldmann, A.E., Foschini, L. Balanced Partitions of Trees and Applications. Algorithmica 71, 354–376 (2015). https://doi.org/10.1007/s00453-013-9802-3

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