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Improved Approximation Algorithms for Balanced Partitioning Problems

Authors Harald Räcke, Richard Stotz

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Harald Räcke
Richard Stotz

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Harald Räcke and Richard Stotz. Improved Approximation Algorithms for Balanced Partitioning Problems. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 58:1-58:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


We present approximation algorithms for balanced partitioning problems. These problems are notoriously hard and we present new bicriteria approximation algorithms, that approximate the optimal cost and relax the balance constraint. In the first scenario, we consider Min-Max k-Partitioning, the problem of dividing a graph into k equal-sized parts while minimizing the maximum cost of edges cut by a single part. Our approximation algorithm relaxes the size of the parts by (1+epsilon) and approximates the optimal cost by O(log^{1.5}(n) * log(log(n))), for every 0 < epsilon < 1. This is the first nontrivial algorithm for this problem that relaxes the balance constraint by less than 2. In the second scenario, we consider strategies to find a minimum-cost mapping of a graph of processes to a hierarchical network with identical processors at the leaves. This Hierarchical Graph Partitioning problem has been studied recently by Hajiaghayi et al. who presented an (O(log(n)),(1+epsilon)(h+1)) approximation algorithm for constant network heights h. We use spreading metrics to give an improved (O(log(n)),(1+epsilon)h) approximation algorithm that runs in polynomial time for arbitrary network heights.
  • graph partitioning
  • dynamic programming
  • scheduling


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  1. Konstantin Andreev and Harald Räcke. Balanced graph partitioning. Theory Comput. Syst., 39(6):929-939, 2006. URL:
  2. Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph (Seffi) Naor, and Roy Schwartz. Min-max graph partitioning and small set expansion. In Proc. of the 52nd FOCS, pages 17-26, 2011. URL:
  3. Marcin Bienkowski, Miroslaw Korzeniowski, and Harald Räcke. A practical algorithm for constructing oblivious routing schemes. In Proc. of the 15th SPAA, pages 24-33, 2003. URL:
  4. Chandra Chekuri and Sanjeev Khanna. On multidimensional packing problems. SIAM J. Comput., 33(4):837-851, 2004. URL:
  5. Guy Even, Joseph (Seffi) Naor, Satish Rao, and Baruch Schieber. Fast approximate graph partitioning algorithms. SIAM J. Comput., 28(6):2187-2214, 1999. Also in Proc. 8th SODA, 1997, pp. 639-648. URL:
  6. Andreas E. Feldmann and Luca Foschini. Balanced partitions of trees and applications. In Proc. of the 29th STACS, pages 100-111, 2012. URL:
  7. Mohammad Taghi Hajiaghayi, Theodore Johnson, Mohammad Reza Khani, and Barna Saha. Hierarchical graph partitioning. In Proc. of the 26th SPAA, pages 51-60, 2014. URL:
  8. Chris Harrelson, Kirsten Hildrum, and Satish Rao. A polynomial-time tree decomposition to minimize congestion. In Proc. of the 15th SPAA, pages 34-43, 2003. URL:
  9. Robert Krauthgamer, Joseph (Seffi) Naor, and Roy Schwartz. Partitioning graphs into balanced components. In Proc. of the 20th SODA, pages 942-949, 2009. URL:
  10. Robert Krauthgamer, Joseph (Seffi) Naor, Roy Schwartz, and Kunal Talwar. Non-uniform graph partitioning. In Proc. of the 25th SODA, pages 1229-1243, 2014. URL:
  11. Konstantin Makarychev and Yuri Makarychev. Nonuniform graph partitioning with unrelated weights. In Proc. of the 41st ICALP, pages 812-822, 2014. URL:
  12. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proc. of the 40th STOC, pages 255-264, 2008. Google Scholar
  13. Harald Räcke and Chintan Shah. Improved guarantees for tree cut sparsifiers. In Proc. of the 22nd ESA, pages 774-785, 2014. URL:
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