Towards Resistance Sparsifiers

Authors Michael Dinitz, Robert Krauthgamer, Tal Wagner

Thumbnail PDF


  • Filesize: 0.52 MB
  • 18 pages

Document Identifiers

Author Details

Michael Dinitz
Robert Krauthgamer
Tal Wagner

Cite AsGet BibTex

Michael Dinitz, Robert Krauthgamer, and Tal Wagner. Towards Resistance Sparsifiers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 738-755, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+epsilon)-resistance sparsifier of size ~O(n/epsilon), and conjecture this bound holds for all graphs on n nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Omega(n/epsilon^2) edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein (JMLR, 2014) leads to the aforementioned resistance sparsifiers.
  • edge sparsification
  • spectral sparsifier
  • graph expansion
  • effective resistance
  • commute time


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Alexandr Andoni, Robert Krauthgamer, and David P. Woodruff. The sketching complexity of graph cuts. CoRR, abs/1403.7058, 2014. URL:
  2. Joshua D. Batson, Daniel A. Spielman, and Nikhil Srivastava. Twice-ramanujan sparsifiers. SIAM J. Comput., 41(6):1704-1721, 2012. URL:
  3. A. A. Benczúr and D. R. Karger. Approximating s-t minimum cuts in Õ(n^2) time. In 28th Annual ACM Symposium on Theory of Computing, pages 47-55. ACM, 1996. URL:
  4. Béla Csaba, Daniela Kühn, Allan Lo, Deryk Osthus, and Andrew Treglown. Proof of the 1-factorization and Hamilton decomposition conjectures. ArXiv e-prints, abs/1401.4159, 2014. URL:
  5. M. M. Deza and M. Laurent. Geometry of cuts and metrics. Springer-Verlag, Berlin, 1997. Google Scholar
  6. Michael Kapralov and Rina Panigrahy. Spectral sparsification via random spanners. In 3rd Innovations in Theoretical Computer Science Conference, pages 393-398. ACM, 2012. URL:
  7. Rohit Khandekar, Subhash A. Khot, Lorenzo Orecchia, and Nisheeth K. Vishnoi. On a cut-matching game for the sparsest cut problem. Technical Report UCB/EECS-2007-177, EECS Department, University of California, Berkeley, 2007. URL:
  8. Rohit Khandekar, Satish Rao, and Umesh V. Vazirani. Graph partitioning using single commodity flows. J. ACM, 56(4), 2009. URL:
  9. Daniela Kühn and Deryk Osthus. Decompositions of complete uniform hypergraphs into hamilton berge cycles. J. Comb. Theory, Ser. A, 126:128-135, 2014. URL:
  10. L. Lovász. Random walks on graphs: a survey. In Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages 353-397. János Bolyai Math. Soc., Budapest, 1996. Google Scholar
  11. D. Peleg and A. A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL:
  12. L. Perkovic and B. Reed. Edge coloring regular graphs of high degree. In Discrete Math.,165/166, pages 567-578, 1997. Google Scholar
  13. D. A. Spielman and S.-H. Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In 36th Annual ACM Symposium on Theory of Computing, pages 81-90. ACM, 2004. URL:
  14. Daniel A. Spielman and Nikhil Srivastava. Graph sparsification by effective resistances. SIAM J. Comput., 40(6):1913-1926, December 2011. URL:
  15. Daniel A. Spielman and Shang-Hua Teng. Spectral sparsification of graphs. SIAM J. Comput., 40(4):981-1025, July 2011. URL:
  16. Ulrike von Luxburg, Agnes Radl, and Matthias Hein. Hitting and commute times in large random neighborhood graphs. Journal of Machine Learning Research, 15(1):1751-1798, 2014. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail