Approximating the Caro-Wei Bound for Independent Sets in Graph Streams

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10856)


The Caro-Wei bound states that every graph \(G=(V, E)\) contains an independent set of size at least \(\beta (G) := \sum _{v \in V} \frac{1}{\deg _G(v) + 1}\), where \(\deg _G(v)\) denotes the degree of vertex v. Halldórsson et al. [1] gave a randomized one-pass streaming algorithm that computes an independent set of expected size \(\beta (G)\) using \(\mathrm {O}(n \log n)\) space. In this paper, we give streaming algorithms and a lower bound for approximating the Caro-Wei bound itself.

In the edge arrival model, we present a one-pass c-approximation streaming algorithm that uses \(\mathrm {O}({\overline{d} \log (n) /c^2})\) space, where \(\overline{d}\) is the average degree of G. We further prove that space \(\varOmega ({\overline{d}/c^2})\) is necessary, rendering our algorithm almost optimal. This lower bound holds even in the vertex arrival model, where vertices arrive one by one together with their incident edges that connect to vertices that have previously arrived. In order to obtain a poly-logarithmic space algorithm even for graphs with arbitrarily large average degree, we employ an alternative notion of approximation: We give a one-pass streaming algorithm with space \(\mathrm {O}(\log ^3 n)\) in the vertex arrival model that outputs a value that is at most a logarithmic factor below the true value of \(\beta \) and no more than the maximum independent set size.


Independent Set Streaming Graph Arrival Vertex Streaming Algorithm Lower Bound 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank an anonymous reviewer whose comments helped us simplify Theorem 1. The work of GC is supported in part by European Research Council grant ERC-2014-CoG 647557; JD is supported by a Microsoft EMEA scholarship and the Alan Turing Institute under the EPSRC grant EP/N510129/1; CK is supported by EPSRC grant EP/N011163/1.


  1. 1.
    Halldórsson, B.V., Halldórsson, M.M., Losievskaja, E., Szegedy, M.: Streaming algorithms for independent sets in sparse hypergraphs. Algorithmica 76(2), 490–501 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations. IRSS, pp. 85–103. Plenum Press, New York (1972). Scholar
  3. 3.
    Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). Acta Math. 182(1), 105–142 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Feige, U.: Approximating maximum clique by removing subgraphs. SIAM J. Discret. Math. 18(2), 219–225 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Halldórsson, M., Radhakrishnan, J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. In: STOC, pp. 439–448 (1994)Google Scholar
  7. 7.
    Wei, V.: A lower bound on the stability number of a simple graph. Technical report, Bell Labs (1981)Google Scholar
  8. 8.
    Griggs, J.R.: Lower bounds on the independence number in terms of the degrees. J. Comb. Theory Ser. B 34(1), 22–39 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caro, Y.: New results on the independence number. Technical report, Tel Aviv Univ (1979)Google Scholar
  10. 10.
    Halldórsson, M.M., Konrad, C.: Distributed large independent sets in one round on bounded-independence graphs. In: Moses, Y. (ed.) DISC 2015. LNCS, vol. 9363, pp. 559–572. Springer, Heidelberg (2015). Scholar
  11. 11.
    Halldórsson, M.M., Sun, X., Szegedy, M., Wang, C.: Streaming and communication complexity of clique approximation. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7391, pp. 449–460. Springer, Heidelberg (2012). Scholar
  12. 12.
    Assadi, S., Khanna, S., Li, Y.: On estimating maximum matching size in graph streams. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 1723–1742 (2017)Google Scholar
  13. 13.
    Cormode, G., Jowhari, H., Monemizadeh, M., Muthukrishnan, S.: The sparse awakens: streaming algorithms for matching size estimation in sparse graphs. In: ESA (2017)Google Scholar
  14. 14.
    Cabello, S., Pérez-Lantero, P.: Interval selection in the streaming model. In: Dehne, F., Sack, J.-R., Stege, U. (eds.) WADS 2015. LNCS, vol. 9214, pp. 127–139. Springer, Cham (2015). Scholar
  15. 15.
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58(1), 137–147 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Woodruff, D.P.: Frequency moments. In: Liu, L., Özsu, M.T. (eds.) Encyclopedia of Database Systems, pp. 1169–1170. Springer, Boston (2009). Scholar
  17. 17.
    Braverman, V., Chestnut, S.R.: Universal sketches for the frequency negative moments and other decreasing streaming sums. In: APPROX/RANDOM, pp. 591–605 (2015)Google Scholar
  18. 18.
    Turán, P.: On an extremal problem in graph theory. Matematikai és Fizikai Lapok 48(436–452), 137 (1941)Google Scholar
  19. 19.
    Henzinger, M., Raghavan, P., Rajagopalan, S.: Computing on data streams. Technical report SRC 1998–011, DEC Systems Research Centre (1998)Google Scholar
  20. 20.
    McGregor, A.: Graph stream algorithms: a survey. SIGMOD Rec. 43(1), 9–20 (2014)CrossRefGoogle Scholar
  21. 21.
    Gonen, M., Ron, D., Shavitt, Y.: Counting stars and other small subgraphs in sublinear-time. SIAM J. Discrete Math. 25(3), 1365–1411 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Eden, T., Ron, D., Seshadhri, C.: Sublinear time estimation of degree distribution moments: the arboricity connection. CoRR abs/1604.03661 (2016)Google Scholar
  23. 23.
    Aliakbarpour, M., Biswas, A.S., Gouleakis, T., Peebles, J., Rubinfeld, R., Yodpinyanee, A.: Sublinear-time algorithms for counting star subgraphs with applications to join selectivity estimation. CoRR abs/1601.04233 (2016)Google Scholar
  24. 24.
    Jowhari, H., Sağlam, M., Tardos, G.: Tight bounds for Lp samplers, finding duplicates in streams, and related problems. In: ACM Principles of Database Systems (2011)Google Scholar
  25. 25.
    Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discrete Math. 5(4), 545–557 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science, Centre for Discrete Mathematics and its Applications (DIMAP)University of WarwickCoventryUK

Personalised recommendations