On Set Expansion Problems and the Small Set Expansion Conjecture

  • Rajiv GandhiEmail author
  • Guy Kortsarz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


We study two problems related to the Small Set Expansion Conjecture [14]: the Maximum weight \(m'\) -edge cover (MWEC) problem and the Fixed cost minimum edge cover (FCEC) problem. In the MWEC problem, we are given an undirected simple graph \(G=(V,E)\) with integral vertex weights. The goal is to select a set \(U\subseteq V\) of maximum weight so that the number of edges with at least one endpoint in \(U\) is at most \(m'\). Goldschmidt and Hochbaum [8] show that the problem is NP-hard and they give a \(3\)-approximation algorithm for the problem. The approximation guarantee was improved to \(2+\epsilon \), for any fixed \(\epsilon > 0\) [12]. We present an approximation algorithm that achieves a guarantee of \(2\). Interestingly, we also show that for any constant \(\epsilon > 0\), a \((2-\epsilon )\)-ratio for MWEC implies that the Small Set Expansion Conjecture [14] does not hold. Thus, assuming the Small Set Expansion Conjecture, the bound of 2 is tight. In the FCEC problem, we are given a vertex weighted graph, a bound \(k\), and our goal is to find a subset of vertices \(U\) of total weight at least \(k\) such that the number of edges with at least one edges in \(U\) is minimized. A \(2(1+\epsilon )\)-approximation for the problem follows from the work of Carnes and Shmoys [3]. We improve the approximation ratio by giving a \(2\)-approximation algorithm for the problem and show a \((2-\epsilon )\)-inapproximability under Small Set Expansion Conjecture conjecture. Only the NP-hardness result was known for this problem [8]. We show that a natural linear program for FCEC has an integrality gap of \(2-o(1)\). We also show that for any constant \(\rho >1\), an approximation guarantee of \(\rho \) for the FCEC problem implies a \(\rho (1+o(1))\) approximation for MWEC. Finally, we define the Degrees density augmentation problem which is the density version of the FCEC problem. In this problem we are given an undirected graph \(G=(V,E)\) and a set \(U\subseteq V\). The objective is to find a set \(W\) so that \((e(W)+e(U,W))/deg(W)\) is maximum. This problem admits an LP-based exact solution [4]. We give a combinatorial algorithm for this problem.



We thank V. Chakravarthy for introducing the FCEC problem to us. We also thank V. Chakravarthy and S. Roy for useful discussions. Thanks also to U. Feige for bringing the Small Set Expansion Conjecture to our attention.


  1. 1.
    Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: Avrachenkov, K., Donato, D., Litvak, N. (eds.) WAW 2009. LNCS, vol. 5427, pp. 25–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A: Detecting high log-densities: an O\((n^{1/4})\) approximation for densest k-subgraph. In: STOC, pp. 201–210 (2010)Google Scholar
  3. 3.
    Carnes, T., Shmoys, D.: Primal-dual schema for capacitated covering problems. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 288–302. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Chakravarthy, V.\({\rm {\tilde{T}}}\)., Modani, N., Natarajan, S.\({\rm {\tilde{R}}}\), Roy, S., Sabharwal, Y.: Density functions subject to co-matroid constraint. In: Foundations of Software Technology and Theoretical Computer Science, pp. 236–248 (2012)Google Scholar
  5. 5.
    Feige, U., Kortsarz, G., Peleg, D.: The dense \(k\)-subgraph problem. Algorithmica 29(3), 410–421 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gajewar, A., Sarma, A.: Multi-skill collaborative teams based on densest subgraphs. In: SDM, pp. 165–176 (2012)Google Scholar
  7. 7.
    Goldberg, A.V.: Finding a maximum density subgraph. Technical report UCB/CSD-84-171, EECS Department, University of California, Berkeley (1984)Google Scholar
  8. 8.
    Goldschmidt, O., Hochbaum, D.S.: k-edge subgraph problems. Discrete Appl. Math. 74(2), 159–169 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khot, S.: Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput. 36(4), 1025–1071 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khuller, S., Saha, B.: On finding dense subgraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 597–608. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Lawler, Eugene L.: Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston, New York (1976)Google Scholar
  12. 12.
    Liang, H.: On the \(k\)-edge-incident subgraph problem and its variants. Discrete Appl. Math. 161(18), 2985–2991 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  14. 14.
    Raghavendra, P., Steurer, D.: Graph expansion and the unique games conjecture. In: STOC, pp. 755–764 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceRutgers University-CamdenCamdenUSA

Personalised recommendations