Abstract
The notion of recoverable value was advocated in work of Feige, Immorlica, Mirrokni and Nazerzadeh [Approx 2009] as a measure of quality for approximation algorithms. There this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable value ρ, if for every graph it recovers an independent set of size at least max I ∑ v ∈ I min [1,ρ/(d(v) + 1)], where d(v) is the degree of vertex v, and I ranges over all independent sets in G. Hence, in a sense, from every vertex v in the maximum independent set the algorithm recovers a value of at least ρ/(d v + 1) towards the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. It easily follows from known results that some simple algorithms for MIS ensure ρ ≥ 1. We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least ρ ≥ 7/3. In addition, we show that approximating MIS in graphs with a given k-coloring within a ratio larger than 2/k is unique games hard. This rules out a natural approach for obtaining ρ ≥ 2.
Keywords
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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Spencer, J.: The Probablistic Method. Wiley, Chichester (2008)
Austrin, P., Khot, S., Safra, S.: Inapproximability of vertex cover and independent set in bounded degree graphs. In: CCC (2009)
Berman, P., Fujito, T.: On approximation properties of the independent set problem for low degree graphs. Theory of Computing Systems
Chlebík, M., Chlebíková, J.: On approximability of the independent set problem for low degree graphs. In: Kralovic, R., Sýkora, O. (eds.) SIROCCO 2004. LNCS, vol. 3104, pp. 47–56. Springer, Heidelberg (2004)
Feige, U., Immorlica, N., Mirrokni, V.S., Nazerzadeh, H.: Pass approximation. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302. Springer, Heidelberg (2010)
Feige, U., Reichman, D.: Recoverable values for independent sets (Detailed version of current paper), http://www.arxiv.org/PS_cache/arxiv/pdf/1103/1103.5609v1.pdf
Griggs, J.: Lower bounds on the independence number in terms of the degrees. Journal of Combinatorial Theory, Series B 34(1), 22–39 (1983)
Guruswami, V., Kemal Sinop, A.: The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number. In: SODA (2011)
Guruswami, V., Saket, R.: On the inapproximability of vertex cover on k-partite k-uniform hypergraphs. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 360–371. Springer, Heidelberg (2010)
Halldorsson, M.M.: Approximations of weighted independent set and hereditary subset problems. Journal of Graph Algorithms and Applications 4, 1–16 (2000)
Halldorsson, M.M., Radhakrishnan, J.: Greed is good: Approximating independent sets in sparse and bounded-degree graphs. Algorithmica 18, 145–163 (1997)
Hastad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)
Hochbaum, D.: Efficient bounds for the stable set, vertex cover, and set packing problems. Discrete Appl. Math 6, 243–254 (1983)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. In: CCC (2003)
Nemhauser, G., Trotter, L.: Vertex packings: Structural properties and algorithms. Math. Programming 8, 232–248 (1975)
Sakai, M., Togasaki, M., Yamazaki, K.: A note on greedy algorithms for the maximum weighted independent set problem. Discrete Applied Mathematics 126, 313–322 (1999)
Wei, V.K.: A lower bound on the stability number of a simple graph. Bell Laboratories Technical Memorandum 8 I- I 12 17.9 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Feige, U., Reichman, D. (2011). Recoverable Values for Independent Sets. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_41
Download citation
DOI: https://doi.org/10.1007/978-3-642-22006-7_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22005-0
Online ISBN: 978-3-642-22006-7
eBook Packages: Computer ScienceComputer Science (R0)