A note on greedy algorithms for the maximum weighted independent set problem
Discrete Applied Mathematics
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Inapproximability of Vertex Cover and Independent Set in Bounded Degree Graphs
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
On the inapproximability of vertex cover on k-partite k-uniform hypergraphs
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
The complexity of finding independent sets in bounded degree (hyper)graphs of low chromatic number
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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 maxI Σv∈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/(dv +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.