Approximating independent sets in sparse graphs
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
We consider the maximum independent set problem on sparse graphs with maximum degree d. The best known result for the problem is an SDP based O(d log log d/ log d) approximation due to Halperin. It is also known that no o(d/ log 2 d) approximation exists assuming the Unique Games Conjecture. We show the following two results: (i) The natural LP formulation for the problem strengthened by O(log 4 (d)) levels of the mixed-hierarchy has an integrality gap ofÕ(d/ log 2 d), whereÕ(·) ignores some
... (·) ignores some log log d factors. However, our proof is non-constructive, in particular it uses an entropy based approach due to Shearer, and does not give aÕ(d/ log 2 d) approximation algorithm with sub-exponential running time. (ii) We give an O(d/ log d) approximation based on polylog(d)-levels of the mixed hierarchy that runs in n O(1) exp(log O(1) d) time, improving upon Halperin's bound by a modest log log d factor. Our algorithm is based on combining Halperin's approach together with an idea used by Ajtai, Erdős, Komlós and Szemerédi to show that Kr-free, degree-d graphs have independent sets of size Ωr(n log log d/d).