Note on the independence number of triangle-free graphs, II
Journal of Combinatorial Theory Series A
Computing independent sets in graphs with large girth
Discrete Applied Mathematics
On randomized greedy matchings
Random Structures & Algorithms
Approximating maximum independent set in bounded degree graphs
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
The ζ (2) limit in the random assignment problem
Random Structures & Algorithms
Improved approximations of independent sets in bounded-degree graphs via subgraph removal
Nordic Journal of Computing
Random Structures & Algorithms
Large independent sets in regular graphs of large girth
Journal of Combinatorial Theory Series B
PTAS for maximum weight independent set problem with random weights in bounded degree graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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We derive new results for the performance of a simple greedy algorithm for finding large independent sets and matchings in constant-degree regular graphs. We show that for r-regular graphs with n nodes and girth at least g, the algorithm finds an independent set of expected cardinality \[ f(r)n-O\biggl(\frac{(r-1)^{\frac{g}{2}}}{ \frac{g}{2}!} n\biggr), \] where f(r) is a function which we explicitly compute. A similar result is established for matchings. Our results imply improved bounds for the size of the largest independent set in these graphs, and provide the first results of this type for matchings. As an implication we show that the greedy algorithm returns a nearly perfect matching when both the degree r and girth g are large. Furthermore, we show that the cardinality of independent sets and matchings produced by the greedy algorithm in arbitrary bounded-degree graphs is concentrated around the mean. Finally, we analyse the performance of the greedy algorithm for the case of random i.i.d. weighted independent sets and matchings, and obtain a remarkably simple expression for the limiting expected values produced by the algorithm. In fact, all the other results are obtained as straightforward corollaries from the results for the weighted case.