Asymptotic performance of the Grimmett–McDiarmid heuristic

Yuval Filmus

In a paper on graph coloring, Grimmett and McDiarmid described a heuristic that finds a large clique in a $G(n,1/2)$ random graph. The heuristic simply scans the vertices in arbitrary order, adding any vertex adjacent to all vertices previously chosen.

Grimmett and McDiarmid showed that with high probability this produces a clique whose size is asymptotically $\log_2 n$, compared to the maximum clique whose size is asymptotically $2\log_2 n$.

We determine the asymptotic distribution of the size of the clique produced by the algorithm, which is obtained by taking the logarithm of an infinite sum of exponential random variables.

Prodinger mentions that the size of the clique has the same distribution as that of the Morris counter, analyzed by Flajolet. In particular, our formulas appear (in a different form) in Flajolet’s paper.


 author = {Yuval Filmus},
 title = {Asymptotic performance of the {G}rimmett--{M}c{D}iarmid heuristic},
 journal = {Manuscript}
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