We present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized. Our goal is to use the concept of competitive ratio-which is a typical worst case notion-also within an average case analysis. We show that the classic SEPT scheduling strategy with (n) worst-case competitive ratio achieves an average of O(1) under several natural distributions, among them the exponential distribution. Our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio. Thus, our result bridges the gap between worst case and average case performance guarantee.

In Combinatorica 17(2), 1997, Kohayakawa, Luczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi's regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph Gn,p. In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi's regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K4, thereby providing a conceptually simple proof for the main result in the paper cited above.

Abstract. In this paper we survey some results concerning balls-intobins-games and the power of two choices. We present a unified and rather elementary analysis for models in the parallel as well as in the sequential setting which is based on witness trees.

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