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We consider the problem of how to schedule t similar and independent tasks to be performed in a synchronous distributed system of p stations communicating via multiple-access channels. Stations are prone to crashes whose patterns of occurrence are specified by adversarial models. Work, defined as the number of the available processor steps, is the complexity measure. We consider only reliable algorithms that perform all the tasks as long as at least one station remains operational. It is shown that every reliable algorithm has to perform work (t + p √ t) even when no failures occur. An optimal deterministic algorithm for the channel with collision detection is developed, which performs work O(t + p √ t). Another algorithm, for the channel without collision detection, performs work O(t + p √ t + p min{ f, t}), where f < p is the number of failures. This algorithm is proved to be optimal, provided that the adversary is restricted in failing no more than f stations. Finally, we consider the question if randomization helps against weaker adversaries for the channel without collision detection. A randomized algorithm is developed which performs the expected minimum amount O(t + p √ t) of work, provided that the adversary may fail a constant fraction of stations and it has to select failure-prone stations prior to the start of an execution of the algorithm.

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We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let H be a fixed graph with k vertices and an independent set of size s. 1. Detecting if an n-vertex graph contains a (not necessarily induced) subgraph isomorphic to H can be done in time O(n ω((k−s)/2 ,1, (k−s)/2)), where ω(p, q, r) is the exponent of fast arithmetic matrix multiplication of an n p × n q matrix by an n q × n r matrix. 2. When s = 2, counting the number of (not necessarily induced) subgraphs isomorphic to H can be done in the same time, i.e., in time O(n ω((k−2)/2 ,1, (k−2)/2)). It follows in particular that we can count the number of subgraphs isomorphic to any H on four vertices that is not K 4 in time O(n ω), where ω = ω(1, 1, 1) is known to be smaller than 2.373. Similarly, we can count the number of subgraphs isomorphic to any H on five vertices that is not K 5 in time O(n ω(2,1,1)), where ω(2, 1, 1) is known to be smaller than 3.257. Finally, we derive inputsensitive variants of our time upper bounds. They are partially expressed in terms of the number m of edges of the input graph and do not rely on fast matrix multiplication.