Technical Report CS0666

Authors: O. Biran, S. Moran, S. Zaks
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Abstract: The question of which distributed tasks can be solved by asynchronous protocols in the presence of crash failures has been widely investigated in recent years. Most of these studies investigate the solvability of specific tasks, providing both positive and negative results. However, the general question whether there is an algorithm that receives as an input a specification of a distributed task \ $T$\ for \ $N$\ processors and a number \ $t$, and decides whether \ $T$\ is solvable in the presence of at most \ $t$\ crash failures, was less studied. In fact, we are not aware of any published proof that this problem is decidable, even when restricted to finite tasks (which are given by a finite list of input-output relation). In this paper we provide partial results concerning this question. For this, we use a result in [BMZ], which associates with each distributed task certain graphs, that enable us to determine whether this task can be solved in the presence of one faulty processor. We use that characterization to show that the above problem is NP complete for \ $t=1$\ and\ $N \geq 3$, and NP hard for any fixed \ $t$\ and\ $N > 2t$.
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