Minimal Obstructions for the Coordinated Attack Problem and Beyond

We consider the well known Coordinated Attack Problem, where two generals have to decide on a common attack, when their messengers can be captured by the enemy. Informally, this problem represents the difficulties to agree in the present of communication faults. We consider here only omission faults (loss of message), but contrary to previous studies, we do not to restrict the way messages can be lost, ie. we use no specific failure metric. Our contribution is threefold. First, we introduce the study of arbitrary patterns of failure ("omission schemes"), proposing notions and notations that revealed very convenient to handle. In the large subclass of omission schemes where the double simultaneous omission can never happen, we characterize which one are obstructions for the Coordinated Attack Problem. We present then some interesting applications. We show for the first time that the well studied omission scheme, where at most one message can be lost at each round, is a kind of least worst case environment for the Coordinated Attack Problem. We also extend our study to networks of arbitrary size. In particular, we address an open question of Santoro and Wid mayer about the Consensus Problem in communication networks with omission faults.