In this paper, we introduce and develop the field of algebraic communication complexity, the theory dealing with the least number of messages to be exchanged between two players in order to compute the value of a polynomial or rational function depending on an input distributed between the two players. We define a general algebraic model, where the involved functions can be computed with the natural operations additions, multiplications and divisions and possibly with comparisons. We provide various lower bound techniques, mainly for fields of characteristic 0.
We then apply this general theory to problems from distributed mechanism design, in particular to the multicast cost sharing problem, and study the number of messages that need to be exchanged to compute the outcome of the mechanism. This addresses a question raised by Feigenbaum, Papadimitriou, and Shenker .
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