Communication complexity
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IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
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This paper considers the communication complexity of approximating common voting rules. Both upper and lower bounds are presented. For n voters and m alternatives, it is shown that for all ε ε (0, 1), the communication complexity of obtaining a 1 - ε approximation to Borda is O(log (1/ε)nm). A lower bound of Ω(nm) is provided for fixed small values of ε. The communication complexity of computing the true Borda winner is Ω(nm log(m)) [5]. Thus, in the case of Borda, one can obtain arbitrarily good approximations with less communication overhead than is required to compute the true Borda winner. For other voting rules, no such 1±ε approximation scheme exists. In particular, it is shown that the communication complexity of computing any constant factor approximation, ρ, to Bucklin is Ω(nm/ρ2). Conitzer and Sandholm [5] show that the communication complexity of computing the true Bucklin winner is O(nm). However, we show that for all δ ε (0, 1), the communication complexity of computing a mδ approximate winner in Bucklin elections is O(nm1 - δ log(m)). For δ ε (1/2, 1), a lower bound of ω(nm1-2δ) is also provided. Similar lower bounds are presented on the communication complexity of computing approximate winners in Copeland elections.