We study a communication variant of local search. There is some fixed, commonly known graph G. Alice holds f A and Bob holds f B , both are functions that specify a value for each vertex. The goal is to find a local maximum of f A + f B with respect to G, i.e., a vertex v for which ( Our main result is that finding a local maximum requires polynomial (in the number of vertices) bits of communication. The result holds for the following families of graphs: three dimensional grids, hypercubes, odd graphs, and degree 4 graphs. Moreover, we prove an optimal communication bound of Ω( ⌋︂ N ) for the hypercube, and for a constant dimension grid, where N is the number of vertices in the graph. We provide applications of our main result in two domains, exact potential games and combinatorial auctions. Each one of the results demonstrates an exponential separation between the non-deterministic communication complexity and the randomized communication complexity of a total search problem. First, we show that finding a pure Nash equilibrium in 2-player N -action exact potential games requires poly(N ) communication. We also show that finding a pure Nash equilibrium in n-player 2-action exact potential games requires exp(n) communication. The second domain that we consider is combinatorial auctions, in which we prove that finding a local maximum in combinatorial auctions requires exponential (in the number of items) communication even when the valuations are submodular. • Theory of computation → Communication complexity; Exact and approximate computation of equilibria.
Babichenko et al. REF prove communication complexity lower bounds for finding a pure equilibrium in the restricted class of potential games.
The communication complexity of local search
STOC 2019
Computer Science Mathematics