The communication complexity of local search

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 (f_A + f_B)(v) ≥ (f_A + f_B)(u) for each neighbor u of v. 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.