  We study the following 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
$(f_A+f_B)(v)\geq (f_A+f_B)(u)$ for every 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 provide an \emph{optimal} communication bound of
$\Omega(\sqrt{N})$ for the hypercube, and for a constant dimensional greed,
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. First, we show that finding a pure Nash
equilibrium in $2$-player $N$-action exact potential games requires polynomial
(in $N$) communication. We also show that finding a pure Nash equilibrium in
$n$-player $2$-action exact potential games requires exponential (in $n$)
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.
  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.
