  Worst-case hardness results for most equilibrium computation problems have
raised the need for beyond-worst-case analysis. To this end, we study the
smoothed complexity of finding pure Nash equilibria in Network Coordination
Games, a PLS-complete problem in the worst case. This is a potential game where
the sequential-better-response algorithm is known to converge to a pure NE,
albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial)
smoothed complexity when the underlying game graph is a complete (resp.
arbitrary) graph, and every player has constantly many strategies. We note that
the complete graph case is reminiscent of perturbing all parameters, a common
assumption in most known smoothed analysis results.
  Second, we define a notion of smoothness-preserving reduction among search
problems, and obtain reductions from $2$-strategy network coordination games to
local-max-cut, and from $k$-strategy games (with arbitrary $k$) to
local-max-cut up to two flips. The former together with the recent result of
[BCC18] gives an alternate $O(n^8)$-time smoothed algorithm for the
$2$-strategy case. This notion of reduction allows for the extension of
smoothed efficient algorithms from one problem to another.
  For the first set of results, we develop techniques to bound the probability
that an (adversarial) better-response sequence makes slow improvements on the
potential. Our approach combines and generalizes the local-max-cut approaches
of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful
definition of the matrix which captures the increase in potential, a tighter
union bound on adversarial sequences, and balancing it with good enough rank
bounds. We believe that the approach and notions developed herein could be of
interest in addressing the smoothed complexity of other potential and/or
congestion games.
