On the Smoothed Complexity of Combinatorial Local Search

We propose a unifying framework for smoothed analysis of combinatorial local optimization problems and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to identify key structural properties, and corresponding parameters, that determine the smoothed running time of local search dynamics. We formalize this via a black-box tool that provides concrete bounds on the expected maximum number of steps needed until local search reaches an exact local optimum. This bound is particularly strong, in the sense that it holds for any starting feasible solution, any choice of pivoting rule, and does not rely on the choice of specific noise distributions that are applied on the input, but it is parameterized by just a global upper bound $\phi$ on the probability density. We then demonstrate the power of this tool by instantiating it for various PLS-hard problems to derive efficient smoothed running times. This not only unifies, and greatly simplifies, prior existing positive results, but also allows us to extend or improve them. Notable problems on which we provide such a contribution are Max-Cut, the Travelling Salesman problem, and Network Coordination Games. Additionally, in this paper we propose novel smoothed analysis formulations, and prove polynomial smoothed running times, for important local optimization problems that have not been studied before from this perspective. Importantly, we provide an extensive study of the problem of finding pure Nash equilibria in general and Network Congestion Games under various representation models, including explicit, step-function, and polynomial latencies. We show that all the problems we study can be solved by their standard local search algorithms in polynomial smoothed time on PLS-hard instances in which these algorithms have exponential worst-case running time.