Parameterized Distributed Algorithms

In this work, we initiate a thorough study of parameterized graph optimization problems in the distributed setting. In a parameterized problem, an algorithm decides whether a solution of size bounded by a \emph{parameter} $k$ exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds. Our results extend beyond the scope of parameterized problems. We show that any LOCAL $(1+\epsilon)$-approximation algorithm for the above problems must take $\Omega(\epsilon^{-1})$ rounds. Joined with the algorithm of [GKM17] and the $\Omega(\sqrt{\frac{\log n}{\log\log n}})$ lower bound of [KMW16], this settles the complexity of $(1+\epsilon)$-approximating MVC, MaxM and MaxIS at $(\epsilon^{-1}\log n)^{\Theta(1)}$. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first deterministic $o(n^2)$ rounds CONGEST algorithms that approximate MVC and MaxM within a factor strictly smaller than $2$.