An Automatic Speedup Theorem for Distributed Problems

Recently, Brandt et al. [STOC'16] proved a lower bound for the distributed Lov\ász Local Lemma, which has been conjectured to be tight for sufficiently relaxed LLL criteria by Chang and Pettie [FOCS'17]. At the heart of their result lies a speedup technique that, for graphs of girth at least $2t+2$, transforms any $t$-round algorithm for one specific LLL problem into a $(t-1)$-round algorithm for the same problem. We substantially improve on this technique by showing that such a speedup exists for any locally checkable problem $\Pi$, with the difference that the problem $\Pi_1$ the inferred $(t-1)$-round algorithm solves is not (necessarily) the same problem as $\Pi$. Our speedup is automatic in the sense that there is a fixed procedure that transforms a description for $\Pi$ into a description for $\Pi_1$ and reversible in the sense that any $(t-1)$-round algorithm for $\Pi_1$ can be transformed into a $t$-round algorithm for $\Pi$. In particular, for any locally checkable problem $\Pi$ with exact deterministic time complexity $T(n, \Delta) \leq t$ on graphs with $n$ nodes, maximum node degree $\Delta$, and girth at least $2t+2$, there is a sequence of problems $\Pi_1, \Pi_2, \dots$ with time complexities $T(n, \Delta)-1, T(n, \Delta)-2, \dots$, that can be inferred from $\Pi$. As a first application of our generalized speedup, we solve a long-standing open problem of Naor and Stockmeyer [STOC'93]: we show that weak $2$-coloring in odd-degree graphs cannot be solved in $o(\log^* \Delta)$ rounds, thereby providing a matching lower bound to their upper bound.