Linear-in-Δ lower bounds in the LOCAL model

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O(Delta) rounds, independently of n; here Delta is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o(Delta) rounds, independently of n. Our work gives the first linear-in-Delta lower bound for a natural graph problem in the standard model of distributed computing-prior lower bounds for a wide range of graph problems have been at best logarithmic in Delta.