Linear-in- ∆ 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(∆) rounds, independently of n; here ∆ 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 distrib-uted algorithm that finds a maximal fractional matching in o(∆) rounds, independently of n. Our work gives the first linear-in- ∆ lower bound for a natural graph problem in the standard LOCAL model of distributed computing—prior lower bounds for a wide range of graph problems have been at best logarithmic in ∆