Abstract

We make a step towards understanding the distributed complexity of global optimization problems. We give bounds on the trade-off between locality and achievable approximation ratio of distributed algorithms for packing and covering problems. Extending a result of [9], we show that in k communication rounds, maximum matching and therefore packing problems cannot be approximated better than Ω(nc/k2/k) and Ω(∆1/k /k) where c is a small constant and n and ∆ denote the number of nodes and the maximum degree of the network graph, respectively. This means that in order to obtain a constant or polylogarithmic approximation, there are graphs with n nodes and graphs with maximum degree ∆ on which Ω ( √ log n / log log n) and Ω(log ∆ / log log ∆) rounds are needed, respectively. On the positive side, we prove that maximum matching and minimum vertex cover (the dual problem) can be approximated by O(∆1/k) in O(k) rounds, showing that the given lower bound is almost tight. We also give a distributed algorithm which approximates any packing or covering LP by O(n1/k) in O(k) rounds. 1

Citations