We give time lower bounds for the distributed approximation of minimum vertex cover (MVC) and related problems such as minimum dominating set (MDS). In k communication rounds, MVC and MDS can only be approximated by factors Ω(n c/k where n and ∆ denote the number of nodes and the largest degree in the graph. The number of rounds required in order to achieve a constant or even only a polylogarithmic approximation ratio is at least Ω( Ô log n/ log log n) and Ω(log ∆/ log log ∆). By a simple reduction, the latter lower bounds also hold for the construction of maximal matchings and maximal independent sets.
In addition, to achieve a poly-logarithmic approximation ratio for minimum dominating set at least max{Ω( log |G|/ log log |G|)Ω(log ∆/ log log ∆)} rounds are required ( REF ).
What cannot be computed locally!
PODC '04
Mathematics Computer Science