Nisan N. (2002) The Communication Complexity of Approximate Set Packing and Covering. In: Widmayer P., Eidenbenz S., Triguero F., Morales R., Conejo R., Hennessy M. (eds) Automata, Languages and Programming. ICALP 2002. Lecture Notes in Computer Science, vol 2380. Springer, Berlin, Heidelberg
We consider a setting where k players are each holding some collection of subsets of 1..n. We consider the communication complexity of approximately solving two problems: The cover number: the minimal number of sets (in the union of their collections) whose union is 1...n and the packing number: the maximum number of sets (in the union of their collections) that are pair-wise disjoint.
We prove that while computing a (ln n)-approximation for the cover number and an min(k, O(√n))-approximation for the packing number can be done with polynomial (in n) amount of communication, getting a (1/2 − ε) logn approximation for the cover number or a better than min(k, n1/2 − ε)-approximation for the packing number requires exponential communication complexity.