The Communication Complexity of Approximate Set Packing and Covering
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, n 1/2 − ε)-approximation for the packing number requires exponential communication complexity.
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