Communication complexity of inevitable intersection

Many known methods for analysing the communication complexity of unstructured search are based on the hardness of the set disjointness problem. Such techniques may “hide” important aspects of the original problem. Intuitively, search is a much simpler task: while set disjointness is hard even for NP, successful search necessarily results in a short witness, which makes it easy for NP. Accordingly, the possibility to deduce hardness of search problems from that of set disjointness can be viewed as an artefact of specific definitions. We construct a natural variation of the intersection-search problem, where the input comes from a product distribution, and nevertheless, every pair of input subsets share at least one element. We call this problem inevitable intersection, its analysis seems to require a new, more subtle approach – in particular, not relying on the hardness of set disjointness. To prove a lower bound on the communication complexity of inevitable intersection, we identify such properties of large rectangles that make it impossible to partition the input matrix into a small number of “nearly-monochromatic” rectangles (the analysis is “non-local”: it addresses not the existence of a large rectangle of low discrepancy, but the possibility to partition the whole matrix into such rectangles). We believe that this technique provides new insight into the combinatorial structure of “search-like” communication problems.