Worst-case interactive communication. II. Two messages are not optimal

For pt.I see ibid., vol.36, no.5, p.1111-26, (1990). The author defines the chromatic-decomposition number of a hypergraph and shows that, under general conditions, it determines the two message complexity. This result is then used to provide that two messages are not optimal. Protocols, complexities, and the characteristic hypergraph of (X,Y) are defined. The playoffs problem is described. Although similar in appearance to the league problem given in an example, it is shown that its two-message complexity is about twice as high as its three-message complexity. The author proves a high lower bound on the chromatic-decomposition number of the playoffs problem's characteristic hypergraph showing that the problem has a high two-message complexity, and that allowing more than two messages may decrease the number of transmitted bits by a factor of two. A technique that improves the lower bound for the chromatic-decomposition number of the playoffs problem is described. However, this improved lower bound does not suffice to increase the provable gap between two and three message complexities.<
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