Abstrakt
Let F = {F-1, F-2, ..., F-n} be a family of n sets on a ground set S, such as a family of balls in R-d. For every finite measure mu on S, such that the sets of F are measurable, the classical inclusion-exclusion formula asserts that mu(F-1 boolean OR F-2 boolean OR . . . boolean OR F-n) = Sigma(I:phi not equal I subset of[n]) (-1)(|I|+1)mu(boolean AND F-i is an element of I(i)), that is, the measure of the union is expressed using measures of various intersections.
The number of terms in this formula is exponential in n, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families F. We provide an upper bound valid for an arbitrary F: we show that every system F of n sets with m non-empty fields in the Venn diagram admits an inclusion-exclusion formula with m(O(log2 n)) terms and with +/- 1 coefficients, and that such a formula can be computed in m(O(log2 n)) expected time.
For every epsilon > 0 we also construct systems with Venn diagram of size m for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least Omega(m(2-epsilon)).