Inclusion-exclusion: Exact and approximate

Abstract

It is often required to find the probability of the union of given n events A1,...,An.The answer is provided, of course, by the inclusion-exclusion formula: Pr(UAi) = ~i Pr(Ai) -~i<j Pr(AiNAj )• .... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, ~P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of all j-wise intersections of the events for j = 1,...k. They developed a method to approximate Pr(UAI) , from the above data with an additive error of exp(-O(k/V~)). In the present article we develop an expression that can be computed in polynomial time, that, given the sums ~lSl=j Pr(niesAi) for j = 1,... k, approximates Pr(UAi) with an additive error of exp(-~(k2/n)). This error is optimal, up to the logarithmic factor implicit in the ~ notation.The problem of enumerating satisfying assignments of a boolean formula in DNF form F= VTlnC i is an instance of the general problem that had been extensively studied [7]. Here A i is the set of assignments that satisfy Ci, and Pr(AiEsAi)=as/2 n where AiEsC i is satisfied by a S assignments. Judging from the general results, it is hard to expect a decent approximation of F's number of satisfying assignments, without knowledge of the numbers a S for, say, all cardinalities 1 <: IS] < v~. Quite surprisingly, already the numbers a S over IS I < log(n+ 1) uniquely determine the number of satisfying assignments for" F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0, 1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Mathematics Subject Classification (1991): 05 A 15, 05 A 16, 05 A 20, 60 C 05, 41 A 10,

Visual cryptography

“…The key combinatorial fact used is the following (see [5,6] Consider now a k out k scheme C with parameters m, a and r. Let the two collections be Co and C1. We construct from the collections two sequences of sets A1, A2, 9 9 9 Ak and B1, B2,.…”

EUV-CDA: Pattern shift aware critical density analysis for EUV mask layouts

“…Computation of all the inclusion exclusion terms comprising NX × NY orders (only first two order are shown above), is a #P -complete combinatorial enumeration problem since it requires the computation of 2 N X ×N Y terms [17]. This computational limitation, along with the quantization error incurred due to the assumption that the pattern shift solution space is discrete, make this method unsuitable for estimating mask yield for any realistic layouts.…”

Non-derivable itemset mining

“…∪ S n . It is exactly this type of problems that is studied in approximate inclusion-exclusion (Galambos and Simonelli 1996;Kahn et al 1996;Melkman and Shimony 1997). Melkman and Shimony (1997) have studied the case in which only the count of the number of items in S 1 ∩ .…”