Inclusion-exclusion: Exact and approximate

Abstract

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(∪A _i )=∑ _i −∑ _i<j Pr(A _i ∩A _j )±.... 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 allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(∪A _i ), from the above data with an additive error of exp$( - O(k/\sqrt n ))$. In the present article we develop an expression that can be computed in polynomial time, that, given the sums ∑_|S|=j Pr(∩ _i∈S A _i ) forj=1,...k, approximates Pr(∪A _i ) with an additive error of exp$( - \bar \Omega (k^2 /n))$. This error is optimal, up to the logarithmic factor implicit in the$\bar \Omega$ notation.

The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v ^m_l C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr(∩ _i∈S A _i )=a _S /2ⁿ where ∧ _i∈S C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities$1 \leqslant |S| \leqslant \sqrt m$. Quite surprisingly, already the numbersa _S over |S|≤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.