Advertisement

Combinatorica

, Volume 16, Issue 4, pp 465–477 | Cite as

Inclusion-exclusion: Exact and approximate

  • Jeff Kahn
  • Nathan Linial
  • Alex Samorodnitsky
Article

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|=jPr(∩ 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 l m 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.

Mathematics Subject Classification (1991)

05 A 15 05 A 16 05 A 20 60 C 05 41 A 10 68 R 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Boole:An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities, Dover 1st printing, 1854.Google Scholar
  2. [2]
    J. Galambos, andT. Xu: A new method for generating Bonferroni-type inequalities by iteration,Math. Proc. Cambridge Philos. Soc.,107 (3), (1990), 601–607.Google Scholar
  3. [3]
    M. Kearns, Y. Mansour, D. Ron, R. Rubinfeld, R. E. Schapire andL. Sellie: On the learnability of discrete distributions,STOC,26 (1994), 273–282.Google Scholar
  4. [4]
    N. Linial andN. Nisan: Approximate inclusion-exclusion,Combinatorica,10 (1990), 349–365.Google Scholar
  5. [5]
    L. Lovász: A note on the line reconstruction problem,J. Comb. Theory [B],13 (1972), 309–310.Google Scholar
  6. [6]
    L. Lovász:Combinatorial Problems and Exercises, North Holland, 1979.Google Scholar
  7. [7]
    M. Luby andB. Velickovic: On deterministic approximation of DNF,STOC,23 (1991), 430–438.Google Scholar
  8. [8]
    W. Müller: The edge reconstruction hypothesis is true for graphs with more thannlog2 n edges,J. Comb. Theory [B],22 (1977), 281–283.Google Scholar
  9. [9]
    H. J. Ryser:Combinatorial Mathematics, The Mathematical Association of America, 1963.Google Scholar
  10. [10]
    V. N. Vapnik andA. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theoret. Probl. and Its Appl.,16 (2) (1971), 264–280.Google Scholar

Copyright information

© Akadémiai Kiadó 1996

Authors and Affiliations

  • Jeff Kahn
    • 1
  • Nathan Linial
    • 3
  • Alex Samorodnitsky
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Institute of MathematicsHebrew UniversityJerusalemIsrael
  3. 3.Institute of Computer ScienceHebrew UniversityJerusalemIsrael

Personalised recommendations