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The Dual BKR Inequality and Rudich's Conjecture

Published online by Cambridge University Press:  02 December 2010

JEFF KAHN ,
MICHAEL SAKS  and
CLIFFORD SMYTH
JEFF KAHN
Affiliation:
Mathematics Department, Rutgers University, Piscataway, NJ, USA (e-mail: jkahn@math.rutgers.edu, saks@math.rutgers.edu)
MICHAEL SAKS
Affiliation:
Mathematics Department, Rutgers University, Piscataway, NJ, USA (e-mail: jkahn@math.rutgers.edu, saks@math.rutgers.edu)
CLIFFORD SMYTH
Affiliation:
Mathematics Department, University of North Carolina Greensboro, Greensboro, NC, USA (e-mail: cdsmyth@uncg.edu)
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Abstract

Let be a set of terms over an arbitrary (but finite) number of Boolean variables. Let U() be the set of truth assignments that satisfy exactly one term in . Motivated by questions in computational complexity, Rudich conjectured that there exist ∊, δ > 0 such that, if is any set of terms for which U() contains at least a (1−∊)-fraction of all truth assignments, then there exists a term t ∈ such that at least a δ-fraction of assignments satisfy some term of sharing a variable with t [8].

We prove a stronger version: for any independent assignment of the variables (not necessarily the uniform one), if the measure of U() is at least 1 − ∊, there exists a t ∈ such that the measure of the set of assignments satisfying either t or some term incompatible with t (i.e., having no satisfying assignments in common with t) is at least . (A key part of the proof is a correlation-like inequality on events in a finite product probability space that is in some sense dual to Reimer's inequality [11], a.k.a. the BKR inequality [5], or the van den Berg–Kesten conjecture [3].)

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 257 - 266
Copyright
Copyright © Cambridge University Press 2010

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