Balancing pairs and the cross product conjecture
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In a finite partially ordered set, Prob (x>y) denotes the proportion of linear extensions in which elementx appears above elementy. In 1969, S. S. Kislitsyn conjectured that in every finite poset which is not a chain, there exists a pair (x,y) for which 1/3⩽Prob(x>y)⩽2/3. In 1984, J. Kahn and M. Saks showed that there exists a pair (x,y) with 3/11<Prob(x>y)<8/11, but the full 1/3–2/3 conjecture remains open and has been listed among ORDER's featured unsolved problems for more than 10 years.
In this paper, we show that there exists a pair (x,y) for which (5−√5)/10⩽Prob(x>y)⩽(5+√5)/10. The proof depends on an application of the Ahlswede-Daykin inequality to prove a special case of a conjecture which we call the Cross Product Conjecture. Our proof also requires the full force of the Kahn-Saks approach — in particular, it requires the Alexandrov-Fenchel inequalities for mixed volumes.
We extend our result on balancing pairs to a class of countably infinite partially ordered sets where the 1/3–2/3 conjecture isfalse, and our bound is best possible. Finally, we obtain improved bounds for the time required to sort using comparisons in the presence of partial information.
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- Balancing pairs and the cross product conjecture
Volume 12, Issue 4 , pp 327-349
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- Kluwer Academic Publishers
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- Partially ordered set
- linear extension
- balancing pairs
- cross-product conjecture
- Ahlswede-Daykin inequality
- Industry Sectors
- Author Affiliations
- 1. Department of Mathematics, London School of Economics, Houghton Street, WC2A 2AE, London, U.K.
- 2. Fachbereich Mathematik, Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany
- 3. Department of Mathematics, Arizona State University, 85287, Tempe, AZ, U.S.A.