linear programming duality
Recently Published Documents


TOTAL DOCUMENTS

25
(FIVE YEARS 0)

H-INDEX

4
(FIVE YEARS 0)

10.37236/4769 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Ehud Friedgut ◽  
Jeff Kahn ◽  
Clara Shikhelman

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $\mu_p$ on the power set of $N$, where $\mu_p(\cal{A}) := Pr[N_p \in \cal{A}]$ for $\cal{A} \subseteq 2^N$.In this paper we study monotone (upward-closed) families $\cal{A}$ for which all minimal sets in $cal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $\mu_p(\cal{A}) / p^k $ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables one to identify in $\cal{A}$ either a substantial subfamily $\cal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$.Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using linear programming duality. This is related to the threshold conjecture of Kahn and Kalai.


2014 ◽  
Vol 6 (1) ◽  
pp. 21-32 ◽  
Author(s):  
Allan B. Cruse

Abstract H. G. Landau has characterized those integer-sequences S = (s1, s2, . . . , sn) which can arise as score-vectors in an ordinary round-robin tournament among n contestants [17]. If s1 ≤ s2 ≤ · · · ≤ sn, the relevant conditions are expressed simply by the inequalities:


CATBox ◽  
2009 ◽  
pp. 33-52
Author(s):  
Winfried Hochstättler ◽  
Alexander Schliep

Sign in / Sign up

Export Citation Format

Share Document