### Rutgers Discrete Mathematics Seminar

Title: Nonnegative k-sums, fractional covers, and probability of small deviations

Speaker: **Benny Sudakov**, UCLA

Date: Tuesday, September 27, 2011 2:00pm

Location: Hill Center, Room 425, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:
More than twenty years ago, Manickam, Miklos, and Singhi
conjectured that for any integers $n geq 4k$, every set of $n$ real
numbers with nonnegative sum has at least $n-1 choose k-1$ $k$-element
subsets whose sum is also nonnegative. In this talk we discuss the
connection of this problem with matchings and fractional covers of
hypergraphs, and with the question of Feige and Samuels on estimating
the probability that the sum of nonnegative independent random
variables exceeds its expectation by a given amount. Using these
connections together with some probabilistic techniques, we verify the
conjecture for $n geq 33k^2$. This substantially improves the best
previously known exponential lower bound on $n$. If times permits we
also mention application of our technique to several other problems.

Joint work with N. Alon and H. Huang.

See: http://math.rutgers.edu/seminars/allseminars.php?sem_name=Discrete%20Math