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


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.