TY - GEN
T1 - On the beck-fiala conjecture for random set systems
AU - Ezra, Esther
AU - Lovett, Shachar
N1 - Funding Information: E.E is supported by an NSF CAREER award 1553354. S.L. is supported by an NSF CAREER award 1350481 and a Sloan fellowship.
PY - 2016/9/1
Y1 - 2016/9/1
N2 - Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X, Σ), where each element x € X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X, Σ) is with high probability O(p t log t); and when |X| > |Σ|t the hereditary discrepancy of (X, Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
AB - Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X, Σ), where each element x € X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X, Σ) is with high probability O(p t log t); and when |X| > |Σ|t the hereditary discrepancy of (X, Σ) is with high probability O(1). The first bound combines the Lovász Local Lemma with a new argument based on partial matchings; the second follows from an analysis of the lattice spanned by sparse vectors.
KW - Beck-Fiala conjecture
KW - Discrepancy theory
KW - Random set systems
UR - https://www.scopus.com/pages/publications/84990842332
U2 - 10.4230/LIPIcs.APPROX-RANDOM.2016.59
DO - 10.4230/LIPIcs.APPROX-RANDOM.2016.59
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016
A2 - Jansen, Klaus
A2 - Mathieu, Claire
A2 - Rolim, Jose D. P.
A2 - Umans, Chris
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016
Y2 - 7 September 2016 through 9 September 2016
ER -