 
Summary: Algorithmic Construction of Sets for kRestrictions
Noga Alon
Dana Moshkovitz
Shmuel Safra
Abstract
This work addresses krestriction problems, which unify combinatorial problems of the
following type: The goal is to construct a short list of strings in m
that satisfies a given
set of kwise demands. For every k positions and every demand, there must be at least
one string in the list that satisfies the demand at these positions. Problems of this form
frequently arise in different fields in Computer Science.
The standard approach for deterministically solving such problems is via almost kwise
independence or kwise approximations for other distributions. We offer a generic algo
rithmic method that yields considerably smaller constructions. To this end, we generalize
a previous work of Naor, Schulman and Srinivasan [18]. Among other results, we greatly
enhance the combinatorial objects in the heart of their method, called splitters, and con
struct multiway splitters, using a new discrete version of the topological Necklace Splitting
Theorem [1].
We utilize our methods to show improved constructions for group testing [19] and gen
eralized hashing [3], and an improved inapproximability result for SetCover under the
