Abstract

The subject of this paper is finding small sample spaces for joint distributions of n discrete random variables. Such distributions are often only required to obey a certain limited set of constraints of the form Pr(E)=. We show that the problem of deciding whether there exists any distribution satisfying a given set of constraints is NP-hard. However, if the constraints are consistent, then there exists a distribution satisfying them which is supported by a "small" sample space (one whose cardinalityis equal to the number of constraints). For the important case of independenceconstraints,where the constraints have a certain form and are consistent with a joint distribution of n independent random variables, a small sample space can be constructed in polynomial time. This last result is also useful for de-randomizing algorithms. We demonstrate this technique by an application to the problem of finding large independentsetsin sparse hypergraphs.

Citations