Wansoo T. Rhee

Optimal bin packing with items of random sizes

Consider a probability μ on [0, 1] and i.i.d. random variables X1, X2, …, Xn distributed like μ. Let Qn denote the optimal (minimum) number of unit size bins needed to pack items of size X1, X2, …, Xn. We characterize the class of μ which have the property that limn→∞ Qn/n = E(X1) a.s., or equivalently that the expected level of occupancy of bins converges to one.

A note on the selection of random variables under a sum constraint

Consider an i.i.d. sequence of non-negative random variables ( X 1 , · ··, X n ) with known distribution F. Consider decision rules for selecting a maximum number of the subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. Coffman et al. (1987) proved that there exists such a rule that maximizes the expected number E n ( c ) of variables selected, and determined the asymptotics of E n ( c ) for special distributions. Here we determine the asymptotics of E n ( c n ) for very general choices of sequences ( c n ) and of F, by showing that E n ( c ) is very close to an easily computable number. Our proofs are (somewhat deceptively) very simple, and rely on an appropriate stopping-time argument.

Exact bounds for the stochastic upward matching problem

We draw at random independently and according to the uniform distribution two sets of n points of the unit square We consider a maximum matching of points of the first set with points of the second set with the restriction that a point can be matched only with a point located at its upper right. Then with probability close to one, the number of unmatched points is of order nl/2 (log n)3/4 .

Optimal bin packing with items of random sizes III

Consider a probability measure

Uniform convexity and the distribution of the norm for a Gaussian measure

\mu

Uniform bound in the central limit theorem for Banach space valued dependent random variables

on

Stochastic analysis of the quadratic assignment problem

[0,1]

Some distributions that allow perfect packing

and independent random variables

On-line bin packing of items of random sizes, II

X_1 , \cdots ,X_n

A note on asymptotic properties of the quadratic assignment problem

distributed according to