ABSTRACT:We give polynomial time algorithms for random sampling from a set of contingency tables, which is the set of m = n matrices with given row and column sums, provided the row and column sums are sufficiently large with respect to m, n. We use this to approximately count the number of such matrices. These problems are of interest in

IcePick is a system for computationally selecting diverse sets of molecules. It computes the dissimilarity of the surface-accessible features of two molecules, taking into account conformational flexibility. Then, the intrinsic diversity of an entire set of molecules is calculated from a spanning tree over the pairwise dissimilarities. IcePick's dissimilarity measure is compared against traditional 2D topological approaches, and the spanning tree diversity measure is compared against commonly used variance techniques. The method has proven easy to implement and is fast enough to be used in selection of reactants for numerous production-sized combinatorial libraries.

The use of combinatorial chemistry has become commonplace within the pharmaceutical industry. Less widespread but gaining in popularity is the derivation of activity models from the high-throughput assays of these libraries. Such models are then used as filters during the design of refined daughter libraries. The design of these second generation libraries, which efficiently test and conform to the derived activity model from the large space of virtual possibilities, remains an area of considerable research. We present here a computationally efficient method for the design of optimally dense (in model matching compounds) synthetic matrices from in silico virtual libraries.

This paper describes methods for counting the number of nonnegative integer solutions of the system Ax = b when A is a nonnegative totally unimodular matrix and b an integral vector of fixed dimension. The complexity (under a unit cost arithmetic model) is strong in the sense that it depends only on the dimensions of A and not on the size of the entries of b. For the special case of ‘contingency tables’ the run-time is 2O(√dlogd) (where d is the dimension of the polytope). The method is complementary to Barvinok's in that our algorithm is effective on problems of high dimension with a fixed number of (non-sign) constraints, whereas Barvinok's algorithms are effective on problems of low dimension and an arbitrary number of constraints.

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