On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.Comment: 26 pages, proofs simplified, results strengthene

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

On the number of different permanents of some sparse (0,1) circulant matrices

Starting from known results about the number of possible values for the permanents of $(0,1)$-circulant matrices with three nonzero entries per row, and whose dimension $n$ is prime, we prove corresponding results for $n$ power of a prime, $n$ product of two distinct primes, and $n=2cdot 3^h$. Supported by some experimental results, we also conjecture that the number of different permanents of $ntimes n$ $(0,1)$-circulant matrices with $k$ nonzero per row is asymptotically equal to $n^{k-2}/k!+O(n^{k-3}).

Independence Models for Integer Points of Polytopes.

The integer points of a high-dimensional polytope P are generally difficult to count or sample uniformly. We consider a class of low-complexity random models for these points which arise from an entropy maximization problem. From these models, by way of "anti-concentration" results for sums of independent random variables, we derive general, efficiently computable upper bounds on the number of integer points of P. We make a detailed study of contingency tables with bounded entries, which are the integer points of a transportation polytope truncated by a cuboid. We provide efficiently computable estimates for the logarithm of the number of m by n tables with specified row and column sums r_1, ..., r_m, c_1, ..., c_n and bounds on the entries. These estimates are asymptotic as m and n go to infinity simultaneously, given that no r_i (resp., c_j) is allowed to exceed a fixed multiple of the average row sum (resp., column sum). As an application, we consider a random, uniformly selected table with entries in {0, 1, ..., kappa} having a given sum. Responding to questions raised by Diaconis and Efron in the context of statistical significance testing, we show that the occurrence of row sums r_1, ..., r_m is positively correlated with the occurrence of column sums c_1, ..., c_n when kappa > 1 and r_1, ..., r_m, c_1, ..., c_n are sufficiently extreme. We give evidence that the opposite is true for near-average values of r_1, ..., r_m, c_1, ..., c_n.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86295/1/auspex_1.pd