Combinatorial and Geometric Approaches to Counting Problems on Linear Matroids, Graphic Arrangements, and Partial Orders

For typical #P-hard problems on graphs, we have recently proposed an approach to solve those problems of moderate size rigorously by means of the binary decision diagram, BDD [12, 13]. This paper extends this approach to counting problems on linear matroids, graphic arrangements and partial orders, most of which are already known to be #P-hard, with using geometric properties. Efficient algorithms are provided to the following problems. Computing the BDD representing all bases of a binary or ternary matroid in an output-size sensitive manner; by using this BDD, the Tutte polynomial of the matroid and the weight enumeration of an (n, k) linear code over GF(2) and GF(3) can be computed in time proportional to the size of the BDD. Computing the Tutte polynomial of a linear matroid over the reals via the arrangement construction algorithm in computational geometry. Computing the number of acyclic orientations of a graph, i.e., the number of cells in the corresponding graphic arrangement, and further the number of its lower-dimensional faces. Computing the number of ideals in a partially ordered set, i.e., the number of some faces of the corresponding cone in the graphic arrangement