Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

A computationally intractable problem on simplicial complexes

AbstractWe analyze the problem of computing the minimum number er(C>) of internal simplexes that need to be removed from a simplicial 2-complex C so that the remaining complex can be nulled by deleting a sequence of external simplexes. We show that the decision version of this problem is NP-complete even when C is embeddable in 3-dimensional space. Since the Betti numbers of C can be computed in polynomial time, this implies that there is no polynomial time computable formula for er(C) in terms of the Betti numbers of the complex, unless p = NP. The problem can be solved in linear time for 1-complexes (graphs).Our reduction can also be used to show that the corresponding approximation problem is at least as difficult as the one for the minimum cardinality vertex cover, and what is worse, as difficult as the minimum set cover problem. Thus simple heuristics may generate solutions that are arbitrarily far from optimal

Algebraic Statistics

Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this thesis, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including: - An output-sensitive algorithm that computes the maxima for a set of n points in two dimensions using 1n log(h) + O(n sqrt(log(h))) comparisons, where h is the size of the output. - A randomized algorithm that computes the maxima in three dimensions that uses 1n log(n) + O(n sqrt(log(n))) expected number of comparisons. - A randomized output-sensitive algorithm that computes the maxima in three dimensions that uses 1n log(h) + O(n log^(2/3)(h)) expected number of comparisons, where h is the size of the output. - An output-sensitive algorithm that computes the convex hull for a set of n points in two dimensions using 1n log(h) + O(n sqrt(log(h))) comparisons and O(n sqrt(log(h))) sidedness tests, where h is the size of the output. - A randomized algorithm for detecting whether of a set of n horizontal and vertical line segments in the plane intersect that uses 1n log(n) +O(n sqrt(log(n))) expected number of comparisons. - A data structure for point location among n axis-aligned disjoint boxes in three dimensions that answers queries using at most (3/2)log(n)+ O(log(log(n))) comparisons. The data structure can be extended to higher dimensions and uses at most (d/2)log(n)+ O(log(log(n))) comparisons. - A data structure for point location among n axis-aligned disjoint boxes that form a space-filling subdivision in three dimensions that answers queries using at most (4/3)log(n)+ O(sqrt(log(n))) comparisons. The data structure can be extended to higher dimensions and uses at most ((d+1)/3)log(n)+ O(sqrt(log(n))) comparisons. Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees