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

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we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes.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

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4