Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square

An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an $n\times n\times n$ ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page

The 2-hop spanning tree problem

Given a graph G with a specified root node r. A spanning tree in G where each node has distance at most 2 from r is called a 2-hop spanning tree. For given edge weights the 2-hop spanning tree problem is to find a minimum weight 2-hop spanning tree. The problem is NP-hard and has some interesting applications. We study a polytope associated with a directed model of the problem give a completeness result for wheels and a vertex description of a linear relaxation. Some classes of valid inequalities for the convex hull of incidence vectors of 2-hop spanning trees are derived by projection techniques

Majorization polytopes

AbstractWe study polytopes related to the concept of matrix majorization: for two real matrices A and B having m rows we say that A majorizes B if there is a row-stochastic matrix X with AX=B. In that case we write A≻B and the associated majorization polytope M(A≻B) is the set of row stochastic matrices X such that AX=B. We investigate some properties of M(A≻B) and obtain e.g., generalizations of some results known for vector majorization. Relations to transportation polytopes and network flow theory are discussed. A complete description of the vertices of majorization polytopes is found for some special cases

Polytopes related to interval vectors and incidence matrices

AbstractIn this short note we investigate polytopes associated with families of interval vectors, i.e., (0,1)-vectors with consecutive ones. Using a linear transformation we show a connection to “extended” incidence matrices of acyclic directed graphs and the convex hull of their columns. This leads to complete linear descriptions of the corresponding polytopes

Combinatorial Fiedler Theory and Graph Partition

Partition problems in graphs are extremely important in applications, as shown in the Data science and Machine learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue $a(G)$ of the Laplacian matrix $L_G$ of the graph $G$. This problem corresponds to the minimization of a quadratic form associated with $L_G$, under certain constraints involving the $\ell_2$-norm. We introduce and investigate a similar problem, but using the $\ell_1$-norm to measure distances. This leads to a new parameter $b(G)$ as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for $b(G)$ for trees. We also comment on an $\ell_{\infty}$-norm version of the problem

Sign-restricted matrices of 00's, 11's, and −1-1's

We study {\em sign-restricted matrices} (SRMs), a class of rectangular $(0, \pm 1)$-matrices generalizing the alternating sign matrices (ASMs). In an SRM each partial column sum, starting from row 1, equals 0 or 1, and each partial row sum, starting from column 1, is nonnegative. We determine the maximum number of nonzeros in SRMs and characterize the possible row and column sum vectors. Moreover, a number of results on interchange operations are shown, both for SRMs and, more generally, for $(0, \pm 1)$-matrices. The Bruhat order on ASMs can be extended to SRMs with the result a distributive lattice. Also, we study polytopes associated with SRMs and some relates decompositions