arXiv subjects

Marcos C. Vargas

Publications and source records attributed to Marcos C. Vargas.

4 recordsLinked to original sources

arXiv2020

Computing sandpile configurations using integer linear programming

It is well known that recurrent sandpile configurations can be characterized as the optimal solution of certain optimization problems. In this article, we present two new integer linear programming models, one that computes recurrent configurations and other that computes the order of the configuration. Finally, by using duality of linear programming, we are able to compute the identity configuration for the cone of a regular graph.

math.CO↗

arXiv2019

Eigenvalues, Smith normal form and determinantal ideals

Determinantal ideals of graphs generalize, among others, the spectrum and the Smith normal form (SNF) of integer matrices associated to graphs. In this work we investigate the relationship of the spectrum and the SNF with the determinantal ideals. We show that an eigenvalue divides the $k$-th invariant factor of its SNF if the eigenvalue belongs to a variety of the $k$-th univariate integer determinantal ideal of the matrix. This result has as a corollary a theorem of Rushanan. We also study graphs having the same determinantal ideals with at most one indeterminate; the socalled codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. We establish a necessary and sufficient condition for graphs to be codeterminantal on $\mathbb{R}[x]$, and we present some computational results on codeterminantal graphs up to 9 vertices. Finally, we show that complete graphs and star graphs are determined by the SNF of its distance Laplacian matrix.

math.CO↗

arXiv2018

The equivalence between two classic algorithms for the assignment problem

We give a detailed review of two algorithms that solve the minimization case of the assignment problem. The Bertsekas' auction algorithm and the Goldberg & Kennedy algorithm. We will show that these algorithms are equivalent in the sense that both perform equivalent steps in the same order. We also present experimental results comparing the performance of three algorithms for the assignment problem. They show the auction algorithm performs and scales better in practice than algorithms that are harder to implement but have better theoretical time complexity.

math.OC↗

arXiv2014

Optimum matchings in weighted bipartite graphs

Given an integer weighted bipartite graph $\{G=(U\sqcup V, E), w:E\rightarrow \mathbb{Z}\}$ we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum weight perfect matchings. Moreover, we construct a subgraph $G_{cs}$ of $G$ which depends on an $ε$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$ that allows us to reduced this problems to their unweighed variants on $G_{cs}$. For instance, when $G$ has a perfect matching and we have an $ε$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$, we solve the problem of finding all the edges that occur in some minimum weight perfect matching in linear time on the number of edges. Therefore, starting from scratch we get an algorithm that solves this problem in time $O(\sqrt{n}m\log(nW))$, where $n=|U|\geq |V|$, $m=|E|$, and $W={\rm max}\{|w(e)|\, :\, e\in E\}$.

math.CO↗