Carolina Mejía

Linear network coding and the model theory of linear rank inequalities

Let n ≥ 4. Can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We showed, in a companion paper, that if it were the case one could solve many algorithmic problems coming from network coding, index coding and secret sharing. Unfortunately, it seems that the entropic regions are not semialgebraic. Are the Ingleton regions semialgebraic sets? We provide some evidence showing that the Ingleton regions are semialgebraic. Furthermore, we show that if the Ingleton regions are semialgebraic, then one can solve many algorithmic problems coming from Linear Network Coding.

On the complexity of sandpile critical avalanches

In this work, we study The Abelian Sandpile Model from the point of view of computational complexity. We begin by studying the length distribution of sandpile avalanches triggered by the addition of two critical configurations: we prove that those avalanches are long on average, their length is bounded below by a constant fraction of the length of the longest critical avalanche which is, in most of the cases, superlinear. At the end of the paper we take the point of view of computational complexity, we analyze the algorithmic hardness of the problem consisting in computing the addition of two critical configurations, we prove that this problem is P complete, and we prove that most algorithmic problems related to The Abelian Sandpile Model are NC reducible to it.

On the Complexity of Sandpile Prediction Problems

In this work we study the complexity of Sandpile prediction problems on several classes of directed graphs. We focus our research on low-dimensional directed lattices. We prove some upper and lower bounds for those problems. Our approach is based on the analysis of some reachability problems related to sandpiles.

Network coding and the model theory of linear information inequalities

Let n ≥ 4, can the entropic region of order n be defined by a finite list of polynomial inequalities? This question was first asked by Chan and Grant. We show that if it were the case one could solve many algorithmic problems coming from Network Coding, Index Coding and Secret Sharing. Unfortunately, it seems that the entropic regions of order larger than four are not semialgebraic. Actually, we guess that it is the case and we provide strong evidence supporting our conjecture.

On the linear algebra of biological homochirality

We look for structural properties of chemical networks giving place to homochiral phenomena. We found a necessary condition for homochirality that we call Frank inequality, and which is a linear inequality related to the entries of the jacobian matrices that occur at racemic steady states. We also investigate the existence of stronger conditions that can be formulated in a similar algebraic way. Those investigations lead us to introduce a homochirality degree for the racemic states of chiral neworks, which is intended to measure the probability of observing homochiral dynamics after perturbing those states. It is important to stress that all the introduced concepts and degrees are effective. The later fact allows us to develop an algorithm that can be used to, given a chiral network as input, compute large samples of steady states of different degrees.

Defining the almost-entropic regions by algebraic inequalities

We study the definability of the almost-entropic regions by finite lists of algebraic inequalities. First, we study linear information inequalities and polyhedrality, we present a proof of a theorem of Matus, which claims that the almost-entropic regions are not polyhedral. Then, we study polynomial inequalities and semialgebraicity, we show that the semialgebracity of the almost-entropic regions is something that depends on the essentially conditionality of a certain class of conditional information inequalities. Those results suggest that the almost-entropic regions are not semialgebraic. We conjecture that those regions are not decidable.

Some Remarks Concerning the Algorithmic Analysis of Gene Regulatory Networks

In this work we study an algorithmic problem related to gene regulatory networks. This problem is the counting of fixed points in boolean networks. We focus our attention on monomial networks, and we prove that the counting of fixed points is #P complete even in this restricted case.

The complexity of three-dimensional critical avalanches

In this work we study the complexity of the three-dimensional sandpile avalanches triggered by the addition of two critical configurations. We prove that the algorithmic problem consisting in predicting the evolution of three dimensional critical avalanches is the hardness core of the three-dimensional Abelian Sandpile Model. On the other hand we prove that three-dimensional critical avalanches are superlinear long on average. It suggests that the prediction problem is superlinear-hard on average.