# Combinatorics of Networks and Computation

## Periodic Reporting for period 1 - CONNECT (Combinatorics of Networks and Computation)

Reporting period: 2017-01-01 to 2018-12-31

Networks are a fundamental part of modern society; they can be the byproduct of the interaction of agents spread worldwide; they might be the vast communication and transportation networks we have built during the past century; and they appear in the most unexpected forms. In the Combinatorics of Networks and Computation (CONNECT) project we study the fundamental properties of networks. We use tools from diverse areas of Mathematics and Computer Science to obtain new insights into these fascinating objects. The consortium involves researchers and graduate students of 14 universities from Austria, Belgium, Canada, Chile, Czech Republic, Italy, Mexico, and Spain.

The activities of the project are divided into 5 work packages.

Geometric Networks(WP1)
Many networks have a geometric component; their nodes might have geographical location or the links between two nodes might be straight line segments. Work package 1 studies how these geometric constrains affect the properties of networks.

Randomness and Imprecise Knowledge in Networks(WP2)
Randomness and uncertainty are facts of life. In the case of networks, the location of a node in a network might be random; or our information on a real life network might be uncertain. Work package 2 studies how networks behave under uncertainty and randomness assumptions; we also design algorithms to deal with this uncertainty.

Restricted Orientation Geometry(WP3)
Manufacturing imposes geometric restrictions. For example, in a circuit each link is often laid out in one of two possible orientations. Work package 3 studies how geometric restrictions affect the properties of networks.

Graph-based algorithms for UAVs and for Musical Information Retrieval(WP4)
Graph-Based Models have been considered in many application areas. For instance, consider in the robotic area a search problem in which a cooperating team of agents is searching for a target in a bounded region. Typically, the approach is to reduce the continuous problem to an optimization on a finite graph and the paths are computed on this graph. In WP4, the goal is to take advance of our expertise in graph theory and algorithms in graphs for the design of efficient algorithms for two application areas, Aerial Robotics (AR) and MIR (Musical Information Retrieval).

Dissemination and Gender Equality Promotion(WP5)
We believe that social outreach is a crucial part of the scientific endeavor. We seek to make our results accessible to a wider audience.
It is of particular interest to promote gender equality in Mathematics. Women continue to be underrepresented in the field of mathematical research. In this work package we take specific actions to address this issue.
At an individual level, research has been carried out through visits between the project participants. As a collective effort, we have organized 9 workshops and conferences. More than 50 joint publications will be produced as a direct result of these actions. Given the volume of results obtained, we are forced to mention only some of them.

WP1

How many different plane networks can be defined on n sites in the plane, provided that each link must be a straight line segment? We gave a new lower bound on this number.

We have improved the upper bound on the minimum number of pairs of links that must cross in complete straight line network. This is an important problem in Combinatorial Geometry. This was made possible with new software developed by us. The software has been made publicly available.

WP2

We have been working in combinatorial optimization problems on stochastic elements. We have studied the algorithmic problem of finding the maximum enclosing box on points in the line or the plane, where each point has assigned a weight and a probability of being present.

WP3

The convex hull of a point set is the minimum polygon that encloses the point set. A variant called the rectilinear convex hull requires that the polygon has vertical or horizontal sides. We obtained an algorithm for maintaining the rectilinear convex hull while the point set is rotated.

We gave an algorithm for finding the angle of rotation of a polygon so that it encloses as many points as possible from a given set of points. Applications of polygon placement problems include global localization of mobile robots, pattern matching, and geometric tolerance.

We computed the region (called the kernel) of a polygon from which every other point of the polygon can be seen, subject to restrictions on this visibility. This has applications in surveillance scenarios.

WP4

We designed algorithms to assign flight plans for drones equipped with cameras, in order to film a prescribed set of scenes. The real life application of this algorithm is for sport events; in particular for bicycle races; in which the center of attention is constantly in motion. A key feature of the algorithms is that they use algorithms for computing flows in networks to find the flight plans.

Using algorithms for network partitioning, we designed an algorithm to classify flamenco melodies.

Using algorithms for shorted paths in directed networks, we designed an algorithm to compute melodic templates for flamenco songs.

WP5

Dissemination of results has followed several lines. First, several academic talks have spread results of the project to a specialized audience in universities, workshops, and conferences. Second, dissemination talks covering results of the project were included in activities like the “Pint of Science” festival, the MATRIX conference, or a satellite activity of the Meeting of the Spanish Royal Mathematical Society. Third, dissemination articles have been written at the Mapping Ignorance journal (ISSN 2529-8992). Fourth, and final, dissemination articles on newspapers (like ABC or The Conversation) have helped to reach a broad public explaining mathematics to some extent related to the project.

To promote gender equality, we are organizing a summer school for women in mathematics. The main objective is to encourage female students to continue with graduate studies in mathematics. Various advanced courses and talks have been scheduled. The event will take place the city of Zacatecas, Mexico in 2019.
We mention some of the real world applications that derive from our research.

We have made significant progress on the study of crossings in drawings of networks with straight links. In the case of complete networks in which the links can be drawn in two layers, the only crossings that matter are those between links of the same layer. In transportation networks a layer might consist of roads and another layer of underground rails. Our obtained methods can be streamlined and applied to any straight line network. This can have important applications in optimization of real life networks.

The algorithms designed to extract melodies from flamenco music have been implemented. With extra work an app can be created that uses these algorithms as a backend. Our algorithms could then be made available to the general consumer.

Dissemination is an important part of CONNECT. Various actions have taken place to reach the general public. We hope that this allows for our results to be better known outside the scientific community. We hope that the events reared towards students and particulary towards women will help to make sure that more of them stay in academia.