## Section: New Results

### Algorithmic Foundations

Computational geometry, Computational topology, Voronoi diagrams, $\alpha $-shapes, Morse theory.

#### Mass Transportation Problems with Connectivity Constraints

Participant : Frédéric Cazals.

In collaboration with D. Mazauric (Inria Sophia Antipolis Méditerranée, Geometrica).

Given two graphs, the supply and the demand graphs, we analyze the mass transportation problem between their vertices, under connectivity constraints [16] . More precisely, for every subset of supply nodes inducing a connected component of the supply graph, we require that the set of demand nodes receiving non-zero flow from this subset induces a connected component of the demand graph. As opposed to the classical problem, a.k.a the earth mover distance (EMD), which is amenable to linear programming (LP), this new problem is very difficult to solve, and we make four contributions. First, we formally introduce two optimal transportation problems, namely minimum-cost flow under connectivity constraints problem (EMD-CC) and maximum-flow under cost and connectivity constraints problem (EMD-CCC). Second, we prove that the decision version of EMD-CC is NP-complete even for very simple classes of instances. We deduce that the decision version of EMD-CCC is NP-complete, and also prove that EMD-CC is not in APX even for simple classes of instances. Third, we develop a greedy heuristic algorithm returning admissible solutions, of time complexity $O\left({n}^{3}{m}^{2}\right)$ with n and m the numbers of vertices of the supply and demand graphs, respectively. Finally, on the experimental side, we compare the transport plans computed by our greedy method against those produced by the aforementioned LP. Using synthetic landscapes (Voronoi landscapes), we show that our greedy algorithm is effective for graphs involving up to 1000 nodes. We also show the relevance of our algorithms to compare energy landscapes of biophysical systems (protein models).

#### Ciruvis: a web-based Tool for Rule Networks and Interaction Detection using Rule-based Classifiers

Participant : Simon Marillet.

In collaboration with J. Komorowski and S. Bornelöv (Uppsala University).

The use of classification algorithms is becoming increasingly important for the field of computational biology. However, not only the quality of the classification, but also its biological interpretation is important. This interpretation may be eased if interacting elements can be identified and visualized, something that requires appropriate tools and methods.

We developed a new approach to detecting interactions in complex systems based on classification [12] . Using rule-based classifiers, we previously proposed a rule network visualization strategy that may be applied as a heuristic for finding interactions. We now complement this work with Ciruvis, a web-based tool for the construction of rule networks from classifiers made of IF-THEN rules. Simulated and biological data served as an illustration of how the tool may be used to visualize and interpret classifiers. Furthermore, we used the rule networks to identify feature interactions, compared them to alternative methods, and computationally validated the findings. Rule networks enable a fast method for model visualization and provide an exploratory heuristic to interaction detection. The tool is made freely available on the web and may thus be used to aid and improve rule-based classification.