On the complexity of optimal homotopies

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

Inductive queries for a drug designing robot scientist

It is increasingly clear that machine learning algorithms need to be integrated in an iterative scientific discovery loop, in which data is queried repeatedly by means of inductive queries and where the computer provides guidance to the experiments that are being performed. In this chapter, we summarise several key challenges in achieving this integration of machine learning and data mining algorithms in methods for the discovery of Quantitative Structure Activity Relationships (QSARs). We introduce the concept of a robot scientist, in which all steps of the discovery process are automated; we discuss the representation of molecular data such that knowledge discovery tools can analyse it, and we discuss the adaptation of machine learning and data mining algorithms to guide QSAR experiments

Elastic Registration of Geodesic Vascular Graphs

Vascular graphs can embed a number of high-level features, from morphological parameters, to functional biomarkers, and represent an invaluable tool for longitudinal and cross-sectional clinical inference. This, however, is only feasible when graphs are co-registered together, allowing coherent multiple comparisons. The robust registration of vascular topologies stands therefore as key enabling technology for group-wise analyses. In this work, we present an end-to-end vascular graph registration approach, that aligns networks with non-linear geometries and topological deformations, by introducing a novel overconnected geodesic vascular graph formulation, and without enforcing any anatomical prior constraint. The 3D elastic graph registration is then performed with state-of-the-art graph matching methods used in computer vision. Promising results of vascular matching are found using graphs from synthetic and real angiographies. Observations and future designs are discussed towards potential clinical applications

Designing antibiotic cycling strategies by determining and understanding local adaptive landscapes

The evolution of antibiotic resistance among bacteria threatens our continued ability to treat infectious diseases. The need for sustainable strategies to cure bacterial infections has never been greater. So far, all attempts to restore susceptibility after resistance has arisen have been unsuccessful, including restrictions on prescribing [1] and antibiotic cycling [2,3]. Part of the problem may be that those efforts have implemented different classes of unrelated antibiotics, and relied on removal of resistance by random loss of resistance genes from bacterial populations (drift). Here, we show that alternating structurally similar antibiotics can restore susceptibility to antibiotics after resistance has evolved. We found that the resistance phenotypes conferred by variant alleles of the resistance gene encoding the TEM {\beta}-lactamase (blaTEM) varied greatly among 15 different {\beta}-lactam antibiotics. We captured those differences by characterizing complete adaptive landscapes for the resistance alleles blaTEM-50 and blaTEM-85, each of which differs from its ancestor blaTEM-1 by four mutations. We identified pathways through those landscapes where selection for increased resistance moved in a repeating cycle among a limited set of alleles as antibiotics were alternated. Our results showed that susceptibility to antibiotics can be sustainably renewed by cycling structurally similar antibiotics. We anticipate that these results may provide a conceptual framework for managing antibiotic resistance. This approach may also guide sustainable cycling of the drugs used to treat malaria and HIV