Autonomous and Optimized Ship Routing

We present state-of-the-art computational methods which are instrumental in autonomous maritime operations, and optimization of routing, scheduling as well as loading. Our aim is to survey mature algorithmic approaches developed within the Lab of Geometric and Algebraic Algorithms, towards exploiting intelligence and automation in modern shipping and, in particular, in various aspects of routing. We showcase our advances in two main axes: (a) geometric computing for collision avoidance in complex environments, thus allowing for semi-autonomous and fully autonomous navigation, and (b) optimization for routing under time constraints of the carrier ship, time windows of availability at the ports of call, and capacity constraints of various compartments within a vessel.

Gradient Boosts the Approximate Vanishing Ideal

In the last decade, the approximate vanishing ideal and its basis construction algorithms have been extensively studied in computer algebra and machine learning as a general model to reconstruct the algebraic variety on which noisy data approximately lie. In particular, the basis construction algorithms developed in machine learning are widely used in applications across many fields because of their monomial-order-free property; however, they lose many of the theoretical properties of computer-algebraic algorithms. In this paper, we propose general methods that equip monomial-order-free algorithms with several advantageous theoretical properties. Specifically, we exploit the gradient to (i) sidestep the spurious vanishing problem in polynomial time to remove symbolically trivial redundant bases, (ii) achieve consistent output with respect to the translation and scaling of input, and (iii) remove nontrivially redundant bases. The proposed methods work in a fully numerical manner, whereas existing algorithms require the awkward monomial order or exponentially costly (and mostly symbolic) computation to realize properties (i) and (iii). To our knowledge, property (ii) has not been achieved by any existing basis construction algorithm of the approximate vanishing ideal.

Algebraic Algorithms for Even Circuits in Graphs

We present an algebraic algorithm to detect the existence of and to list all indecomposable even circuits in a given graph. We also discuss an application of our work to the study of directed cycles in digraphs.

Computing all border bases for ideals of points

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

Shedding New Light on Common Algorithms: What can we Learn from Vedic Mathematics?

In Sanskrit, the ancient Hinduism language, ‘Vedas’ means ‘knowledge’. The Vedas are a corpus of more than 1,000,000 ancient philosophical writings divided into Sutras, some of which deal with mathematics. These mathematics Sutras, termed ‘Vedic Mathematics’, concern various fields of mathematics. The Vedic methods are coherent, logical and simple, and students enjoy practicing them. Besides 'spicing up' the regular mathematics lessons by integrating some of the Vedic algorithms, engaging students in proving them supports the development of their insights regarding the rationale underlying the formal rules and algorithms included in the curriculum. In this workshop, we present some of the basic Vedic arithmetic and algebraic algorithms, involve the participants in proving the them and discuss the advantages and disadvantages of integrating Vedic mathematics into classes at different age groups and study levels.