Evolutionary algorithms, being problem-independent and randomized heuristics, are generally believed to be robust to dynamic changes and noisy access to the problem instance. We propose a new method to obtain rigorous runtime results for such settings. In contrast to many previous works, our new approach mostly relies on general parameters of the dynamics or the noise models, such as the expected change of the dynamic optimum or the probability to have a dynamic change in one iteration. Consequently, we obtain bounds which are valid for large varieties of such models. Despite this generality, for almost all particular models regarded in the past our bounds are stronger than those given in previous works. As one particular result, we prove that the (1 + λ) EA can optimize the OneMax benchmark function efficiently despite a constant rate of 1-bit flip noise. For this, a logarithmic size offspring population suffices (the previous-best result required a super-linear value of λ). Our results suggest that the typical way to find the optimum in such adverse settings is not via a steady approach of the optimum, but rather via an exceptionally fast approach after waiting for a rare phase of low dynamic changes or noise.

Abstract In this paper, we consider the problem of information transfer across shapes and propose an extension to the widely used functional map representation. Our main observation is that in addition to the vector space structure of the functional spaces, which has been heavily exploited in the functional map framework, the functional algebra (i.e., the ability to take pointwise products of functions) can significantly extend the power of this framework. Equipped with this observation, we show how to improve one of the key applications of functional maps, namely transferring real‐valued functions without conversion to point‐to‐point correspondences. We demonstrate through extensive experiments that by decomposing a given function into a linear combination consisting not only of basis functions but also of their pointwise products, both the representation power and the quality of the function transfer can be improved significantly. Our modification, while computationally simple, allows us to achieve higher transfer accuracy while keeping the size of the basis and the functional map fixed. We also analyze the computational complexity of optimally representing functions through linear combinations of products in a given basis and prove NP‐completeness in some general cases. Finally, we argue that the use of function products can have a wide‐reaching effect in extending the power of functional maps in a variety of applications, in particular by enabling the transfer of high‐frequency functions without changing the representation size or complexity.

We consider the problem of non-rigid shape matching, and specifically the functional maps framework that was recently proposed to find correspondences between shapes. A key step in this framework is to formulate descriptor preservation constraints that help to encode the information (e.g., geometric or appearance) that must be preserved by the unknown map. In this paper, we show that considering descriptors as linear operators acting on functions through multiplication, rather than as simple scalar-valued signals, allows to extract significantly more information from a given descriptor and ultimately results in a more accurate functional map estimation. Namely, we show that descriptor preservation constraints can be formulated via commutativity with respect to the unknown map, which can be conveniently encoded by considering relations between matrices in the discrete setting. As a result, when the vector space spanned by the descriptors has a dimension smaller than that of the reduced basis, our optimization may still provide a fully-constrained system leading to accurate point-to-point correspondences, while previous methods might not. We demonstrate on a wide variety of experiments that our approach leads to significant improvement for functional map estimation by helping to reduce the number of necessary descriptor constraints by an order of magnitude, even given an increase in the size of the reduced basis.

The Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to retailers through a shared warehouse. Each shipment involves transporting goods from the supplier to the warehouse, at a fixed cost C, followed by a redistribution of these goods from the warehouse to the retailers that ordered them, where transporting goods to a retailer ρ has a fixed cost cρ. In addition, we incur waiting costs for each order, possibly an arbitrary non-decreasing function of time, different for each order. The objective is to minimize the overall cost of satisfying all orders, namely the sum of all shipping and waiting costs. JRP has been well studied in Operations Research and, more recently, in the area of approximation algorithms. For arbitrary waiting cost functions, the best known approximation ratio is 1.8. This ratio can be reduced to ≈ 1.574 for the JRP-D model, where there is no cost for waiting but orders have deadlines. As for hardness results, it is known that the problem is ℙ -hard and that the natural linear program for JRP has integrality gap at least 1.245. Both results hold even for JRP-D. In the online scenario, the best lower and upper bounds on the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64 applies even to the restricted version of JRP, denoted JRP-L, where the waiting cost function is linear. We provide several new approximation results for JRP. In the offline case, we give an algorithm with ratio ≈ 1.791, breaking the barrier of 1.8. We also show that the integrality gap of the linear program for JRP-L is at least 12/11 ≈ 1.09. In the online case, we show a lower bound of ≈ 2.754 on the competitive ratio for JRP-L (and thus JRP as well), improving the previous bound of 2.64. We also study the online version of JRP-D, for which we prove that the optimal competitive ratio is 2.