No abstract

In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others.In the (min, +)-convolution problem, the goal is to compute a sequence (c. This can easily be done in O(n 2 ) time, but no O(n 2−ε ) algorithm is known for ε > 0. In this paper, we undertake a systematic study of the (min, +)-convolution problem as a hardness assumption.First, we establish the equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min, +)-convolution problem has been used as a building block in algorithms for many problems, notably problems in stringology. It has also appeared as an ad hoc hardness assumption. Second, we investigate some of these connections and provide new reductions and other results. We also explain why replacing this assumption with the SETH might not be possible for some problems.

The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of , even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only . Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis Gharan et al. (FOCS, 550–559, 2011), and then by Mömke and Svensson (FOCS, 560–569, 2011). In this paper, we provide an improved analysis of the approach presented in Mömke and Svensson (FOCS, 560–569, 2011) yielding a bound of on the approximation factor, as well as a bound of for any ε>0 for a more general Travelling Salesman Path Problem in graphic metrics.

Photo identification is an important tool for estimating abundance and monitoring population trends over time. However, manually matching photographs to known individuals is time‐consuming. Motivated by recent developments in image recognition, we hosted a data science challenge on the crowdsourcing platform Kaggle to automate the identification of endangered North Atlantic right whales (Eubalaena glacialis). The winning solution automatically identified individual whales with 87% accuracy with a series of convolutional neural networks to identify the region of interest on an image, rotate, crop, and create standardized photographs of uniform size and orientation and then identify the correct individual whale from these passport‐like photographs. Recent advances in deep learning coupled with this fully automated workflow have yielded impressive results and have the potential to revolutionize traditional methods for the collection of data on the abundance and distribution of wild populations. Presenting these results to a broad audience should further bridge the gap between the data science and conservation science communities.

In the shortest superstring problem, we are given a set of strings {s 1 , . . . , s k } and want to find a string that contains all s i as substrings and has minimum length. This is a classical problem in approximation and the best known approximation factor is 2 In this paper we give an algorithm that achieves an approximation ratio of 2 11 23 , breaking through the long-standing bound of 2 1 2 . We use the standard reduction of Shortest-Superstring to Max-ATSP-Path. The new, somewhat surprising, algorithmic idea is to take the better of the two solutions obtained by using: (a) the currently best