In this paper, we propose TauRieL and target Traveling Salesman Problem (TSP) since it has broad applicability in theoretical and applied sciences. TauRieL utilizes an actor-critic inspired architecture that adopts ordinary feedforward nets to obtain a policy update vector @math . Then, we use @math to improve the state transition matrix from which we generate the policy. Also, the state transition matrix allows the solver to initialize from precomputed solutions such as nearest neighbors. In an online learning setting, TauRieL unifies the training and the search where it can generate near-optimal results in seconds. The input to the neural nets in the actor-critic architecture are raw 2-D inputs, and the design idea behind this decision is to keep neural nets relatively smaller than the architectures with wide embeddings with the tradeoff of omitting any distributed representations of the embeddings. Consequently, TauRieL generates TSP solutions two orders of magnitude faster per TSP instance as compared to state-of-the-art offline techniques with a performance impact of 6.1 in the worst case.
A wide range of TSP variants have been applied to a wide range of domains. Examples include the use of TSP in music for conjunct melody generation in computer-aided composition, as well as for forming automatic playlists and track artist suggestions - now used by Spotify and Tidal @cite_34 . TSP heuristics have also been used for diffractometer guidance in X-ray crystallography @cite_33 ; and for Telescope scheduling in exoplanet discovery @cite_10 as well as in galaxy imaging @cite_24 . It can also be applied to bioinformatics as a genome ordering problem by posing the ordering as a path traveling through each gene marker @cite_3 or as a clustering problem to solve gene expression clustering @cite_18 .
@cite_18 @cite_33 @cite_3 @cite_24 @cite_34 @cite_10
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We present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5 3-approximation algorithm for this problem, and this asymptotically tight bound in fact had been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5 2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP. For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. Motivation: During the last decade, improvements in high-throughput sequencing have generated a wealth of genomic data. Functionally interpreting these sequences and finding the biological signals that are hallmarks of gene function and regulation is currently mostly done using automated genome annotation platforms, which mainly rely on integrated machine learning frameworks to identify different functional sites of interest, including splice sites. Splicing is an essential step in the gene regulation process, and the correct identification of splice sites is a major cornerstone in a genome annotation system. Results: In this paper, we present SpliceRover, a predictive deep learning approach that outperforms the state-of-the-art in splice site prediction. SpliceRover uses convolutional neural networks (CNNs), which have been shown to obtain cutting edge performance on a wide variety of prediction tasks. We adapted this approach to deal with genomic sequence inputs, and show it consistently outperforms already existing approaches, with relative improvements in prediction effectiveness of up to 80.9 when measured in terms of false discovery rate. However, a major criticism of CNNs concerns their 'black box' nature, as mechanisms to obtain insight into their reasoning processes are limited. To facilitate interpretability of the SpliceRover models, we introduce an approach to visualize the biologically relevant information learnt. We show that our visualization approach is able to recover features known to be important for splice site prediction (binding motifs around the splice site, presence of polypyrimidine tracts and branch points), as well as reveal new features (e.g. several types of exclusion patterns near splice sites). Preface Introduction Molecular Biology Mathematics, Statistics, and Computer Science Some Molecular Biology DNA and Proteins The Central Dogma The Genetic Code Transfer RNA and Protein Sequences Genes Are Not Simple Biological Chemistry Restriction Maps Introduction Graphs Interval Graphs Measuring Fragment Sizes Multiple Maps Double Digest Problem Classifying Multiple Solutions Algorithms for DDP Algorithms and Complexity DDP is N P-Complete Approaches to DDP Simulated Annealing: TSP and DDP Mapping with Real Data Cloning and Clone Libraries A Finite Number of Random Clones Libraries by Complete Digestion Libraries by Partial Digestion Genomes per Microgram Physical Genome Maps: Oceans, Islands, and Anchors Mapping by Fingerprinting Mapping by Anchoring An Overview of Clone Overlap Putting It Together Sequence Assembly Shotgun Sequencing Sequencing by Hybridization Shotgun Sequencing Revisited Databases and Rapid Sequence Analysis DNA and Protein Sequence Databases A Tree Representation of a Sequence Hashing a Sequence Repeats in a Sequence Sequence Comparison by Hashing Sequence Comparison with at most l Mismatches Sequence Comparison by Statistical Content Dynamic Programming Alignment of Two Sequences The Number of Alignments Shortest and Longest Paths in a Network Global Distance Alignment Global Similarity Alignment Fitting One Sequence into Another Local Alignment and Clumps Linear Space Algorithms Tracebacks Inversions Map Alignment Parametric Sequence Comparisons Multiple Sequence Alignment The Cystic Fibrosis Gene Dynamic Programming in r-Dimensions Weighted-Average Sequences Profile Analysis Alignment by Hidden Markov Models Consensus Word Analysis Probability and Statistics for Sequence Alignment Global Alignment Local Alignment Extreme Value Distributions The Chein-Stein Method Poisson Approximation and Long Matches Sequence Alignment with Scores Probability and Statistics for Sequence Patterns A Central Limit Theorem Nonoverlapping Pattern Counts Poisson Approximation Site Distributions RNA Secondary Structure Combinatorics Minimum Free-energy Structures Consensus folding Trees and Sequences Trees Distance Parsimony Maximum Likelihood Trees Sources and Perspectives Molecular Biology Physical Maps and Clone Libraries Sequence Assembly Sequence Comparisons Probability and Statistics RNA Secondary Structure Trees and Sequences References Problem Solutions and Hints Mathematical Notation Algorithm Index Author Index Subject Index