While there are optimal TSP solvers, as well as recent learning-based approaches, the generalization of the TSP to the Multiple Traveling Salesmen Problem is much less studied. Here, we design a neural network solution that treats the salesmen, cities and depot as three different sets of varying cardinalities. We apply a novel technique that combines elements from recent architectures that were developed for sets, as well as elements from graph networks. Coupled with new constraint enforcing output layers, a dedicated loss, and a search method, our solution is shown to outperform all the meta-heuristics of the leading solver in the field.
From the technical perspective, our architecture relates to an emerging body of work, which encodes inputs that are given as sets . Similar to such methods, we employ pooling in order to obtain a representation that is invariant to the order of the elements, followed by a local computation at each element. As shown by @cite_1 and @cite_21 , under mild conditions, this is the only way to achieve permutation invariance. As mentioned above, our network is adapted to process multiple sets. In addition, since the mTSP has properties that are common to both set and graph problems, our network is also inspired by the recent advancements in graph-networks, commonly called . These networks operate on graphs, and use the underlying metric between nodes to propagate information in a hierarchical manner. While @cite_4 used the distance metric as an explicit input to each local computation and @cite_3 used it implicitly in the propagation rule, we use a learned transformation from distances to local weights, which integrates the graph properties to our sets-specialized architecture.
@cite_3 @cite_21 @cite_1 @cite_4
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It is widely believed that the success of deep convolutional networks is based on progressively discarding uninformative variability about the input with respect to the problem at hand. This is supported empirically by the difficulty of recovering images from their hidden representations, in most commonly used network architectures. In this paper we show that this loss of information is not a necessary condition to learn representations that generalize well on complicated problems, such as ImageNet. Via a cascade of homeomorphic layers, we build the i-RevNet, a network that can be fully inverted up to the final projection onto the classes, i.e. no information is discarded. Building an invertible architecture is difficult, for example, because the local inversion is ill-conditioned, we overcome this by providing an explicit inverse. An analysis of i-RevNet’s learned representations suggests an explanation of the good accuracy by a progressive contraction and linear separation with depth. To shed light on the nature of the model learned by the i-RevNet we reconstruct linear interpolations between natural images representations. In this work, we propose a method for simultaneously learning features and a corresponding similarity metric for person re-identification. We present a deep convolutional architecture with layers specially designed to address the problem of re-identification. Given a pair of images as input, our network outputs a similarity value indicating whether the two input images depict the same person. Novel elements of our architecture include a layer that computes cross-input neighborhood differences, which capture local relationships between the two input images based on mid-level features from each input image. A high-level summary of the outputs of this layer is computed by a layer of patch summary features, which are then spatially integrated in subsequent layers. Our method significantly outperforms the state of the art on both a large data set (CUHK03) and a medium-sized data set (CUHK01), and is resistant to over-fitting. We also demonstrate that by initially training on an unrelated large data set before fine-tuning on a small target data set, our network can achieve results comparable to the state of the art even on a small data set (VIPeR). Two architectures that generalize convolutional neural networks (CNNs) for the processing of signals supported on graphs are introduced. We start with the selection graph neural network (GNN), which replaces linear time invariant filters with linear shift invariant graph filters to generate convolutional features and reinterprets pooling as a possibly nonlinear subsampling stage where nearby nodes pool their information in a set of preselected sample nodes. A key component of the architecture is to remember the position of sampled nodes to permit computation of convolutional features at deeper layers. The second architecture, dubbed aggregation GNN, diffuses the signal through the graph and stores the sequence of diffused components observed by a designated node. This procedure effectively aggregates all components into a stream of information having temporal structure to which the convolution and pooling stages of regular CNNs can be applied. A multinode version of aggregation GNNs is further introduced for operation in large-scale graphs. An important property of selection and aggregation GNNs is that they reduce to conventional CNNs when particularized to time signals reinterpreted as graph signals in a circulant graph. Comparative numerical analyses are performed in a source localization application over synthetic and real-world networks. Performance is also evaluated for an authorship attribution problem and text category classification. Multinode aggregation GNNs are consistently the best-performing GNN architecture. Given a network represented by a graph @math G=V,E, we consider a dynamical process of influence diffusion in G that evolves as follows: Initially only the nodes of a given @math S⊆V are influenced; subsequently, at each round, the set of influenced nodes is augmented by all the nodes in the network that have a sufficiently large number of already influenced neighbors. The question is to determine a small subset of nodes Sa target set that can influence the whole network. This is a widely studied problem that abstracts many phenomena in the social, economic, biological, and physical sciences. It is known [6] that the above optimization problem is hard to approximate within a factor of @math 2log1-∈|V|, for any @math ∈>0. In this paper, we present a fast and surprisingly simple algorithm that exhibits the following features: 1 when applied to trees, cycles, or complete graphs, it always produces an optimal solution i.e., a minimum size target set; 2 when applied to arbitrary networks, it always produces a solution of cardinality matching the upper bound given in [1], and proved therein by means of the probabilistic method; 3 when applied to real-life networks, it always produces solutions that substantially outperform the ones obtained by previously published algorithms for which no proof of optimality or performance guarantee is known in any class of graphs.