GIS and Network Analysis

Both geographic information systems (GIS) and network analysis are burgeoning fields, characterised by rapid methodological and scientific advances in recent years. A geographic information system (GIS) is a digital computer application designed for the capture, storage, manipulation, analysis and display of geographic information. Geographic location is the element that distinguishes geographic information from all other types of information. Without location, data are termed to be non-spatial and would have little value within a GIS. Location is, thus, the basis for many benefits of GIS: the ability to map, the ability to measure distances and the ability to tie different kinds of information together because they refer to the same place (Longley et al., 2001). GIS-T, the application of geographic information science and systems to transportation problems, represents one of the most important application areas of GIS-technology today. While traditional GIS formulation's strengths are in mapping display and geodata processing, GIS-T requires new data structures to represent the complexities of transportation networks and to perform different network algorithms in order to fulfil its potential in the field of logistics and distribution logistics. This paper addresses these issues as follows. The section that follows discusses data models and design issues which are specifically oriented to GIS-T, and identifies several improvements of the traditional network data model that are needed to support advanced network analysis in a ground transportation context. These improvements include turn-tables, dynamic segmentation, linear referencing, traffic lines and non-planar networks. Most commercial GIS software vendors have extended their basic GIS data model during the past two decades to incorporate these innovations (Goodchild, 1998). The third section shifts attention to network routing problems that have become prominent in GIS-T: the travelling salesman problem, the vehicle routing problem and the shortest path problem with time windows, a problem that occurs as a subproblem in many time constrained routing and scheduling issues of practical importance. Such problems are conceptually simple, but mathematically complex and challenging. The focus is on theory and algorithms for solving these problems. The paper concludes with some final remarks.

SAT Modulo Monotonic Theories

We define the concept of a monotonic theory and show how to build efficient SMT (SAT Modulo Theory) solvers, including effective theory propagation and clause learning, for such theories. We present examples showing that monotonic theories arise from many common problems, e.g., graph properties such as reachability, shortest paths, connected components, minimum spanning tree, and max-flow/min-cut, and then demonstrate our framework by building SMT solvers for each of these theories. We apply these solvers to procedural content generation problems, demonstrating major speed-ups over state-of-the-art approaches based on SAT or Answer Set Programming, and easily solving several instances that were previously impractical to solve

Supervised regionalization methods, a survey.

This paper reviews almost four decades of contributions on the subject of supervised regionalization methods. These methods aggregate a set of areas into a predefined number of spatially contiguous regions while optimizing certain aggregation criteria. The authors present a taxonomic scheme that classifies a wide range of regionalization methods into eight groups, based on the strategy applied for satisfying the spatial contiguity constraint. The paper concludes by providing a qualitative comparison of these groups in terms of a set of certain characteristics, and by suggesting future lines of research for extending and improving these methods.regionalization, constrained clustering, analytical regions.

Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard Knapsack Problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a Knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum Knapsack solution is sufficient to find optimum Knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed Knapsack solution: for Knapsack instances consisting of $n$ items, an RNN of depth five and width $w$ computes a solution of value at least $1-\mathcal{O}(n^2/\sqrt{w})$ times the optimum solution value. Our results build upon a classical dynamic programming formulation of the Knapsack Problem as well as a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the Knapsack Problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202

Convergent Duality for the Traveling Salesman Problem

A constructive method is presented for optimizing exactly the Traveling Salesman Problem as a sequence of shortest route problems. The method combines group theoretic and Lagrangean relaxation constructions. Key Words: Traveling Salesman Problem, Lagrangean relaxation, shortest route problem, generalized linear programming, group theory