Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Bilevel facility location problems: theory and applications.

In this doctoral thesis we focus on studying facility location problems considering customer preferences. In these problems, there is a set of customers or users who demand a service or product that must be supplied by one or more facilities. By facilities it is understood some object or structure that offers some service to customers. One of the most important assumptions is that customers have established their own preferences over the facilities and should be taken into account in the customer-facility assignment. In real life, customers choose facilities based on costs, preferences, a predetermined contract, or a loyalty coefficient, among others. That is, they are free to choose the facilities that will serve them. The situation described above is commonly modeled by bilevel programming, where the upper level corresponds to location decisions to optimize a predefined criteria, such as, minimize location and distribution costs or maximize the demand covered by the facilities; and the lower level is associated to -customer allocation- to optimize customer preferences. The hierarchy among both levels is justified because the decision taken in the upper level directly affects the decision’s space in the lower level

The Complexity of Partial Function Extension for Coverage Functions

Coverage functions are an important subclass of submodular functions, finding applications in machine learning, game theory, social networks, and facility location. We study the complexity of partial function extension to coverage functions. That is, given a partial function consisting of a family of subsets of [m] and a value at each point, does there exist a coverage function defined on all subsets of [m] that extends this partial function? Partial function extension is previously studied for other function classes, including boolean functions and convex functions, and is useful in many fields, such as obtaining bounds on learning these function classes. We show that determining extendibility of a partial function to a coverage function is NP-complete, establishing in the process that there is a polynomial-sized certificate of extendibility. The hardness also gives us a lower bound for learning coverage functions. We then study two natural notions of approximate extension, to account for errors in the data set. The two notions correspond roughly to multiplicative point-wise approximation and additive L_1 approximation. We show upper and lower bounds for both notions of approximation. In the second case we obtain nearly tight bounds

Systematic approaches for retail service location decisions

Text includes handwritten formulas. No pages 18, 92This thesis investigates systems applications to community facility planning by focusing on the use of models in locating retail facilities. This approach was taken because a number of major concepts employed in retail location are directly transferable to most types of urban services where consumers may choose to utilize a number of different locations. A general decision making process for locating retail services is described. Review of the types of information needed by a retail location planner finds that the central issue he faces is estimating the sales volume of a proposed site. The effect of other competing locations, consumer preferences and accessibility make this task difficult without some type of systematic approach. This could be called the classical "problem" of retail location. An extensive search was made of the work of others related to this problem. A number of approaches were found which attempted to represent the interrelated elements of consumers, access, and retailers which constitute a retail system. No dominant theory has been developed in the area; instead, a number of individual lines of inquiry were found with similarities between. Several selected location models are then reviewed in application to specific problems. The major criticism provided focusses on the degree of difficulty model authors have in representing consumer-retailer behavior and the type of information required to support the modelling. It was found that no one type of model can be regarded as superior since each may have been developed for different planning applications which vary in type of retail service and geographic area represented. There are other steps in retail location decision making where further applications of systems approaches may be valuable. These include population and income forecasting for a small area and economic evaluation of location alternatives once gross sales have been estimated. Further development of these areas in conjunction with the retail models described is suggested. Finally, a number of concepts found in various approaches to retail location may have direct benefit in the successful application of planning standards commonly used by architects and urban designers. Insight gained through certain theoretical approaches to retail location imply that increased care should be taken in the derivation and application of meaningful planning standards

Symmetry in Applied Mathematics

Applied mathematics and symmetry work together as a powerful tool for problem reduction and solving. We are communicating applications in probability theory and statistics (A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested, The Asymmetric Alpha-Power Skew-t Distribution), fractals - geometry and alike (Khovanov Homology of Three-Strand Braid Links, Volume Preserving Maps Between p-Balls, Generation of Julia and Mandelbrot Sets via Fixed Points), supersymmetry - physics, nanostructures -chemistry, taxonomy - biology and alike (A Continuous Coordinate System for the Plane by Triangular Symmetry, One-Dimensional Optimal System for 2D Rotating Ideal Gas, Minimal Energy Configurations of Finite Molecular Arrays, Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations), algorithms, programs and software analysis (Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory, On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems, On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives) to specific subjects (Facility Location Problem Approach for Distributed Drones, Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data). Diverse topics are thus combined to map out the mathematical core of practical problems

Production localization factors: an industrial and literature based review

Decision are commonly based on the available or easily accessible information; this is also true for more complex assessments like production localization. Where to locate production is often a key strategic decisions that has great impact on a company’s profitability for a long time; insufficient business intelligence may therefore have grave consequences. Six production localization factor studies have been assessed to see if they are focusing on the same issues and if there are any gaps. A new approach for structuring localization factors and the localization process is then presented and assessed with regards to some previously identified critical issues

Location analysis - possibilities of use in public administration

V článku jsou stručně popsána teoretická východiska lokační teorie a možnosti využití v oblasti veřejné správy jako je navrhování sítí a lokace různých zařízení v geografickém prostoru regionů.The paper under consideration describes key theoretical issues of continuous/discrete location theory and possibilities of applications in the area of public administration activities such as networks design and location of different facilities in geographical area of regions