(Open) packing number of some graph products

The packing number of a graph G is the maximum number of closed neighborhoods of vertices in G with pairwise empty intersections. Similarly, the open packing number of G is the maximum number of open neighborhoods in C with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Geometric Inhomogeneous Random Graphs for Algorithm Engineering

The design and analysis of graph algorithms is heavily based on the worst case. In practice, however, many algorithms perform much better than the worst case would suggest. Furthermore, various problems can be tackled more efficiently if one assumes the input to be, in a sense, realistic. The field of network science, which studies the structure and emergence of real-world networks, identifies locality and heterogeneity as two frequently occurring properties. A popular model that captures these properties are geometric inhomogeneous random graphs (GIRGs), which is a generalization of hyperbolic random graphs (HRGs). Aside from their importance to network science, GIRGs can be an immensely valuable tool in algorithm engineering. Since they convincingly mimic real-world networks, guarantees about quality and performance of an algorithm on instances of the model can be transferred to real-world applications. They have model parameters to control the amount of heterogeneity and locality, which allows to evaluate those properties in isolation while keeping the rest fixed. Moreover, they can be efficiently generated which allows for experimental analysis. While realistic instances are often rare, generated instances are readily available. Furthermore, the underlying geometry of GIRGs helps to visualize the network, e.g.,~for debugging or to improve understanding of its structure. The aim of this work is to demonstrate the capabilities of geometric inhomogeneous random graphs in algorithm engineering and establish them as routine tools to replace previous models like the Erd\H{o}s-R{\\u27e}nyi model, where each edge exists with equal probability. We utilize geometric inhomogeneous random graphs to design, evaluate, and optimize efficient algorithms for realistic inputs. In detail, we provide the currently fastest sequential generator for GIRGs and HRGs and describe algorithms for maximum flow, directed spanning arborescence, cluster editing, and hitting set. For all four problems, our implementations beat the state-of-the-art on realistic inputs. On top of providing crucial benchmark instances, GIRGs allow us to obtain valuable insights. Most notably, our efficient generator allows us to experimentally show sublinear running time of our flow algorithm, investigate the solution structure of cluster editing, complement our benchmark set of arborescence instances with a density for which there are no real-world networks available, and generate networks with adjustable locality and heterogeneity to reveal the effects of these properties on our algorithms

Generalized limited packings of some graphs with a limited number of P4-partners

By using modular decomposition and handling certain graph operations such as join and union, we show that the Generalized Limited Packing Problem—NP-complete in general—can be solved in polynomial time in some graph classes with a limited number of P4-partners; specifically P4-tidy graphs, which contain cographs and P4-sparse graphs. In particular, we describe an algorithm to compute the associated numbers in polynomial time within these graph classes. In this way, we generalize some of the previous results on the subject. We also make some progress on the study of the computational complexity of the Generalized Multiple Domination Problem in graphs.Fil: Dobson, M. P.. Universidad Nacional de Rosario. Facultad de Cs.exactas Ingenieria y Agrimensura. Escuela de Cs.exactas y Naturales. Departamento de Matematica; ArgentinaFil: Hinrichsen, E.. Universidad Nacional de Rosario. Facultad de Cs.exactas Ingenieria y Agrimensura. Escuela de Cs.exactas y Naturales. Departamento de Matematica; ArgentinaFil: Leoni, Valeria Alejandra. Universidad Nacional de Rosario. Facultad de Cs.exactas Ingenieria y Agrimensura. Escuela de Cs.exactas y Naturales. Departamento de Matematica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Cientifico Tecnológico Rosario; Argentin

Workforce scheduling and planning : a combinatorial approach

This thesis investigates solution methodologies for concrete combinatorial problems in scheduling and planning. In all considered problems, it is assumed that the available information does not change over time; hence these problems have a deterministic structure. The problems studied in this thesis are divided into two groups; \workforce scheduling" and \planning". In workforce scheduling, the center problem is to build a schedule of tasks and technicians. It is assumed that the time line is split into workdays. In every workday, tasks must be grouped as sequences, each being performed by a team of technicians. Skill requirements of every task in a sequence must be met by the assigned team. This scheduling problem with some other aspects is di??cult to solve quickly and e??ciently. We developed a Mixed Integer Programming (MIP) based heuristic approach to tackle this complex scheduling problem. Our MIP model is basically a formulation of the matching problem on bipartite graphs and it enabled us to have a global way of assigning technicians to tasks. It is capable of revising technician-task allocations and performs very well, especially in the case of rare skills. A workday schedule of the aforementioned problem includes many-to-one type workforce assignments. As the second problem in workforce scheduling, stability of these workforce assignments is investigated. The stability de??nition of Gale-Shapley on the Marriage model is extended to the setting of multi-skill workforce assignments. It is shown that ??nding stable assignments is NP-hard. In some special cases stable assignments can be constructed in polynomial time. For the general case, we give linear inequalities of binary variables that describe the set of stable assignments. We propose a MIP model including these linear inequalities. To the best of our knowledge, the Gale-Shapley stability is not studied under the multi-skill workforce scheduling framework so far in the literature. The closed form description of stable assignments is also the ??rst embedding of the Gale-Shapley stability concept into an NP-complete problem. In the second problem group, two vehicle related problems are studied; the "dial a ride problem" and the "vehicle refueling problem". In the former, the goal is to check whether a list of pick-up and delivery tasks can be achieved under several timing constraints. It is shown this feasibility testing can be done in linear time using interval graphs. In the vehicle refueling problem, the goal is to make refueling decisions to reach a destination such that the cost of the travel is minimized. A greedy algorithm is presented to ??nd optimal refueling decisions. Moreover, it is shown that the vehicle refueling problem is equivalent to a ow problem on a special network

On a spontaneous decentralized market process

We examine a spontaneous decentralized market process widely observed in real life labor markets. This is a natural random decentralized dynamic competitive process. We show that this process converges globally and almost surely to a competitive equilibrium. This result is surprisingly general by assuming only the existence of an equilibrium. Our findings have also meaningful policy implications

Human Resource Management in Emergency Situations

The dissertation examines the issues related to the human resource management in emergency situations and introduces the measures helping to solve these issues. The prime aim is to analyse complexly a human resource management, built environment resilience management life cycle and its stages for the purpose of creating an effective Human Resource Management in Emergency Situations Model and Intelligent System. This would help in accelerating resilience in every stage, managing personal stress and reducing disaster-related losses. The dissertation consists of an Introduction, three Chapters, the Conclusions, References, List of Author’s Publications and nine Appendices. The introduction discusses the research problem and the research relevance, outlines the research object, states the research aim and objectives, overviews the research methodology and the original contribution of the research, presents the practical value of the research results, and lists the defended propositions. The introduction concludes with an overview of the author’s publications and conference presentations on the topic of this dissertation. Chapter 1 introduces best practice in the field of disaster and resilience management in the built environment. It also analyses disaster and resilience management life cycle ant its stages, reviews different intelligent decision support systems, and investigates researches on application of physiological parameters and their dependence on stress. The chapter ends with conclusions and the explicit objectives of the dissertation. Chapter 2 of the dissertation introduces the conceptual model of human resource management in emergency situations. To implement multiple criteria analysis of the research object the methods of multiple criteria analysis and mahematics are proposed. They should be integrated with intelligent technologies. In Chapter 3 the model developed by the author and the methods of multiple criteria analysis are adopted by developing the Intelligent Decision Support System for a Human Resource Management in Emergency Situations consisting of four subsystems: Physiological Advisory Subsystem to Analyse a User’s Post-Disaster Stress Management; Text Analytics Subsystem; Recommender Thermometer for Measuring the Preparedness for Resilience and Subsystem of Integrated Virtual and Intelligent Technologies. The main statements of the thesis were published in eleven scientific articles: two in journals listed in the Thomson Reuters ISI Web of Science, one in a peer-reviewed scientific journal, four in peer-reviewed conference proceedings referenced in the Thomson Reuters ISI database, and three in peer-reviewed conference proceedings in Lithuania. Five presentations were given on the topic of the dissertation at conferences in Lithuania and other countries