A new DSATUR-based algorithm for exact vertex coloring

This paper describes a new exact algorithm PASS for the vertex coloring problem based on the well known DSATUR algorithm. At each step DSATUR maximizes saturation degree to select a new candidate vertex to color, breaking ties by maximum degree w.r.t. uncolored vertices. Later Sewell introduced a new tiebreaking strategy, which evaluated available colors for each vertex explicitly. PASS differs from Sewell in that it restricts its application to a particular set of vertices. Overall performance is improved when the new strategy is applied selectively instead of at every step. The paper also reports systematic experiments over 1500 random graphs and a subset of the DIMACS color benchmark.

Max-Coloring Paths: Tight Bounds and Extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V, E) with a non-negative weight function w on V such that Sigma(k)(i=1) max(v epsilon Ci) w(v(i)) is minimized, where C-1, ... , C-k are the various color classes. Max-coloring general graphs is as hard as the classical vertex coloring problem, a special case where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring abroad class of trees and show it can be solved in time O(vertical bar V vertical bar+time for sorting the vertex weights). When vertex weights belong to R, we show a matching lower bound of Omega(vertical bar V vertical bar log vertical bar V vertical bar) in the algebraic computation tree model.

Modelli e Algoritmi per Problemi di Coloring con il Linguaggio AMPL

Come il titolo suggerisce, due sono gli aspetti oggetto di questo elaborato: i modelli e gli algoritmi (con le rispettive criticità), e il linguaggio di modellazione AMPL. Il filo conduttore che integra le due parti, nonché mezzo ultimo per un’applicazione pratica dell’attività di modellazione, è l’ottimizzatore, la ”macchina” che effettua la risoluzione vera e propria dei suddetti modelli.

Random graph coloring:Statistical physics approach

The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte Carlo simulations. We critically discuss the merits and shortcomings of the various methods, and interpret the results obtained. We present an exact analytical expression for the two-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges. ©2002 The American Physical Society.

On the Vertex Folkman Numbers Fv(2,...,2;q)

2000 Mathematics Subject Classification: 05C55.

Effective Learning-Based Hybrid Search for Bandwidth Coloring

International audience

An Efficient Simulated Annealing With a Valid Solution Mechanism For TDMA Brodcast Scheduling problem

This paper presents an efficient Simulated Annealing with valid solution mechanism for finding an optimum conflict-free transmission schedule for a broadcast radio network. This is known as a Broadcast Scheduling Problem (BSP) and shown as an NP-complete problem, in earlier studies. Because of this NP-complete nature, earlier studies used genetic algorithms, mean field annealing, neural networks, factor graph and sum product algorithm, and sequential vertex coloring algorithm to obtain the solution. In our study, a valid solution mechanism is included in simulated annealing. Because of this inclusion, we are able to achieve better results even for networks with 100 nodes and 300 links. The results obtained using our methodology is compared with all the other earlier solution methods.

An Algorithm for Group Formation and Maximal Independent Set in an Amorphous Computer

Amorphous computing is the study of programming ultra-scale computing environments of smart sensors and actuators cite{white-paper}. The individual elements are identical, asynchronous, randomly placed, embedded and communicate locally via wireless broadcast. Aggregating the processors into groups is a useful paradigm for programming an amorphous computer because groups can be used for specialization, increased robustness, and efficient resource allocation. This paper presents a new algorithm, called the clubs algorithm, for efficiently aggregating processors into groups in an amorphous computer, in time proportional to the local density of processors. The clubs algorithm is well-suited to the unique characteristics of an amorphous computer. In addition, the algorithm derives two properties from the physical embedding of the amorphous computer: an upper bound on the number of groups formed and a constant upper bound on the density of groups. The clubs algorithm can also be extended to find the maximal independent set (MIS) and $Delta + 1$ vertex coloring in an amorphous computer in $O(log N)$ rounds, where $N$ is the total number of elements and $Delta$ is the maximum degree.

Modelagem matemática e aplicações do problema de coloração em grafos

Pós-graduação em Matemática - IBILCE

Contributions to exact approaches in combinatorial optimization

The focus of this thesis is to contribute to the development of new, exact solution approaches to different combinatorial optimization problems. In particular, we derive dedicated algorithms for a special class of Traveling Tournament Problems (TTPs), the Dial-A-Ride Problem (DARP), and the Vehicle Routing Problem with Time Windows and Temporal Synchronized Pickup and Delivery (VRPTWTSPD). Furthermore, we extend the concept of using dual-optimal inequalities for stabilized Column Generation (CG) and detail its application to improved CG algorithms for the cutting stock problem, the bin packing problem, the vertex coloring problem, and the bin packing problem with conflicts. In all approaches, we make use of some knowledge about the structure of the problem at hand to individualize and enhance existing algorithms. Specifically, we utilize knowledge about the input data (TTP), problem-specific constraints (DARP and VRPTWTSPD), and the dual solution space (stabilized CG). Extensive computational results proving the usefulness of the proposed methods are reported.

A framework for mixed-criticality partitioned real-time embedded systems

Los sistemas empotrados son cada día más comunes y complejos, de modo que encontrar procesos seguros, eficaces y baratos de desarrollo software dirigidos específicamente a esta clase de sistemas es más necesario que nunca. A diferencia de lo que ocurría hasta hace poco, en la actualidad los avances tecnológicos en el campo de los microprocesadores de los últimos tiempos permiten el desarrollo de equipos con prestaciones más que suficientes para ejecutar varios sistemas software en una única máquina. Además, hay sistemas empotrados con requisitos de seguridad (safety) de cuyo correcto funcionamiento depende la vida de muchas personas y/o grandes inversiones económicas. Estos sistemas software se diseñan e implementan de acuerdo con unos estándares de desarrollo software muy estrictos y exigentes. En algunos casos puede ser necesaria también la certificación del software. Para estos casos, los sistemas con criticidades mixtas pueden ser una alternativa muy valiosa. En esta clase de sistemas, aplicaciones con diferentes niveles de criticidad se ejecutan en el mismo computador. Sin embargo, a menudo es necesario certificar el sistema entero con el nivel de criticidad de la aplicación más crítica, lo que hace que los costes se disparen. La virtualización se ha postulado como una tecnología muy interesante para contener esos costes. Esta tecnología permite que un conjunto de máquinas virtuales o particiones ejecuten las aplicaciones con unos niveles de aislamiento tanto temporal como espacial muy altos. Esto, a su vez, permite que cada partición pueda ser certificada independientemente. Para el desarrollo de sistemas particionados con criticidades mixtas se necesita actualizar los modelos de desarrollo software tradicionales, pues estos no cubren ni las nuevas actividades ni los nuevos roles que se requieren en el desarrollo de estos sistemas. Por ejemplo, el integrador del sistema debe definir las particiones o el desarrollador de aplicaciones debe tener en cuenta las características de la partición donde su aplicación va a ejecutar. Tradicionalmente, en el desarrollo de sistemas empotrados, el modelo en V ha tenido una especial relevancia. Por ello, este modelo ha sido adaptado para tener en cuenta escenarios tales como el desarrollo en paralelo de aplicaciones o la incorporación de una nueva partición a un sistema ya existente. El objetivo de esta tesis doctoral es mejorar la tecnología actual de desarrollo de sistemas particionados con criticidades mixtas. Para ello, se ha diseñado e implementado un entorno dirigido específicamente a facilitar y mejorar los procesos de desarrollo de esta clase de sistemas. En concreto, se ha creado un algoritmo que genera el particionado del sistema automáticamente. En el entorno de desarrollo propuesto, se han integrado todas las actividades necesarias para desarrollo de un sistema particionado, incluidos los nuevos roles y actividades mencionados anteriormente. Además, el diseño del entorno de desarrollo se ha basado en la ingeniería guiada por modelos (Model-Driven Engineering), la cual promueve el uso de los modelos como elementos fundamentales en el proceso de desarrollo. Así pues, se proporcionan las herramientas necesarias para modelar y particionar el sistema, así como para validar los resultados y generar los artefactos necesarios para el compilado, construcción y despliegue del mismo. Además, en el diseño del entorno de desarrollo, la extensión e integración del mismo con herramientas de validación ha sido un factor clave. En concreto, se pueden incorporar al entorno de desarrollo nuevos requisitos no-funcionales, la generación de nuevos artefactos tales como documentación o diferentes lenguajes de programación, etc. Una parte clave del entorno de desarrollo es el algoritmo de particionado. Este algoritmo se ha diseñado para ser independiente de los requisitos de las aplicaciones así como para permitir al integrador del sistema implementar nuevos requisitos del sistema. Para lograr esta independencia, se han definido las restricciones al particionado. El algoritmo garantiza que dichas restricciones se cumplirán en el sistema particionado que resulte de su ejecución. Las restricciones al particionado se han diseñado con una capacidad expresiva suficiente para que, con un pequeño grupo de ellas, se puedan expresar la mayor parte de los requisitos no-funcionales más comunes. Las restricciones pueden ser definidas manualmente por el integrador del sistema o bien pueden ser generadas automáticamente por una herramienta a partir de los requisitos funcionales y no-funcionales de una aplicación. El algoritmo de particionado toma como entradas los modelos y las restricciones al particionado del sistema. Tras la ejecución y como resultado, se genera un modelo de despliegue en el que se definen las particiones que son necesarias para el particionado del sistema. A su vez, cada partición define qué aplicaciones deben ejecutar en ella así como los recursos que necesita la partición para ejecutar correctamente. El problema del particionado y las restricciones al particionado se modelan matemáticamente a través de grafos coloreados. En dichos grafos, un coloreado propio de los vértices representa un particionado del sistema correcto. El algoritmo se ha diseñado también para que, si es necesario, sea posible obtener particionados alternativos al inicialmente propuesto. El entorno de desarrollo, incluyendo el algoritmo de particionado, se ha probado con éxito en dos casos de uso industriales: el satélite UPMSat-2 y un demostrador del sistema de control de una turbina eólica. Además, el algoritmo se ha validado mediante la ejecución de numerosos escenarios sintéticos, incluyendo algunos muy complejos, de más de 500 aplicaciones. ABSTRACT The importance of embedded software is growing as it is required for a large number of systems. Devising cheap, efficient and reliable development processes for embedded systems is thus a notable challenge nowadays. Computer processing power is continuously increasing, and as a result, it is currently possible to integrate complex systems in a single processor, which was not feasible a few years ago.Embedded systems may have safety critical requirements. Its failure may result in personal or substantial economical loss. The development of these systems requires stringent development processes that are usually defined by suitable standards. In some cases their certification is also necessary. This scenario fosters the use of mixed-criticality systems in which applications of different criticality levels must coexist in a single system. In these cases, it is usually necessary to certify the whole system, including non-critical applications, which is costly. Virtualization emerges as an enabling technology used for dealing with this problem. The system is structured as a set of partitions, or virtual machines, that can be executed with temporal and spatial isolation. In this way, applications can be developed and certified independently. The development of MCPS (Mixed-Criticality Partitioned Systems) requires additional roles and activities that traditional systems do not require. The system integrator has to define system partitions. Application development has to consider the characteristics of the partition to which it is allocated. In addition, traditional software process models have to be adapted to this scenario. The V-model is commonly used in embedded systems development. It can be adapted to the development of MCPS by enabling the parallel development of applications or adding an additional partition to an existing system. The objective of this PhD is to improve the available technology for MCPS development by providing a framework tailored to the development of this type of system and by defining a flexible and efficient algorithm for automatically generating system partitionings. The goal of the framework is to integrate all the activities required for developing MCPS and to support the different roles involved in this process. The framework is based on MDE (Model-Driven Engineering), which emphasizes the use of models in the development process. The framework provides basic means for modeling the system, generating system partitions, validating the system and generating final artifacts. The framework has been designed to facilitate its extension and the integration of external validation tools. In particular, it can be extended by adding support for additional non-functional requirements and support for final artifacts, such as new programming languages or additional documentation. The framework includes a novel partitioning algorithm. It has been designed to be independent of the types of applications requirements and also to enable the system integrator to tailor the partitioning to the specific requirements of a system. This independence is achieved by defining partitioning constraints that must be met by the resulting partitioning. They have sufficient expressive capacity to state the most common constraints and can be defined manually by the system integrator or generated automatically based on functional and non-functional requirements of the applications. The partitioning algorithm uses system models and partitioning constraints as its inputs. It generates a deployment model that is composed by a set of partitions. Each partition is in turn composed of a set of allocated applications and assigned resources. The partitioning problem, including applications and constraints, is modeled as a colored graph. A valid partitioning is a proper vertex coloring. A specially designed algorithm generates this coloring and is able to provide alternative partitions if required. The framework, including the partitioning algorithm, has been successfully used in the development of two industrial use cases: the UPMSat-2 satellite and the control system of a wind-power turbine. The partitioning algorithm has been successfully validated by using a large number of synthetic loads, including complex scenarios with more that 500 applications.

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). Let Delta = Delta(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K-n,K-n. Alon, McDiarmid and Reed observed that a'(K-p-1,K-p-1) = p for every prime p. In this paper we prove that a'(K-p,K-p) <= p + 2 = Delta + 2 when p is prime. Basavaraju, Chandran and Kummini proved that a'(K-n,K-n) >= n + 2 = Delta + 2 when n is odd, which combined with our result implies that a'(K-p,K-p) = p + 2 = Delta + 2 when p is an odd prime. Moreover we show that if we remove any edge from K-p,K-p, the resulting graph is acyclically Delta + 1 = p + 1-edge-colorable. (C) 2009 Elsevier B.V. All rights reserved.

Acyclic edge coloring of 2-degenerate graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a'(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contains seriesparallel graphs, outerplanar graphs, non - regular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a'(G)<=Delta + 2, where Delta = Delta(G) denotes the maximum degree of the graph. We prove the conjecture for 2-degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2-degenerate graph with maximum degree ?, then a'(G)<=Delta + 1. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 68:1-27, 2011

On the Parameterized Complexity of the Maximum Edge 2-Coloring Problem

We investigate the parameterized complexity of the following edge coloring problem motivated by the problem of channel assignment in wireless networks. For an integer q >= 2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. This problem is NP-hard for q >= 2, and has been well-studied from the point of view of approximation. Our main focus is the case when q = 2, which is already theoretically intricate and practically relevant. We show fixed-parameter tractable algorithms for both the standard and the dual parameter, and for the latter problem, the result is based on a linear vertex kernel.