Combinatorial Algorithms

La suma es resultado de la adición y una técnica fundamental del conteo, cuyas implicaciones van más allá de respuestas a preguntas como ¿cuántos objetos hay sobre la mesa? o ¿cuántos alumnos integran el grupo de tercer grado de educación primaria
en mi escuela? En cada caso, basta con la enumeración de cada elemento para conocer el total. Para esta ocasión analizaremos los procesos aditivos y multiplicativos como técnicas para contar eficientemente.

This paper presents a comparative study of the African Buffalo Optimization algorithm and the Random-ized Insertion Algorithm to solving the asymmetric Travelling Salesman's Problem with the overall objective of determining a better method to solving the asymmetric Travelling Salesman's Problem instances. Our interest in the asymmetric Travelling Salesman's Problem (ATSP) is borne out of the fact that most practical daily-life problems are asymmetric rather than symmetric. The choice of the Random Insertion Algorithm as a comparative algorithm was informed by our desire to investigate the general belief among the scientific community that Heuristics being mostly problem-dependent algorithms are more efficient that metaheuristics that are usually general-purpose algorithms. Moreover, both the metaheuristic, the African Buffalo Optimization and the Heuristic, Randomized Insertion Algorithms hold some of the best results in literature in solving the ATSP. Similarly, both methods employ different search techniques in attempting solutions to the ATSP: while the African Buffalo Optimization uses the modified Karp-Steele technique, the Randomized Insertion employs random insertion mechanism. After investigating all the 19 benchmark ATSP datasets available in TSPLIB, it was discovered that the Randomized Insertion Algorithm achieves slightly better result to the problems but the African Buffalo Optimization is much faster.

Combinatoria La combinatoria trata del estudio de las posibles agrupaciones de objetos. Contar el número de objetos que verifican ciertas propiedades es uno de los objetivos de la combinatoria. Problemas muy diversos, como determinar el número posible de apuestas diferentes en una quiniela, el número posible de posiciones en que unos corredores pueden terminar una carrera, el número posible de matrículas de los coches de un país o las diferentes formas de distribuir una serie de objetos en cajas son problemas que se abordan mediante las técnicas de conteo que veremos en este capítulo. Lo que pretendemos es por tanto, contar los elementos de un conjunto, o más precisamente, determinar su cardinal. Dado un conjunto A denotaremos por |A| a su cardinal. Nosotros aquí trataremos únicamente con conjuntos que tienen un número finito de elementos. En tal caso, se dice que un conjunto A tiene cardinal n si existe una biyección entre el conjunto A y el conjunto {0, 1, · · · , n − 1}. Es claro que si dos conjuntos son biyectivos tienen el mismo cardinal. A la hora de contar ciertos objetos, lo que haremos será identificar estos objetos con los elementos de algún conjunto del cual sepamos determinar su cardinal (es decir, estableceremos una biyección entre el conjunto de objetos que queremos contar y otro conjunto del cual hallaremos su cardinal). Para comenzar, estudiaremos en primer lugar cómo determinar el cardinal de algunos conjuntos. 4.1. Métodos elementales de conteo 4.1.1. Principio de inclusión-exclusión Proposición 4.1.1 (Principio de la suma). Sean A 1 y A 2 dos conjuntos disjuntos (es decir, A 1 ∩A 2 = ∅). Entonces |A 1 ∪ A 2 | = |A 1 | + |A 2 |. Intuitivamente está claro lo que significa este principio. No obstante, si quisiéramos una demostración, ésta se basaría en que los conjuntos {0, 1, · · · , m − 1} y {n, n + 1, · · · , n + m − 1} son biyectivos. El principio puede extenderse a tres o más conjuntos. En tal caso, dice que si A 1 , A 2 , · · · , A n son conjuntos disjuntos dos a dos (es decir, A i ∩ A j = ∅ para i = j) entonces |A 1 ∪ A 2 ∪ · · · ∪ A n | = |A 1 | + |A 2 | + · · · + |A n | El principio de la suma podría enunciarse también como sigue: Si una primera tarea se puede realizar de n 1 formas, y una segunda tarea se puede realizar de n 2 formas, y las dos tareas son incompatibles, entonces hay n 1 + n 2 formas de realizar una de las dos tareas. Este principio de la suma es muy restrictivo, pues requiere que los conjuntos sean disjuntos, o que las tareas sean incompatibles. Sin embargo, en general, la situación es que los conjuntos no sean disjuntos. En este caso se tiene: Proposición 4.1.2. Sean A 1 y A 2 dos conjuntos. Entonces |A 1 ∪ A 2 | = |A 1 | + |A 2 | − |A 1 ∩ A 2 |. La idea de este resultado está clara. Si queremos contar los elementos que están en A 1 ∪ A 2 , contamos por una parte los que están en A 1 y por otra parte los que están en A 2 , lo que nos da |A 1 | + |A 2 |. Sin 97

We describe a general purpose algorithm for counting simple cycles and simple paths 4 of any length on a (weighted di)graph on N vertices and M edges, achieving a time complexity of O(N + M + (L^ω + L∆) |S_L|). In this expression, |S_L| is the number of (weakly) connected induced subgraphs of G on at most L vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where (1 + L^(ω−1)/∆ )|S_L| ≤  |Cycle_L|, with |Cycle_L| the total number of simple cycles of length at most L, including backtracks and self-loops. On Erdös-Rényi random graphs, we find empirically that this happens when the edge probability is larger than circa 4/N. In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on n letters with a time complexity of O(N + M + (n^L L^ω + ∆) |S_L|). A Matlab implementation of the algorithm proposed here is available for download.

Abstract: ATM network optimization problems defined as combinatorial optimization problems are considered. Several approximate algorithms for solving such problems are developed. Results of their comparison by experiments on a set of problems with random input data are presented.

The theory of the computational complexity is a subject of international interest of scientific, technological and enterprise organizations. The computational complexity contains diverse elements such as the classes of problems complexity (P, NP, NP-hard and NP-complete, and others), complexity of algorithms (it is a way to classify how efficient is an algorithm by means the execution time to solve a problem with the worst-case input), complexity of instances (it is computational complexity measures to determine the complex of the problems instances), and others elements. This work is focused on presenting a review of the theory of instances computational complexity. KeyWords: Instances Computational Complexity, Complexity of instances, computational complexity. The theory of the computational complexity is the part of the theory of the computation that studies the resources required during the calculation to solve a problem (Hartmanis and Hopcroft, 1971). The resources commonly stu...

Metaheuristics for a vehicle routing problem on bipartite graphs with distance constraints / R. Aringhieri, M. Bruglieri, F. Malucelli, M. Nonato. ((Intervento presentato al 2005. convegno MIC2005: The Sixth Metaheuristics International Conference tenutosi a Vienna nel Agosto 2005. ... There are no files associated with this item. ... Items in AIR are protected by copyright, with all rights reserved, unless otherwise indicated. ... ICT Support, development & maintenance are provided by the AePIC team @ CILEA. Powered on DSpace Software.

Abstract. The Feedback Vertex Set problem asks whether a graph con-tains q vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if q vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for ...

This paper introduces a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parametrized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition, which enables us to find optimal solutions for several graph problems. While the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. As an example, in this paper, we give an algorithm for the Dominating Set problem that outputs an optimal set in time \documentclass...

One evolutionary strategy based on the physical
kinetic process of rebounding is presented in this work. The
novel algorithm, Simulated Rebounding Algorithm (SRA), is
compared in performance to the well known evolutionary
strategy Simulated Annealing, two implementations of Genetic
Algorithms (GA) and a hybrid GA with some characteristics
taken from SRA. All these evolutionary algorithms (EA) are
applied to the optimal design of stand-alone generation
systems using renewable and conventional energy sources. The
main goal of the optimization is to find a set of optimal
configurations for the optimal expansion planning of the
power system. The optimal set is found minimizing an
objective function (OF), which includes investment,
installation, operation and maintenance costs and the
associated uncertainties. In the OF a penalization due to
energy and power not supplied, originated in unit outages, is
also considered. The OF is evaluated by means of stochastic
correlated simulation, requiring considerable calculation time.
The goal of this paper is to compare the behavior of the
different EAs applied to this complex optimization problem,
and to analyze the empirical performance of the novel
algorithm in comparison to well known EAs regarding
accuracy and computation time

An economic dispatch methodology based on combinatorial optimization is presented in this work. The methodology uses
a high performance evolutionary algorithm -Simulated Rebounding- as optimization engine. The advantage of the methodology is based on 1) the easy consideration of non-linearities and non-convexities without assuming major simplifications, 2) the possibility to evaluate good solutions with minor relaxations and 3) the parallel nature of the omputations. The combinatorial approach finds the optimum by trying dispatches fulfilling the given operation constraints. The spot pricing methodology is based on an ex-post
approach considering the results of the economic dispatch.
The main handicap of any combinatorial approach is, even with current computers, computation time requirement. Nevertheless, the internal parallelism of the methodology proposed allows the natural implementation on parallel computers to speed up drastically the time performance.
Some results of the application of the methodology to power systems are presented, in which the network is considered at the transmission level. A comparison with an economic dispatch based on linear-programming is also presented.

This paper presents a combinatorial (decision tree induc- tion) technique for transparent surface modeling from polarization im- ages. This technique simultaneously uses the object's symmetry, brewster angle, and degree of polarization to select accurate reference points. The reference points contain information about surface's normals position and direction at near occluding boundary. We reconstruct rotationally symmetric objects by rotating these reference points.

The verification and validation of IT systems has long been a crucial issue. Used in the most critical areas, when they are defective, their impact is measured in human lives or colossal material losses. To this end, effective formal verification and validation techniques are required in the design of a system, The most attractive formal verification technique from a cost / performance point of view is that based on the evaluation of models (model checking). Model checking is an automatic verification technique for hardware and software systems. As the number of state variables in the system increases, the size of the system state space grows exponentially, this is so called the “State space explosion problem”. To deal with this problem, we propose a new approch for the distribution of the state space generation on a network of connected machines. But what if the environment was not static ?
Dynamic environment ! which leaves open the question of how to deal with this dynamicity , without losing the purpose of our approach. In our thesis, we contribute to solve the dynamicity issue, adressing both cases of it; when machines turn off and when new ones are integrated into the network architecture. All over this, we was working on preserving the quality of the solutions in term of: The workload balancing, and a lower communication cost (caused by the transitions between states located in different machines).