Information systems are nowadays commonly represented with a graph, which makes them easier to interpret and understand. A graph is an abstract data structure consisting of nodes and edges, where the nodes, sometimes also referred to as vertices, represent entities, and the edges are links that connect the nodes representing relationships among them. From a theoretical perspective a graph is an ordered pair G = (V, E), where V is the set of vertices and E a set of edges, where each edge associates two vertices in V. This type of graphs is usually referred to as undirected and unweighted graphs. Those graphs where edges have orientation (i.e., given to vertices u, v in V there exist an arc between u and v but not the reverse one) are known as directed graphs. Finally, weighted graphs are those where a number (weight) is assigned to each edge. These weights might represent costs, lengths or capacities, depending on the problem at hand. Some authors call such a graph a network.

Graphs are the basic modeling unit in a wide variety of areas within CS and OR, like project management, production scheduling, line balancing, business plans, or software visualization. Graph theory offers a rich source of problems and techniques for optimization, programming and data structure development. The design of good heuristics or approximation algorithms for optimization problems often requires significant specialized knowledge and trial-and-error over a graph. In many real-world applications, it is typically the case that the same optimization problem is solved again and again on a regular basis, maintaining the same problem structure but differing in the data. This provides an opportunity for learning heuristic algorithms that exploit the structure of such recurring problems. We study the reinforcement learning and graph embedding to address this challenge. The learned greedy policy behaves like a meta-algorithm that incrementally constructs a solution, and the action is determined by the output of a graph embedding network capturing the current state of the solution.

The main objective of this project is to solve difficult optimization problems, NP-hard in scientific terms, see Garey and Johnson, (1990), on a graph modeling a real situation. A difficult optimization problem is one for which it is not possible to guarantee that the optimal solution will be found within a “reasonable” computational time. There are many difficult problems in practice which trigger the development of efficient procedures capable of finding “good” solutions even when these solutions could not be proven optimal. These methods, for which solution quality is as important as the computing time needed to find it, are known as heuristics. Nowadays, a family of modern and intelligent heuristics, usually known as metaheuristics (Glover, 1986), are master strategies that guide subordinate algorithms to produce solutions beyond those that are normally obtained in a quest for local optimality. Metaheuristics are among the most prominent and successful techniques to solve a large amount of complex and computationally hard combinatorial and numerical optimization problems. Both research groups have a long experience on the development and application of these methodologies. In this project, we propose the application of the metaheuristic methodologies to solve difficult graph-based problems arising in business and industry.

]]>Cyberspace plays nowadays a key role in modern societies and economies, and its protection is a pivotal challenge in national security strategies. Over the last decade, various technological developments have contributed to make our dependency on cyberspace even greater, including the generation and processing of massive amounts of data, the influence of social networks over all activities of our daily lives, or the trend to

connect to Internet virtually every real-world device.

The scientific program proposed in this project aims at contributing to a more secure cyberspace in our current and future technological context. Our approach identifies three paradigms, each with a varying degree of maturity, that will reshape cybersecurity in the coming years. These are: the processing of massive amounts of information (big data), including those generated by citizens and devices; the embedding of computers in essentially all reallife objects (cyberphysical systems) and their connection to the Internet (IoT, Internet of Things); and the challenges and opportunities associated with the rise of quantum computing. To address these challenges we propose an interdisciplinary work program involving five research groups with proven expertise in the areas of system and application security, data analysis, next generation communication systems, and cryptography.

The project establishes three general objectives closely linked to the three challenges discussed above. First, we will develop advanced data analysis techniques to deal with massive amount of data, with a focus on two key application domains: social networks and new generation networks. Second, we will propose security mechanisms and tools for connected IoT devices and the network services they access. Finally, we will explore cryptographic techniques providing information security services and protecting users privacy. This will be done by leveraging new techniques such as the applications of the blockchain technology, as well as the threat posed by quantum computers.

Expected results include a significant advance in the research lines sketched above. In addition, harmonization between research and innovation tasks is ensured by the presence in the consortium of relevant industry partners (BBVA, Vicomtech y Unisys) with a keen interest in developing marketfitting technologies, together with various extraordinarily relevant institutional partners (INCIBE, Jefatura de Información de la Guardia Civil y Centro Criptológico Nacional). Thus, it is expected that this project’s results will be transferred and exploited as soon as they develop, contributing to an improved national cybersecurity at all levels. Consequently, our expected results have a very high technological relevance since they will provide tools for a more trustworthy cyberspace for citizens, companies, and Public Administrations.

The project goals are perfectly aligned with the Spanish and European priorities for the development of secure environments, which aim at strengthening citizen’s rights and improving the competitiveness of our industries and our defense capabilities.

]]>There are a large amount of problems that are framed within the context of combinatorial optimization characterized by the high interest associated with their practical resolution. This project deals with five distinct families of combinatorial problems. These are:

- Ordering problems: with applications in VLSI design or in the efficient resolution of systems of equations.
- Location problems: with interest in telecommunication applications such as distribution of signal regenerators or network design.
- Graph-based problems: with applications in the distribution of electronic devices in electronic boards or in image segmentation.
- Routing problems: by focusing on multi-objective problems with applications in the transport of hazardous materials or in recommendation systems.
- Selection problems: with applications in the construction of diverse groups or clustering of documents.

The methodology to solve the problems described above are metaheuristics procedures, among which we can highlight evolutionary algorithms, tabu search, variable neighborhood search or GRASP, to name some of the best known. For each combinatorial problem, we will propose the most suitable metaheuristic according to their mathematical structure or model. We will mainly focus on design of novel strategies to obtain high quality solutions. Besides, it is expected to figure out general strategies that can be easily applied to other related problems. We will also focus on the efficient and flexible implementation of those strategies, taking advantage of new language programming features and multi-core microprocessors. Finally we will focus on exploitation through a management platform, which integrates the problems addressed above. Simultaneously, we will develop an application to put into production (in the companies interested in our research project) the algorithms developed during the project.

In addition to solve the problems presented above, a second objective of the project is to develop the metaheuristic methodologies themselves. To successfully meet this challenge the research team has researcher Nenad Mladenovic, which developed in conjunction with Pierre Hansen the variable neighborhood search methodology.

All these problems will be integrated into Optsicom, a software tool that allow the execution of algorithms devoted to solve optimization problems and analyze the associated results. Optsicom can be used at two levels: as an end-user or as a researcher on heuristic methods. In this line, the problems integrated in Optsicom will also be available via the Web at Optsicom Repository, a web platform for comprehensive management of optimization problems. This platform will post all the information associated with an optimization problem. For each problem, it is expected to store the description, algorithms, instances, experimental results and relevant references. Furthermore, the results obtained by executing the algorithms can be compared using different statistical tests that are available as part of the software tool.

]]>El tratamiento y organización de la enorme cantidad de información en formato electrónico de la que se dispone actualmente se han convertido en una necesidad dentro de esta Sociedad de la Información en la que vivimos. En consecuencia, no tendría sentido disponer de grandes repositorios de información de carácter lingüístico en los que no pudiéramos extraer un conocimiento útil. Este proyecto se enmarca dentro del área de la optimización heurística aplicada a problemas del Procesamiento de Lenguaje Natural e Ingeniería Lingüística. En este proyecto se desarrollará un software para la optimización de diferentes problemas de optimización con el objetivo de dar solución a dos de los grandes problemas a los que se enfrenta actualmente el campo del Procesamiento del Lenguaje Natural: la clasificación automática y el agrupamiento, o clustering, de documentos.

Los problemas que se pretenden abordar están basados en modelos estructurados; es decir, en los que se conoce una descripción o formulación matemática completa. Se propondrán diferentes modelos de resolución eficientes basados en procedimientos metaheurísticos. Los métodos que se propongan serán comparados con los mejores métodos de resolución existentes para ese tipo de problemas, tanto en el ámbito académico como en el comercial. Se pretende que esto dé lugar, tanto a una aplicación que proporcione soluciones de gran calidad, como a publicaciones científicas de impacto internacional.

Existe un tipo de problemas de optimización especialmente difíciles de resolver en los que se dispone sólo de información parcial, denominados Sistemas Complejos. En ellos no se tiene una descripción explícita del problema ya que algunos de sus elementos característicos, como son la función objetivo o las restricciones, se obtienen de forma indirecta. Como consecuencia, éstos se tratan como una caja negra.

El proyecto de investigación se centrará en el diseño de un Solver genérico (Context-Independent Solver) para la optimización de sistemas complejos mediante técnicas metaheurísticas. El Solver desarrollado generará soluciones como entrada a la caja negra y posteriormente analizará el resultado devuelto, extrayendo información sobre las soluciones, de tal forma que iterativamente se vayan generando soluciones de mayor calidad

Para diseñar el Solver, en primer lugar se categorizarán los problemas en función de si están descritos mediante variables enteras, permutaciones de elementos o variables continuas. Posteriormente, se diseñará un método basado en metaheurísticas para resolver cada tipo de problema. El último paso del diseño del Solver genérico consistirá en la integración de los tres métodos en un único esquema general que seleccionará el más adecuado para la resolución de cada problema. El Solver se complementará con una implementación del mismo en una herramienta denominada COSYO.

COSYO será un Solver genérico para la optimización de sistemas complejos modelados como una caja negra. Se considerarán dos perfiles de usuario de la herramienta. Por un lado, investigadores o profesionales con conocimientos de optimización (usándolo como librería de programación) y, por otro lado, profesionales que no tengan conocimientos avanzados en optimización (usándolo desde la hoja de cálculo de OpenOffice.org)

Los métodos propuestos en el desarrollo del proyecto se compararán con los mejores métodos existentes para ese tipo de problemas tanto en el ámbito académico como en el comercial. Esto dará lugar tanto a una aplicación que proporcione soluciones de gran calidad como a publicaciones científicas de impacto internacional.

]]>Los sistemas de acceso a la información multimedia que trabajan sobre colecciones de imágenes suelen tener acceso a dos tipos de datos: los descriptores textuales y el contenido visual de las imágenes. Tradicionalmente, estos sistemas han abordado o bien el problema de la recuperación de imágenes analizando la información textual asociada (TextBased Information Retrieval, TBIR) o bien analizando el contenido visual (ContentBased Information Retrieval, CBIR). Hasta hace unos pocos años, las aproximaciones mixtas no aportaban ninguna ventaja a los resultados, además de ser bastante ineficientes.

Por un lado, investigadores de NLP&IRUNED y del grupo de Vision Team de la Universitat de Valencia coordinaron su experiencia previa en recuperación textual y la basada en contenido de imágenes. Fruto de los trabajos de esta colaboración, ha sido una aproximación que no solo se aprovecha de la sinergia entre los aspectos visuales y de las anotaciones textuales conjuntamente, sino que además aporta un método de cálculo eficiente para la búsqueda de imágenes anotadas en grandes colecciones, a partir de una consulta multimedia, ya sea texto y una o varias imágenes. Este trabajo ha generado, además de participaciones en competiciones como ImageCLEF y MediEval, varias publicaciones en actas de congresos, un artículo en la revista IEEE Transactions on Multimedia Journal y una tesis doctoral en el grupo NLP&IRUNED titulada Fusión Multimedia Semántica Tardía aplicada a la Recuperación de Información Multimedia.

Por otro lado, otro equipo mixto formado por integrantes de NLP&IRUNED y GAVABURJC han integrado tecnologías previas para construir un sistema híbrido de búsqueda de imágenes. La propuesta, que combinaba rasgos de contenido y análisis del texto enriquecido con recursos lingüísticos como WordNet, participó en dos ediciones de la competición Photo Annotation Task de ImageCLEF.

]]>There are a large amount of problems that are framed within the context of combinatorial optimization characterized by the high interest associated with their practical resolution. This project deals with five distinct families of combinatorial problems. These are:

- Ordering problems: with applications in VLSI design or in the efficient resolution of systems of equations.
- Location problems: with interest in telecommunication applications such as distribution of signal regenerators or network design.
- Graph-based problems: with applications in the distribution of electronic devices in electronic boards or in image segmentation.
- Routing problems: by focusing on multi-objective problems with applications in the transport of hazardous materials or in recommendation systems.
- Selection problems: with applications in the construction of diverse groups or clustering of documents.

The methodology to solve the problems described above are metaheuristics procedures, among which we can highlight evolutionary algorithms, tabu search, variable neighborhood search or GRASP, to name some of the best known. For each combinatorial problem, we will propose the most suitable metaheuristic according to their mathematical structure or model. We will mainly focus on design of novel strategies to obtain high quality solutions. Besides, it is expected to figure out general strategies that can be easily applied to other related problems. We will also focus on the efficient and flexible implementation of those strategies, taking advantage of new language programming features and multi-core microprocessors. Finally we will focus on exploitation through a management platform, which integrates the problems addressed above. Simultaneously, we will develop an application to put into production (in the companies interested in our research project) the algorithms developed during the project.

In addition to solve the problems presented above, a second objective of the project is to develop the metaheuristic methodologies themselves. To successfully meet this challenge the research team has researcher Nenad Mladenovic, which developed in conjunction with Pierre Hansen the variable neighborhood search methodology.

All these problems will be integrated into Optsicom, a software tool that allow the execution of algorithms devoted to solve optimization problems and analyze the associated results. Optsicom can be used at two levels: as an end-user or as a researcher on heuristic methods. In this line, the problems integrated in Optsicom will also be available via the Web at Optsicom Repository, a web platform for comprehensive management of optimization problems. This platform will post all the information associated with an optimization problem. For each problem, it is expected to store the description, algorithms, instances, experimental results and relevant references. Furthermore, the results obtained by executing the algorithms can be compared using different statistical tests that are available as part of the software tool.

]]>Optsicom project, University of Valencia (Spain)

Let G(V,E) be an undirected graph where *V (n = |V|)* and *E (m = |E|)* are the sets of vertices and edges, respectively. A linear layout φ of the vertices of *G* is a bijection or mapping φ : *V → {1, 2, . . . , n}* in which each vertex receives a unique and different integer between 1 and *n*. For vertex u, let *φ(u)* denote its position or label in layout φ. Let *L(p,φ,G)* be the set of vertices in V with a position in the layout φ lower than or equal to position *p*. Symmetrically, let R(p,φ,G) be the set of vertices with a position in the layout φ larger than position *p*. In mathematical terms,

Since layouts are usually represented in a straight line, where the vertex in position 1 comes first, *L(p,φ,G)* can be simply called the set of left vertices with respect to position *p* and, *R(p,φ,G)* the set of the right vertices w.r.t. *p*.

The Cut-value at position *p* of layout *φ, Cut(p,φ,G)*, is defined as the number of vertices in *L(p,φ,G)* with one or more adjacent vertices in *R(p,φ,G)*, then,

where *N(u) = {v ∈ V : (u, v) ∈ E}*. The Vertex Separation value *(VS)* of layout φ is the maximum of the Cut-value among all positions in layout *φ: VS(φ,G) = *max* _{p} Cut(p,φ,G)*.

Figure 1.a shows an illustrative example of an undirected graph *G* with 7 vertices and 9 edges. Figure 1.b depicts a solution (layout) φ of this graph and the Cut-value of each position *p, Cut(p,φ,G)*. For example, *Cut(1,φ,G) = 1* because *L(1,φ,G) = {D}* and *R(1,φ,G) = {A, F, G, E, B, C}* and there is one vertex in *L*having an adjacent vertex in *R*. Similarly, *Cut(3,φ,G) = 2* where *L(3,φ,G) = {D, A, F}* and *R(3,φ,G) = {G, E, B, C}*. The objective function value, computed as the maximum of these cut values, is *VS(G,φ) = 3* whose related position is *p = 4*.

The decisional version of the VSP was proved to be NP-complete for general graphs (Lengauer, 1981). It is also known that the problem remains NP-complete for planar graphs with maximum degree of three (Monien and Sudborough, 1988), as well as for chordal graphs (Gustedt, 1993), bipartite graphs (Goldberg et al. 1995), grid graphs and unit disk graphs (Diaz et al. 2001).

We can find many different graph problems that, although stated in different terms, are equivalent to the VSP in the sense that a solution to one problem provides a solution to the other one. Some of them are the Path-Width problem (Kinnersley 1992), the Interval Thickness problem (Kirousis and Papadimitriou 1985), the Node Search Number (Kirousis and Papadimitriou 1986} and the Gate Matrix Layout (Kinnersley and Langston 1994). The equivalence between these problems is a consequence of the results presented in (Fellows and Langston, 1994, Kinnersley 1992 and Kirousis and Papadimitriou 1986). For any graph *G* let *VS(G)*, *PW(G)*, *IT(G)*, *SN(G)* and *GML(G)* be the objective function value of the optimal solution for the Vertex Separation, Path-Width, Interval Thickness, Node Search Number and Gate Matrix Layout problems, respectively. These values verify the following relations:

*VS(G) = PW(G) = IT(G) = SN(G) – 1 = GML(G) + 1.*

The VSP appears in the context of finding “good separators” for graphs (Lipton and Tarjan 1979) where a separator is a set of vertices or edges whose removal separates the graph into disconnected subgraphs. This optimization problem has applications in VLSI design for partitioning circuits into smaller subsystems, with a small number of components on the boundary between the subsystems (Leiserson, 1980). The decisional version of the VSP consists of finding a vertex separation value larger than a given threshold. It has applications on computer language compiler design and exponential algorithms. In compiler design, the code to be compiled can be represented as a directed acyclic graph (DAG) where the vertices represent the input values to the code as well as the values computed by the operations within the code. An edge from node *u* to node *v* in this DAG represents the fact that value *u* is one of the inputs to operation *v*. A topological ordering of the vertices of this DAG represents a valid reordering of the code, and the number of registers needed to evaluate the code in a given ordering is precisely the vertex separation number of the ordering (Bodlaender et al., 1998). The decisional version of VSP has also applications in graph theory (Fomin and Hie, 2006). Specifically, if a graph has a vertex separation value, say *w*, then it is possible to find the maximum independent set of *G* in time *O(2 ^{w} n)*. Other practical applications include Graph Drawing and Natural Language Processing (Dujmovic et al., 2008, Miller, 1956).

We have experimented with three sets of instances, totalizing 173 instances:

**HB**: We derived 73 instances from the Harwell-Boeing Sparse Matrix Collection. This collection consists of a set of standard test matrices*M = M*arising from problems in linear systems, least squares, and eigenvalue calculations from a wide variety of scientific and engineering disciplines. The graphs are derived from these matrices by considering an edge_{uv}*(u, v)*for every element*M*. From the original set we have selected the 73 graphs with_{uv}= 0*n ≤ 1000*. The number of vertices and edges range from 24 to 960 and from 34 to 3721, respectively.**Grids:**This set consists of 50 matrices constructed as the Cartesian product of two paths (Raspaud et al., 2009). They are also called two dimensional meshes and the optimal solution of the VSP for squared grids is known by construction, see [7]. Specifically, the vertex separation value of a square grid of size λ × λ is λ. For this set, the vertices are arranged on a square grid with a dimension λ × λ for 5 ≤ λ ≤ 54. The number of vertices and edges range from 5 × 5 = 25 to 54 × 54 = 2916 and from 40 to 5724, respectively.**Trees**: Let*T(λ)*be set of trees with minimum number of nodes and vertex separation equal to λ. As it is stated in Ellis et al. (1994), there is just one tree in*T(1)*, namely the tree with a single edge, and another one in*T(2)*, the tree constructed with a new node acting as root of three subtrees that belong to*T(1)*. In general, to construct a tree with vertex separation λ + 1 it is necessary to select any three members from*T(λ)*and link any one node from each of these to a new node acting as the root of the new tree. The number of nodes,*n(λ)*, of a tree in*T(λ)*can be obtained using the recurrence relation*n(λ) = 3n(λ – 1 ) + 1*where and*n(1) = 2*(see Ellis et al. (1994) for additional details). We consider 50 different trees: 15 trees in*T(3)*, 15 trees in*T(4)*and 20 trees in*T(5)*. The number of vertices and edges range from 22 to 202 and from 21 to 201, respectively.

The VSPLIB contains 173 instances:

Download Vertex Separation Instances (VSPLIB).

All the algorithms were implemented in Java SE 6 and the experiments were conducted on an Intel Core i7 2600 CPU (3.4 GHz) and 4 GB RAM. We have considered the VSPLIB instances described above. The individual results for each instance can be downloaded here in Excel format.

- Bodlaender HL, Gustedt J, Telle JA. Linear-time register allocation for a fixed number of registers.
*Proc. of teh Symposium on Discrete Algorithms*, 1998; 574–583. - Díaz J., Penrose M.D., Petit J., Serna M. Approximating layout problems on random geometric graphs.
*Algorithms*2001; 39(1):78–116. - Dujmovic V, Fellows MR, Kitching M, Liotta G, Mccartin K, Nishimura N, Ragde P, Rosamond FA, Whitesides S, Wood DR.
*On the Parameterized Complexity of Layered Graph Drawing. Algorithmica*2008; 52(2):267–292. - Ellis J.A., Sudborough I.H., Turner J.S.. The vertex separation and search number of a graph.
*Journal Information and Computation*, 1994; 113:50–79. - Fellows MR, Langston MA. On search, decision and the efficiency of polynomial-time algorithms.
*Journal of Computing Systems Sciences*1994; 49(3):769–779 - Fomin FV, Hie K. Pathwidth of cubic graphs and exact algorithms.
*Information Processing Letters*2006; 97:191–196. - Goldberg P.W., Golumbic M.C., Kaplan H., Shamir R. Four strikes against physical mapping of DNA.
*Journal of Computing Biology*1995; 12(1):139–152. - Gustedt J. On the pathwidth of chordal graphs.
*Discrete Applied Mathemastics*1993; 45(3):233–248. - Kinnersley N.G. The vertex separation number of a graph equals its path-width.
*Information Processing Letters*1992; 42(6):345–50. - Kinnersley NG, Langston MA. Obstruction set isolation for the gate matrix layout problem.
*Discrete Applied Mathematics*1994; 54(2-3):169–213. - Kirousis M, Papadimitriou C.H. Interval graphs and searching.
*Discrete Mathematics*1985; 55(2):181–184. - Kirousis M, Papadimitriou C.H. Searching and pebbling.
*Theory of Computation Sciences*1986; 47(2):205–218. - Leiserson CE. Area-Efficient Graph Layouts (for VLSI).
*Proc. of IEEE Symposium on Foundations of Computer Science*, 1980; 270–281. - Lengauer T. Black-White Pebbles and Graph Separation.
*Acta Informatica 1981*; 16:465–475. - Lipton RJ, Tarjan RE. A separator theorem for planar graphs.
*SIAM Journal of Applied Mathematics*1979; 36:177–189. - Miller GA. The Magical Number Seven, Plus or Minus Two.
*SIAM Journal of Applied Mathematics*1956; 13:81–97. - Monien B., Sudborough I.H. Min Cut is NP-Complete for Edge Weighted Treees.
*Theory of Computatiuon Sciences*1988;58:209–229. - Raspaud A., Schröder H., Sýkora O., Török L., and Vrt’o I. Antibandwidth and cyclic antibandwidth of meshes and hypercubes.
*Discrete Mathematics*, 2009; 309:3541–3552.

University of Valencia (Spain), University of Heidelberg (Germany) and University Rey Juan Carlos (Spain)

Let *D*_{n} = (*V*_{n}, *A*_{n}) denote the complete digraph on *n* nodes, where *V*_{n} is the set of nodes and *A*_{n} the set of arcs. A *tournament T* in *A*_{n} consists of a subset of arcs containing for every pair of nodes *i* and *j* either arc (*i*, *j*) or arc (*j*, *i*), but not both. *T* is an *acyclic tournament* if it does not contain any directed cycle. Obviously, an acyclic tournament induces an ordering < *v*_{i1}, *v*_{i2},…,*v*_{in} > of the nodes (and vice versa). Node *v*_{i1} is the one with no entering arcs in *T*, *v*_{i2} has exactly one entering arc, etc., and *v*_{in} is the node with no outgoing arc. Given arc weights *w*_{ij} for every pair *i*, *j* in *V*_{n}, the *linear ordering problem* (lop) consists of finding an acyclic tournament *T* in *A*_{n} such that the sum of the weights of arcs in *T* is maximal, or in other words, of finding an ordering of the nodes such that the sum of the weights of the arcs compatible with this ordering is maximal.

Given an (*n*, *n*) matrix *C* = (*c*_{ij}) the triangulation problem is to determine a simultaneous permutation of the rows and columns of *C* such that the sum of superdiagonal entries becomes as large as possible (or equivalently, the sum of subdiagonal entries is as small as possible). Note, that it does not matter if diagonal entries are taken into account or not. Obviously, by setting arc weights *w*_{ij} = *c*_{ij} for the complete digraph *D*_{n}, the triangulation problem for*C* can be solved as linear ordering problem *D*_{n}. Conversely,a linear odering problem for *D*_{n} can be transformed to a triangulation problem for an (*n*, *n*) matrix *C* by setting *c*_{ij} = *w*_{ij} and the diagonal entries *c*_{ii} = 0 (or to arbitrary values).

The LOP can be formulated as 0/1 linear integer programming problem as follows. We use 0/1 variables *x*_{ij}, for (*i*, *j*) in*A*_{n}, stating whether arc (*i*, *j*) is present in the tournament or not. Taking into account that a tournament is acyclic if and only if it does not contain any dicycle of length 3, it is easily seen that the LOP can be formulated as the 0/1-IP.

The most relevant metaheuristics developed to solve this problem are:

**TS**: Tabu Search. Laguna et al (1999)**MA**: Memetic Algorithm. Schiavinotto and Stützle (2004)**VNS**: Variable Neighbourhood Search. García et al (2006)**SA**: Simulated Annealing. Charon and Hudry (2007)**SS**: Scatter Search. Campos et al (2001)**GRASP**: Greedy ramdomized adaptive search procedure. Campos et al (2001)

We have compiled a comprehensive set of benchmark problems including all problem instances which have so far been used for conducting computational experiments for the \lop. Furthermore we have included new instances. In their original definition, some problem instances are not in normal form. For the computations documented here, all problems have been transformed to normal form. We give a brief description of the origin and the characteristics of the groups of problems.

**Input/Output matrices**: This is a well-known set of instances that contains 50 real-world linear ordering problems generated from input-output tables from various sources (Grötschel et al 1984). They are comparatively easy for nowadays metaheuristics and are thus more of interest for economists than for the assessment of approximate methods for hard problems. The original entries in these tables were not necessarily integral, but for \LOLIB\ they were scaled to integral values. Download.**SGB instances**: These instances are taken from the*Stanford GraphBase*and consist of input-output tables from sectors of the economy of the United States (Knuth 1993). The set has a total of 25 instances with 75 sectors. Download.**Random instances of type A**: This is a set with 175 random problems that has been widely used for experiments. Problems of type I (called*RandomAI*), are generated from a [0,100] uniform distribution. This type of problems was proposed in Reinetl (1985) and generated in Campos et al (2001). Problems were originally generated from a [0, 25000] uniform distribution in Laguna et al (1999) and modified afterwards, sampling from a significatively narrow range ([0,100]) to make them harder to solve. Sizes are*n*= 100, 150 and 200 and there are 25 instances in each set giving a total of 75. We have extended this set including 25 additional instances with size*n*= 500. Download. Problems of type II, which we call*RandomAII*, are generated by counting the number of times a sector appears in a higher position than another in a set of randomly generated permutations. This type of problems was proposed in Chanas and Kobylanski (1996) and generated in Campos el al (2001). For a problem of size*n*,*n*/2 permutations are generated. There are 25 instances with sizes 100, 150 and 200, respectively. Download.**Random instances of type B**: For these random problems, the superdiagonal entries are drawn uniformly distributed from the interval [0,*U*_{1}] and the subdiagonal entries from [0,*U*_{2}], where*U*_{1}>=e*U*_{2}. Download.**Instances of Mitchell and Borchers**: These instances have been used by Mitchell and Borchers for their computational experiments (Mitchell and Borchers, 2000). They are random matrices where the subdiagonal entries are uniformly distributed in [0,99] and the superdiagonal entries are drawn uniformly from [0,39]. Furthermore a certain percentage of the entries was zeroed out. Download.**Instances of Schiavinotto and Stützle**: Some further benchmark instances have been created and used by Schiavinotto and Stützle (2004). These instances were generated from the real-world input-output tables by replicating them to obtain larger problems. Thus, the distribution of numbers in these instances somehow reflects real input-output tables, but otherwise they behave more like random problems. Tha data set has been called XLOLIB, instances with*n*= 150 and*n*= 250 are available. For each original input-output instance, two instances, one of size*n*=150 and another one of size*n*= 250 were generated. Therefore this set contains 98 instances (49 with size 150 and 49 with size 250). We have removed 20 of these instances because there entries were so large that the sum of entries was not representable as 4-byte integer. Therefore, this set finally has 78 instances. Download.**Further special instances**: We added some further problem instances that were used for experiments in some publications.*EX*instances were used in particular in Christof (1997) and in Christof and Reinelt (1996).*econ*instances were generated from the matrix usa79. They turned out not to be solvable as linear program using only 3-dicycle inequalities.*atp*instances were created from the results of ATP tennis tournaments in 1993/1994. Nodes correspond to a selection of players and the weight of an arc (*i*,*j*) is the number of victories of player*i*against player*j*. Paley graphs have been used in Goemans and Hall (1996) to prove results about the acyclic subdigraph polytope. They are a special class of tournaments where adjacency comes from an algebraic definition. They are constructed from the members of a suitable finite field by connecting pairs of elements that differ in a quadratic residue. Download.

We have performed an intensive experimentation with the best know methods (run for 2 hours) to compute the best known values for the instances. We have also reviewed related literature searching for better values than these. The final best known values for the instances can be downloaded here.

We performed a computational comparison of the state of the art methods on the instances. We have considered two different time limits in our comparative: 10 seconds (to measure the aggressiveness) and 10 minutes (to measure the robustness). Both experiments were performed in a Dual Intel Xeon at 3.06 GHz with 3.2 GiB of RAM. It can be downloaded in Excel.

- Campos, V., Glover, F. Laguna, M. and Martí, R.: An experimental evaluation of a scatter search for the linear ordering problem,
*Journal of Global Optimizationz*21 (2001), 397–414 - Chanas, S. and Kobylanski, P.: A new heuristic algorithm solving the linear ordering problem,
*Computational Optimization and Applications*6 (1996), 191–205 - Charon, I. and Hudry, O.: A survey on the linear ordering problem for weighted or unweighted tournaments,
*4OR*5 (2007), 5–60. - Christof, T.: Low-dimensional 0/1-polytopes and branch-and-cut,
*Combinatorial Optimization*,Shaker, 1997 - Christof, T. and Reinelt, G.: Combinatorial optimization and small polytopes,
*Top*4 (1996), 1-64 - García, C.G., Pérez-Brito, D., Campos, V. and Martí, R.: Variable neighborhood search for the linear ordering problem,
*Computers and Operations Research*33 (2006),3549–3565 - Goemans, M.X.and Hall, L.A.: The strongest facets of the acyclic subgraph polytope are unknown,
*Proc. of the 5th Int. IPCO Conference*, LNCS 1084, Springer, 1996, 415-429 - Grötschel, M., Jünger, M. and Reinelt, G.: A cutting plane algorithm for the linear ordering problem,
*Operations Research*32 (1984), 1195–1220. - Knuth, D.E.: The Stanford GraphBase: a platform for combinatorial computing, Addison-Wesley, 1993.
- Laguna, M., Martí R. and Campos, V.: Intensification and diversification with elite tabu search solutions for the linear ordering problem,
*Computers and Operations Research*26 (1999), 1217–1230. - Mitchell, J.E., Borchers, B: Solving linear ordering problems, in: Frenk, H., Roos, K., Terlaky, T.Zhang, s. (eds.), High Performance Optimization, Applied Optimization Vol. 33) Kluwer, 2000, 340–366
- Reinelt, G.: The linear ordering problem: algorithms and applications, research and exposition in mathematics 8, Heldermann, 1985.
- Schiavinotto, T. and Stützle, T.: The linear ordering problem: Instances, search space analysis and algorithms,
*Journal of Mathematical Modelling and algorithms*3 2004, 367–402.

Optsicom project, University of Valencia (Spain)

The Cutwidth Minimization Problem (CMP) is an NP-hard problem (Gavril 1977) and consists of finding a linear layout of a graph so that the maximum linear cut of edges (i.e., the number of edges that cut a line between consecutive vertices) is minimized. The cutwidth minimization problem can be easily described in mathematical terms. Given a graph *G*=*(V,E)* with *n*=|*V*| and *m*=|*E*|, a labeling or linear arrangement *f* of *G* assigns the integers {1,2,…,*n*} to the vertices in *V*, where each vertex receives a different label. The cutwidth of a vertex *v* with respect to *f*, *CW*_{f}(*v*), is the number of edges (*u,w*) ∈ *E*satisfying *f*(*u*)≤*f*(*v*)<*f*(*w*). The cutwidth of the graph, *CW*_{f}(*G*), is the maximum of the cutwidth of its vertices:

Figure 1.a shows an example of an undirected graph with 6 vertices and 7 edges. Figure 1.b shows a labeling, *f*, of the graph in Figure 1.a, setting the vertices in a line with the order of the labeling, as commonly represented in the CMP. In this way, since *f*(*C*)=1, vertex *C* comes first, followed by vertex *A*(*f*(*A*)=2) and so on. We represent *f* with the ordering (*C, A, D, E, B, F*) meaning that vertex *C* is located in the first position (label 1), vertex *A* is located in the second position (label 2) and so on. In Figure 1.b, the cutwidth of each vertex is represented as a dashed line with its corresponding value. For example, the cutwidth of vertex *C* is *CW*_{f}(*C*) = 1, because the edge (*C,B*) has and endpoint in *C* labeled with 1 and the other endpoint in a vertex labeled with a value larger than 1. In a similar way, we can compute the cutwidth of vertex *A*, *CW*_{f}(*A*)=4, by counting the appropriate number of edges ((*C,B*), (*A,B*), (*A,E*), and (*A,D*)). Then, since the cutwidth of the graph *G*, *CW*_{f}(*G*), is the maximum of the cutwidth of all vertices in *V*, in this particular example we obtain *CW*_{f}(*G*)=*CW*_{f}(*D*)=5.

The CMP can be formulated as 0/1 linear integer programming problem as follows (Luttamaguzi et al., 2005):

where *x*_{i}^{k} is a decision binary variable whose indices are *i, k* ∈ {1,2,…,*n*}. This variable specifies whether *i* is placed in position *k* in the ordering. In other words, for all *x*_{i}^{k} (*i, k*={1, 2,…,*n*})they take on value 1 if and only if *i* occupies the position *k* in the ordering; otherwise *x*_{i}^{k} takes on value 0. Constraints (3) and (4) ensure that each vertex is only assigned to one position and one position is only assigned to one vertex respectively. Consequently, contraints (1), (2), (3) and (4) together implies that a solution of the problem is an ordering.

The decision binary variable *y*_{i}^{k}_{j}^{l} is defined as *x*_{i}^{k} ∧ *x*_{j}^{l}, where *i, j* ∈ {1,2,…,*n*}, (*v*_{i}, *v*_{j}) ∈ *E* and *k, l* ∈ {1,2,…,*n*} the labels associated to vertex *v*_{i} and *v*_{j} respectively. In the linear formulation above this conjunction is computed with constraints (5), (6) and (7).

Constraint (8) computes for each position *c* in the ordering, the number of edges whose origin is placed in any position *k* (1 ≤ *k* < c) and destination in any position *l* (*c* < *l* ≤ *n*). The cutwidth problem consists of minimizing the maximum number of cutting edges in any position, *c* ∈{1,…,*n* – 1} of the labeling. Therefore, the objective function *b* must be larger than or equal to this quantity.

The most relevant heuristic methods developed to solve this problem are:

**GRASP+Path Relinking**: Hybrid method that combines GRASP methodology with Path-Relinking. Andrade and Resende (2007a).**GRASP+Evolutionary Path Relinking**: Hybrid method that combines GRASP methodology with Evolutionary Path-Relinking. Andrade and Resende (2007b).

**Simulated Annealing**: Simulated Annealing based on different constructive methods and local search procedures. Cohoon and Sahni (1987)

**Scatter Search**: Scatter Search method based on a GRASP constructive algorithm, a local search strategy based on insertion moves and voting-based combination methods. Pantrigo et al. (2012).

**Tabu Search**: A Tabu Search method used to help a Branch-and-bound algorithm. Palubeckis and Rubliauskas (2012).

**Variable Neighbourhood Search**: A new variant of the Variable Neighbourhood Search framework called Variable Formulation Search for the Cutwidth Minimization Problem. Pardo et al. (2013). (Results)

We have compiled three sets of instances for our experimentation, totalizing 252 instances. The first one, Small, was introduced in Martí et al. (2008), the second one, Grids, was introduced in Rolim et al. (1995) and the third one, Harwell-Boeing, is a subset of the public-domain Matrix Market library (available at http://math.nist.gov/MatrixMarket/data/Harwell-Boeing/). Next, it can be found a brief description of the origin and characteristics of the sets of instances:

**Small**: This data set consists of 84 graphs introduced in the context of the bandwidth reduction problem. The number of vertices ranges from 16 to 24, and the number of edges ranges from 18 to 49.**Grids:**This data set consists of 81 matrices constructed as the Cartesian product of two paths (Raspaud et al., 2009). They are also called two dimensional meshes and, as documented in Raspaud et al. (2009), the optimal solution of the cutwidth problem for these types of instances is known by construction. For this set of instances, the vertices are arranged on a grid with a dimension width x height where width, height are selected from the set {3, 6, 9, 12, 15, 18, 21, 24, 27}.**Harwell-Boeing**: We derived 87 instances from the Harwell-Boeing Sparse Matrix Collection. This collection consists of a set of standard test matrices*M*= (*M*_{ij}) arising from problems in linear systems, least squares, and eigenvalue calculations from a wide variety of scientific and engineering disciplines. Graphs are derived from these matrices by considering an edge (*i,j*) for every element*M*_{ij}≠ 0. From the original set we have selected the 87 graphs with*n*≤ 700. Their number of vertices ranges from 30 to 700 and the number of edges from 46 to 41686.

The CMPLIB contains 252 instances:

Download Cutwidth Instances (CMPLIB).

We have performed an intensive experimentation with the best know methods to compute the best known values for the instances. In addition, we have performed a computational comparison of the state of the art methods on the reported instances. The experiment was performed in a Intel Core 2 Quad CPU Q 8300 with 6 GiB of RAM and Ubuntu 9.04 64 bits OS. Results and best known values for instances can be reviewed in Pantrigo et al. (2010) and Martí et al. (2010) and can be downloaded here.

- Andrade, D.V. and M.G.C. Resende. GRASP with path-relinking for network migration scheduling.
*Proceedings of International Network Optimization Conference, 2007a.* - Andrade, D.V. and M.G.C. Resende. GRASP with evolutionary path-relinking.
*Proceedings of Seventh Metaheuristics International Conference (MIC), 2007b.* - Cohoon, J. and S. Sahni. Heuristics for the Board Permutation Problem.
*Journal of VLSI and Computer Systems*, 2, 37- 61, 1987. - Gavril, F. Some NP-complete problems on graphs.
*In Proceedings of the 11th conference on information Sciences and Systems*, 91-95, 1977. - Luttamaguzi, J., M. Pelsmajer, Z. Shen and B. Yang. Integer Programming Solutions for Several Optimization Problems in Graph Theory.
*Technical report, DIMACS,*2005. - Martí, R., V. Campos and E. Piñana, Branch and Bound for the Matrix Bandwidth Minimization.
*European Journal of Operational Research*, 186:513-528, 2008. - Martí, R., J.J. Pantrigo, A. Duarte and E.G. Pardo. Branch and Bound for Cutwidth Minimization Problem.
*Computers & Operations Research*. Volume 40, Issue 1, Pages 137–149, 2013 - Palubeckis, G., D. Rubliauskas. A branch-and-bound algorithm for the minimum cut linear arrangement problem.
*Journal of Combinatorial Optimization*. Volume 24, Issue 4, Pages 540-563, 2012 - Pantrigo, J.J., R. Martí, A. Duarte and E.G. Pardo. Scatter Search for the Cutwidth Minimization Problem
*Annals of Operations Research.*Volume 199, Issue 1, Pages 285-304, 2012. - Pardo, E.G., Mladenovic N., Pantrigo, J.J., Duarte A. Variable Formulation Search for the Cutwidth Minimization Problem.
*Applied Soft Computing.*Volume 13, Issue 5, Pages 2242–2252, 2013. - Raspaud, A., H. Schröder, O. Sýkora, L. Török, and I. Vrt’o. Antibandwidth and cyclic antibandwidth of meshes and hypercubes.
*Discrete Mathematics.*309:3541-3552, 2009. - Rolim, J., O. Sýkora and I. Vrt’o. Cutwidth of the de Bruijn graph.
*RAIRO Informatique Théorique et Applications*, 29(6):509-514, 1995.