Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem

Given an undirected graph, we study the capacitated vertex separator problem that asks to find a subset of vertices of minimum cardinality, the removal of which induces a graph having a bounded number of pairwise disconnected shores (subsets of vertices) of limited cardinality. The problem is of great importance in the analysis and protection of communication or social networks against possible viral attacks and for matrix decomposition algorithms. In this article, we provide a new bilevel interpretation of the problem and model it as a two-player Stackelberg game in which the leader interdicts the vertices (i.e., decides on the subset of vertices to remove), and the follower solves a combinatorial optimization problem on the resulting graph. This approach allows us to develop a computational framework based on an integer programming formulation in the natural space of the variables. Thanks to this bilevel interpretation, we derive three different families of strengthening inequalities and show that they can be separated in polynomial time. We also show how to extend these results to a min-max version of the problem. Our extensive computational study conducted on available benchmark instances from the literature reveals that our new exact method is competitive against the state-of-the-art algorithms for the capacitated vertex separator problem and is able to improve the best-known results for several difficult classes of instances. The ideas exploited in our framework can also be extended to other vertex/edge deletion/ insertion problems or graph partitioning problems by modeling them as two-player Stackel- berg games and solving them through bilevel optimization

Contributions to Determining Exact Ground-States of Ising Spin-Glasses and to their Physics

In the last decades, much research has focused on a better understanding of so-called spin glasses (e.g., the alloys CuMn and AuFe.) Spin glasses are not yet fully understood. In order to be able to test the different theories that have been proposed for the nature of spin glasses we have to analyze numerically generated data. We consider spin glasses in the Ising model, where the spins (magnetic dipoles) have exactly two possibilities for aligning themselves. We are interested in the low-temperature states of the system, in which the spins are `frozen' and disordered. Determining a state of minimum energy, a ground state, amounts to calculating a maximum cut in a graph. The max-cut problem is a prominent NP-hard problem from combinatorial optimization. Maximum cuts can be determined exactly with a branch-and-cut algorithm. This thesis consists of two parts. In the first part we introduce the max-cut problem and a branch-and-cut algorithm for solving reasonably sized problems. We present several approaches for improving its performance for Ising spin-glass instances. In the second part of this work, we study the physics of spin glasses. We first discuss what is known in the literature. Then we present results for Bethe spin glasses. Finally we study the nature of short-range three-dimensional spin glasses. Results of the former were obtained in collaboration with M. Palassini and A.K. Hartmann, results of the latter in cooperation with M. Palassini and A. Peter Young

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Models and algorithms for decomposition problems

This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms. This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced. The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity

Solving k-way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching.We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k = 3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solving the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far

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

Keyword search in graphs, relational databases and social networks

Keyword search, a well known mechanism for retrieving relevant information from a set of documents, has recently been studied for extracting information from structured data (e.g., relational databases and XML documents). It offers an alternative way to query languages (e.g., SQL) to explore databases, which is effective for lay users who may not be familiar with the database schema or the query language. This dissertation addresses some issues in keyword search in structured data. Namely, novel solutions to existing problems in keyword search in graphs or relational databases are proposed. In addition, a problem related to graph keyword search, team formation in social networks, is studied. The dissertation consists of four parts. The first part addresses keyword search over a graph which finds a substructure of the graph containing all or some of the query keywords. Current methods for keyword search over graphs may produce answers in which some content nodes (i.e., nodes that contain input keywords) are not very close to each other. In addition, current methods explore both content and non-content nodes while searching for the result and are thus both time and memory consuming for large graphs. To address the above problems, we propose algorithms for finding r-cliques in graphs. An r-clique is a group of content nodes that cover all the input keywords and the distance between each pair of nodes is less than or equal to r. Two approximation algorithms that produce r-cliques with a bounded approximation ratio in polynomial delay are proposed. In the second part, the problem of duplication-free and minimal keyword search in graphs is studied. Current methods for keyword search in graphs may produce duplicate answers that contain the same set of content nodes. In addition, an answer found by these methods may not be minimal in the sense that some of the nodes in the answer may contain query keywords that are all covered by other nodes in the answer. Removing these nodes does not change the coverage of the answer but can make the answer more compact. We define the problem of finding duplication-free and minimal answers, and propose algorithms for finding such answers efficiently. Meaningful keyword search in relational databases is the subject of the third part of this dissertation. Keyword search over relational databases returns a join tree spanning tuples containing the query keywords. As many answers of varying quality can be found, and the user is often only interested in seeing the·top-k answers, how to gauge the relevance of answers to rank them is of paramount importance. This becomes more pertinent for databases with large and complex schemas. We focus on the relevance of join trees as the fundamental means to rank the answers. We devise means to measure relevance of relations and foreign keys in the schema over the information content of the database. The problem of keyword search over graph data is similar to the problem of team formation in social networks. In this setting, keywords represent skills and the nodes in a graph represent the experts that possess skills. Given an expert network, in which a node represents an expert that has a cost for using the expert service and an edge represents the communication cost between the two corresponding experts, we tackle the problem of finding a team of experts that covers a set of required skills and also minimizes the communication cost as well as the personnel cost of the team. We propose two types of approximation algorithms to solve this bi-criteria problem in the fourth part of this dissertation

Binary matrix factorisations under Boolean arithmetic

For a binary matrix X, the Boolean rank br(X) is the smallest integer for which X can be factorised into the Boolean matrix product of two binary matrices A and B with inner dimension br(X). The isolation number i(X) of X is the maximum number of 1s no two of which are in a same row, column or a 2 x 2 submatrix of all 1s. In Part I. of this thesis, we continue Anna Lubiw's study of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds for all its submatrices. We show that the stronger concept of superfirmness of X is equivalent to having no odd holes in the rectangle cover graph of X, the graph in which br(X) and i(X) translate to the clique cover number and the independence number, respectively. A binary matrix is minimally non-firm if it is not firm but all of its proper submatrices are. We introduce a matrix operation that leads to generalised binary matrices and, under some conditions, preserves firmness and superfirmness. Then we use this matrix operation to derive several infinite families of minimally non-firm matrices. To the best of our knowledge, minimally non-firm matrices have not been studied before and our constructions provide the first infinite families of them. In Part II. of this thesis, we explore rank-k binary matrix factorisation (k-BMF). In k-BMF, we are given an m x n binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension m x k and k x n respectively, which minimise the distance between X and the Boolean matrix product of A and B in the squared Frobenius norm. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times a zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

Logic learning and optimized drawing: two hard combinatorial problems

Nowadays, information extraction from large datasets is a recurring operation in countless fields of applications. The purpose leading this thesis is to ideally follow the data flow along its journey, describing some hard combinatorial problems that arise from two key processes, one consecutive to the other: information extraction and representation. The approaches here considered will focus mainly on metaheuristic algorithms, to address the need for fast and effective optimization methods. The problems studied include data extraction instances, as Supervised Learning in Logic Domains and the Max Cut-Clique Problem, as well as two different Graph Drawing Problems. Moreover, stemming from these main topics, other additional themes will be discussed, namely two different approaches to handle Information Variability in Combinatorial Optimization Problems (COPs), and Topology Optimization of lightweight concrete structures