A branch-and-cut algorithm for the Edge Interdiction Clique Problem

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP

Benchmark Problems for Exhaustive Exact Maximum Clique Search Algorithms

There are well established widely used benchmark tests to assess the performance of practical exact clique search algorithms. In this paper a family of further benchmark problems is proposed mainly to test exhaustive clique search procedures

Algoritmos para o problema da clique máxima : análise e comparação experimental

Orientador : Prof. Dr. Renato CarmoTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa: Curitiba, 28/09/2017Inclui referências : f. 107-113Resumo: O problema da Clique Máxima (CM) é um problema fundamental e há uma grande motivação pela busca de algoritmos tão eficientes quanto possível para resolvê-lo de forma exata. Como esperado para um problema NP-difícil, os melhores algoritmos com desempenho de pior caso conhecido tem custo de tempo exponencial. Por outro lado, resultados experimentais encontrados na literatura indicam que instâncias de tamanho considerável podem ser resolvidas usando algoritmos baseados na técnica de branch-and-bound. Com isso, observa-se uma distância entre os melhores resultados analíticos e os melhores resultados experimentais. Uma possível explicação para discrepância aparente entre teoria e prática foi encontrada pela análise de instâncias aleatórias. Diversos algoritmos de branch- and-bound para a solução exata do CM foram estudados, analisados e implementados. Com base nos resultados analíticos é proposta uma metodologia para comparação experimental de algoritmos, que tem como principal ponto positivo o fato de que algoritmos podem ser comparados independente de detalhes de implementação e execução. Vários algoritmos foram testados como prova de conceito. Também foram estudadas instâncias de pior caso para algoritmos de branch-and-bound que só utilizam coloração como limitante superior, resultando em um custo exponencial de tempo para estes algoritmos. Uma nova família de algoritmos foi desenvolvida, capaz de resolver tais instâncias em tempo polinomial. Recentemente, técnicas de resolvedores para problemas de satisfatibilidade têm sido aplicadas em algoritmos para CM. Tais técnicas dependem de uma redução entre os dois problemas, mas o significado em termos do grafo fica obscurecido nas descrições originais. Algumas técnicas foram estudadas e convertidas para uma descrição que não usa termos referentes aos problemas de satisfatibilidade. A implementação de vários algoritmos estudados foi disponibilizada em um repositório de acesso público. Palavras-chave: Solução exata. Branch-and-bound. Análise de algoritmos. Comparação experimental.Abstract: e Maximum Clique problem (CM) is a fundamental problem and there is a great motivation for the development of efficient exact algorithms to solve it. As expected for a NP-hard problem, the best algorithms where worst case analyses have been conducted present exponential running times. On the other hand, experimental results available in the literature show that instances of considerable size can be solved by branch and bound algorithms. Therefore, there is an apparent gap between the best theoretical results and the best experimental results. One possible explanation for this discrepancy between theory and practice was found through the analyses of random instances. Several exact branch and bound algorithm for CM were studied, analyzed and implemented. Based on these analytical results, a new methodology for the comparison of algorithms is proposed, where algorithms can be tested and compared regardless of implementation and execution details. Several algorithms were tested as a proof of concept. Worst case instances for some branch and bound algorithms were studied, namely algorithms that adopt only coloring-based bounding techniques to reduce the search space. These algorithms present exponential time cost for the studied instances. A new family of algorithms was developed, which is able to solve the mentioned instances in polinomial time. Recently, techniques from satisfiability solvers have been used in algorithms for CM. Such techniques depend on a reduction between the problems, and the original descriptions in terms of propositional calculus obscures their graph theoretic meaning. Some of these techniques were studied and converted to a description that uses only graph theory terminology. The implementation of several algorithms was made available in a public access repository. Keywords: Exact solution. Branch-and-bound. Analysis of algorithms. Experimental comparison

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem

Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems

Certifying Correctness for Combinatorial Algorithms : by Using Pseudo-Boolean Reasoning

Over the last decades, dramatic improvements in combinatorialoptimisation algorithms have significantly impacted artificialintelligence, operations research, and other areas. These advances,however, are achieved through highly sophisticated algorithms that aredifficult to verify and prone to implementation errors that can causeincorrect results. A promising approach to detect wrong results is touse certifying algorithms that produce not only the desired output butalso a certificate or proof of correctness of the output. An externaltool can then verify the proof to determine that the given answer isvalid. In the Boolean satisfiability (SAT) community, this concept iswell established in the form of proof logging, which has become thestandard solution for generating trustworthy outputs. The problem isthat there are still some SAT solving techniques for which prooflogging is challenging and not yet used in practice. Additionally,there are many formalisms more expressive than SAT, such as constraintprogramming, various graph problems and maximum satisfiability(MaxSAT), for which efficient proof logging is out of reach forstate-of-the-art techniques.This work develops a new proof system building on the cutting planesproof system and operating on pseudo-Boolean constraints (0-1 linearinequalities). We explain how such machine-verifiable proofs can becreated for various problems, including parity reasoning, symmetry anddominance breaking, constraint programming, subgraph isomorphism andmaximum common subgraph problems, and pseudo-Boolean problems. Weimplement and evaluate the resulting algorithms and a verifier for theproof format, demonstrating that the approach is practical for a widerange of problems. We are optimistic that the proposed proof system issuitable for designing certifying variants of algorithms inpseudo-Boolean optimisation, MaxSAT and beyond

Solving hard subgraph problems in parallel

This thesis improves the state of the art in exact, practical algorithms for finding subgraphs. We study maximum clique, subgraph isomorphism, and maximum common subgraph problems. These are widely applicable: within computing science, subgraph problems arise in document clustering, computer vision, the design of communication protocols, model checking, compiler code generation, malware detection, cryptography, and robotics; beyond, applications occur in biochemistry, electrical engineering, mathematics, law enforcement, fraud detection, fault diagnosis, manufacturing, and sociology. We therefore consider both the ``pure'' forms of these problems, and variants with labels and other domain-specific constraints. Although subgraph-finding should theoretically be hard, the constraint-based search algorithms we discuss can easily solve real-world instances involving graphs with thousands of vertices, and millions of edges. We therefore ask: is it possible to generate ``really hard'' instances for these problems, and if so, what can we learn? By extending research into combinatorial phase transition phenomena, we develop a better understanding of branching heuristics, as well as highlighting a serious flaw in the design of graph database systems. This thesis also demonstrates how to exploit two of the kinds of parallelism offered by current computer hardware. Bit parallelism allows us to carry out operations on whole sets of vertices in a single instruction---this is largely routine. Thread parallelism, to make use of the multiple cores offered by all modern processors, is more complex. We suggest three desirable performance characteristics that we would like when introducing thread parallelism: lack of risk (parallel cannot be exponentially slower than sequential), scalability (adding more processing cores cannot make runtimes worse), and reproducibility (the same instance on the same hardware will take roughly the same time every time it is run). We then detail the difficulties in guaranteeing these characteristics when using modern algorithmic techniques. Besides ensuring that parallelism cannot make things worse, we also increase the likelihood of it making things better. We compare randomised work stealing to new tailored strategies, and perform experiments to identify the factors contributing to good speedups. We show that whilst load balancing is difficult, the primary factor influencing the results is the interaction between branching heuristics and parallelism. By using parallelism to explicitly offset the commitment made to weak early branching choices, we obtain parallel subgraph solvers which are substantially and consistently better than the best sequential algorithms

Optimisation de la gestion des avions dans un aéroport : affectation aux points de stationnement, routage au sol et ordonnancement à la piste.

In this thesis, we address the optimization of aircraft ground operations at airports, focusing on three main optimization problems: the stand allocation, the ground routing between stands and runways, and the sequencing of take-offs and landings.These works result from a close collaboration with Amadeus. Our approaches have been tested and validated with real data from European airports.The stand allocation problem is formulated as a Mixed Integer Program (MIP). We show that finding an allocation plan respecting operational requirements is NP-Complete and we strengthen our model in several directions. We obtain better solutions than the literature withing reasonable computation times for an industrial application.The ground routing problem is modeled by a MIP formulation adapted from the literature. We show that the main indicators of the industry are in contradiction with the objective of reducing taxi times and therefore air pollution. We propose new indicators based on take-off times instead of push back times.Lastly, we focus on the integration of the runway sequencing with the ground routing. We highlight that a better integration allows to reduce taxi times while improving the management of the runway. We propose a sequential heuristic based on an innovative MIP formulation of the runway sequencing problem. This heuristic is shown to provide high quality solutions in reasonable computation times, unlike the exact approach from the literature.Le cadre de cette thèse est l'optimisation des opérations aéroportuaires. Nous nous intéressons à trois problèmes de gestion des avions dans un aéroport : l'affectation aux points de stationnement, le routage au sol entre les pistes et les points de stationnement, et l'ordonnancement des décollages et des atterrissages.Ce travail a été réalisée en collaboration étroite avec la société Amadeus. Nos approches ont été testées et validées avec des données réelles provenant d'aéroports européens.Nous proposons une formulation en Programme Linéaire en Nombres Entiers (PLNE) du problème d'affectation aux points de stationnement. Nous montrons que trouver une affectation réalisable est un problème NP-Complet et nous proposons diverses améliorations visant à réduire le temps de résolution de notre modèle. Nous obtenons ainsi des solutions de meilleure qualité que celles de la littérature, tout en conservant un temps de calcul raisonnable.Le problème de routage au sol est modélisé en adaptant un PLNE de la littérature. Nous montrons que les indicateurs de l'industrie sont en contradiction avec l'objectif de réduction du temps de roulage, et donc des émissions de pollutions. Nous proposons de nouveaux indicateurs basés sur l'heure de décollage, et non sur l'heure de départ du point de stationnement.Enfin, nous nous intéressons à l'intégration de l'ordonnancement à la piste avec le routage au sol. Nous montrons qu'une meilleure intégration permet de réduire le temps de roulage et d'améliorer la gestion de la piste. Nous proposons une heuristique séquentielle basée sur une modélisation en PLNE innovante du problème d'ordonnancement à la piste. Nous montrons que cette heuristique fournit des solutions de bonne qualité en temps raisonnable, contrairement à l'approche exacte de la littérature