Theory and Practical Applications of Treewidth Tom C
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Generating Single and Multiple Cooperative Heuristics for the One Dimensional Bin Packing Problem Using a Single Node Genetic Programming Island Model
Generating Single and Multiple Cooperative Heuristics for the One Dimensional Bin Packing Problem Using a Single Node Genetic Programming Island Model Kevin Sim Emma Hart Institute for Informatics and Digital Innovation Institute for Informatics and Digital Innovation Edinburgh Napier University Edinburgh Napier University Edinburgh, Scotland, UK Edinburgh, Scotland, UK [email protected] [email protected] ABSTRACT exhibit superior performance than similar human designed Novel deterministic heuristics are generated using Single Node heuristics and can be used to produce collections of heuris- Genetic Programming for application to the One Dimen- tics that collectively maximise the potential of selective HH. sional Bin Packing Problem. First a single deterministic The papers contribution is two fold. Single heuristics ca- heuristic was evolved that minimised the total number of pable of outperforming a range of well researched deter- bins used when applied to a set of 685 training instances. ministic heuristics are generated by combining features ex- Following this, a set of heuristics were evolved using a form tracted from those heuristics. Secondly an island model[19] of cooperative co-evolution that collectively minimise the is adapted to use multiple SNGP implementations to gener- number of bins used across the same set of problems. Re- ate diverse sets of heuristics which collectively outperform sults on an unseen test set comprising a further 685 prob- any of the single heuristics when used in isolation. Both ap- lem instances show that the single evolved heuristic out- proaches are trained and evaluated using equal divisions of a performs existing deterministic heuristics described in the large set of 1370 benchmark problem instances sourced from literature. -
13. Mathematics University of Central Oklahoma
Southwestern Oklahoma State University SWOSU Digital Commons Oklahoma Research Day Abstracts 2013 Oklahoma Research Day Jan 10th, 12:00 AM 13. Mathematics University of Central Oklahoma Follow this and additional works at: https://dc.swosu.edu/ordabstracts Part of the Animal Sciences Commons, Biology Commons, Chemistry Commons, Computer Sciences Commons, Environmental Sciences Commons, Mathematics Commons, and the Physics Commons University of Central Oklahoma, "13. Mathematics" (2013). Oklahoma Research Day Abstracts. 12. https://dc.swosu.edu/ordabstracts/2013oklahomaresearchday/mathematicsandscience/12 This Event is brought to you for free and open access by the Oklahoma Research Day at SWOSU Digital Commons. It has been accepted for inclusion in Oklahoma Research Day Abstracts by an authorized administrator of SWOSU Digital Commons. An ADA compliant document is available upon request. For more information, please contact [email protected]. Abstracts from the 2013 Oklahoma Research Day Held at the University of Central Oklahoma 05. Mathematics and Science 13. Mathematics 05.13.01 A simplified proof of the Kantorovich theorem for solving equations using scalar telescopic series Ioannis Argyros, Cameron University The Kantorovich theorem is an important tool in Mathematical Analysis for solving nonlinear equations in abstract spaces by approximating a locally unique solution using the popular Newton-Kantorovich method.Many proofs have been given for this theorem using techniques such as the contraction mapping principle,majorizing sequences, recurrent functions and other techniques.These methods are rather long,complicated and not very easy to understand in general by undergraduate students.In the present paper we present a proof using simple telescopic series studied first in a Calculus II class. -
Lipics-WABI-2021-7.Pdf (2
Tree Diet: Reducing the Treewidth to Unlock FPT Algorithms in RNA Bioinformatics Bertrand Marchand # LIX CNRS UMR 7161, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-vallée, France Yann Ponty1 # LIX CNRS UMR 7161, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France Laurent Bulteau1 # LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-vallée, France Abstract Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed- Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the Tree-Diet problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth tw′. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account. Our core result is an FPT dynamic programming algorithm for Tree-Diet, using 2O(tw)n time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when tw′ or tw−tw′ is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics. -
Complexity and Exact Algorithms for Vertex Multicut in Interval and Bounded Treewidth Graphs 1
Complexity and Exact Algorithms for Vertex Multicut in Interval and Bounded Treewidth Graphs 1 Jiong Guo a,∗ Falk H¨uffner a Erhan Kenar b Rolf Niedermeier a Johannes Uhlmann a aInstitut f¨ur Informatik, Friedrich-Schiller-Universit¨at Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany bWilhelm-Schickard-Institut f¨ur Informatik, Universit¨at T¨ubingen, Sand 13, D-72076 T¨ubingen, Germany Abstract Multicut is a fundamental network communication and connectivity problem. It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed- parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth. Key words: Combinatorial optimization, Complexity theory, NP-completeness, Dynamic programming, Graph theory, Parameterized complexity 1 Introduction Motivation and previous results. Multicut in graphs is a fundamental network design problem. It models questions concerning the reliability and ∗ Corresponding author. Tel.: +49 3641 9 46325; fax: +49 3641 9 46002 E-mail address: [email protected]. 1 An extended abstract of this work appears in the proceedings of the 32nd Interna- tional Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2006) [19]. Now, in particular the proof of Theorem 5 has been much simplified. Preprint submitted to Elsevier Science 2 February 2007 robustness of computer and communication networks. -
Three-Dimensional Bin-Packing Approaches, Issues, and Solutions
Three Dimensional Bin-Packing Issues and Solutions Seth Sweep University of Minnesota, Morris [email protected] Abstract This paper reviews the three-dimensional version of the classic NP-hard bin-packing, optimization problem along with its theoretical relevance and practical importance. Also, new approximation solutions are introduced based on prior research. The three-dimensional bin-packing optimization problem concerns placing box-shaped objects of arbitrary size and number into a box-shaped, three-dimensional space efficiently. Approximation algorithms become a guide that attempts to place objects in the least amount of space and time. As a NP-hard problem, three-dimensional bin- packing holds much academic interest to computer scientists. However, this problem also has relevance in industrial settings such as shipping cargo and efficiently designing machinery with replaceable parts, such as automobiles. Industry relies on algorithms to provide good solutions to versions of this problem. This research project adapted common approaches to one-dimensional bin-packing, such as next fit and best fit, to three dimensions. Adaptation of these algorithms is possible by dividing the space of a bin. Then, by paying attention strictly to the width of each object, the one-dimensional approaches attempt to efficiently pack into the divided compartments of the original bin. Through focusing entirely on width, these divisions effectively become multiple one-dimensional bins. However, entirely disregarding the volume of the bin by ignoring height and depth may generate inefficient packings. As this paper details in more depth, generally cubical objects may pack well but exceptionally high or long objects leave large gaps of empty space unpacked. -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
Exactly Solving Packing Problems with Fragmentation
Exactly solving packing problems with fragmentation M. Casazza, A. Ceselli, Università degli Studi di Milano, Dipartimento di Informatica, OptLab – Via Bramante 65, 26013 – Crema (CR) – Italy {marco.casazza, alberto.ceselli}@unimi.it April 13, 2015 In packing problems with fragmentation a set of items of known weight is given, together with a set of bins of limited capacity; the task is to find an assignment of items to bins such that the sum of items assigned to the same bin does not exceed its capacity. As a distinctive feature, items can be split at a price, and fractionally assigned to different bins. Arising in diverse application fields, packing with fragmentation has been investigated in the literature from both theoretical, modeling, approximation and exact optimization points of view. We improve the theoretical understanding of the problem, and we introduce new models by exploiting only its combinatorial nature. We design new exact solution algorithms and heuristics based on these models. We consider also variants from the literature arising while including different objective functions and the option of handling weight overhead after splitting. We present experimental results on both datasets from the literature and new, more challenging, ones. These show that our algorithms are both flexible and effective, outperforming by orders of magnitude previous approaches from the literature for all the variants considered. By using our algorithms we could also assess the impact of explicitly handling split overhead, in terms of both solutions quality and computing effort. Keywords: Bin Packing, Item Fragmentation, Mathematical Programming, Branch&Price 1 1 Introduction Logistics has always been a benchmark for combinatorial optimization methodologies, as practitioners are traditionally familiar with the competitive advantage granted by optimized systems. -
Solving Packing Problems with Few Small Items Using Rainbow Matchings
Solving Packing Problems with Few Small Items Using Rainbow Matchings Max Bannach Institute for Theoretical Computer Science, Universität zu Lübeck, Lübeck, Germany [email protected] Sebastian Berndt Institute for IT Security, Universität zu Lübeck, Lübeck, Germany [email protected] Marten Maack Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Matthias Mnich Institut für Algorithmen und Komplexität, TU Hamburg, Hamburg, Germany [email protected] Alexandra Lassota Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Malin Rau Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, 38000 Grenoble, France [email protected] Malte Skambath Department of Computer Science, Universität Kiel, Kiel, Germany [email protected] Abstract An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no “small” items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4k · k! · nO(1). To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. -
A Width Parameter Useful for Chordal and Co-Comparability Graphs
A width parameter useful for chordal and co-comparability graphs Dong Yeap Kang1, O-joung Kwon2, Torstein J. F. Strømme3, and Jan Arne Telle3 ? 1 Department of Mathematical Sciences, KAIST, Daejeon, South Korea 2 Institute of Software Technology and Theoretical Computer Science, TU Berlin, Germany 3 Department of Informatics, University of Bergen, Norway [email protected], [email protected], [email protected], [email protected] Abstract. In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, also known as (σ; ρ)- problems or LC-VSVP problems, on many special graph classes. In this paper, we focus on chordal graphs and co-comparability graphs, that strictly contain interval graphs and permutation graphs respectively. First, we show that mim-width is unbounded on these classes, thereby settling an open problem from 2012. Then, we introduce two graphs Kt Kt and Kt St to restrict these graph classes, obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respec- tively, by adding a perfect matching. We prove that (Kt St)-free chordal graphs have mim-width at most t − 1, and (Kt Kt)-free co-comparability graphs have mim-width at most t − 1. From this, we obtain several algorithmic consequences, for instance, while Dominating Set is NP-complete on chordal graphs, like all LC-VSVP problems it can be solved in time O(nt) on chordal graphs where t is the maximum among induced sub- graphs Kt St in the given graph. -
Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗
Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity∗ Erik D. Demaine, Martin L. Demaine MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, {edemaine,mdemaine}@mit.edu Dedicated to Jin Akiyama in honor of his 60th birthday. Abstract. We show that jigsaw puzzles, edge-matching puzzles, and polyomino packing puzzles are all NP-complete. Furthermore, we show direct equivalences between these three types of puzzles: any puzzle of one type can be converted into an equivalent puzzle of any other type. 1. Introduction Jigsaw puzzles [37,38] are perhaps the most popular form of puzzle. The original jigsaw puzzles of the 1760s were cut from wood sheets using a hand woodworking tool called a jig saw, which is where the puzzles get their name. The images on the puzzles were European maps, and the jigsaw puzzles were used as educational toys, an idea still used in some schools today. Handmade wooden jigsaw puzzles for adults took off around 1900. Today, jigsaw puzzles are usually cut from cardboard with a die, a technology that became practical in the 1930s. Nonetheless, true addicts can still find craftsmen who hand-make wooden jigsaw puzzles. Most jigsaw puzzles have a guiding image and each side of a piece has only one “mate”, although a few harder variations have blank pieces and/or pieces with ambiguous mates. Edge-matching puzzles [21] are another popular puzzle with a similar spirit to jigsaw puzzles, first appearing in the 1890s. In an edge-matching puzzle, the goal is to arrange a given collection of several identically shaped but differently patterned tiles (typically squares) so that the patterns match up along the edges of adjacent tiles. -
Graceful Labeling of Graphs
GRACEFUL LABELING OF GRAPHS Rodrigo Ming Zhou Dissertação de Mestrado apresentada ao Programa de Pós-graduação em Engenharia de Sistemas e Computação, COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Mestre em Engenharia de Sistemas e Computação. Orientadores: Celina Miraglia Herrera de Figueiredo Vinícius Gusmão Pereira de Sá Rio de Janeiro Dezembro de 2016 GRACEFUL LABELING OF GRAPHS Rodrigo Ming Zhou DISSERTAÇÃO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE MESTRE EM CIÊNCIAS EM ENGENHARIA DE SISTEMAS E COMPUTAÇÃO. Examinada por: Prof. Celina Miraglia Herrera de Figueiredo, D.Sc. Prof. Vinícius Gusmão Pereira de Sá, D.Sc. Prof. Rosiane de Freitas Rodrigues, D.Sc. Prof. Raphael Carlos Santos Machado, D.Sc. RIO DE JANEIRO, RJ – BRASIL DEZEMBRO DE 2016 Zhou, Rodrigo Ming Graceful Labeling of Graphs/Rodrigo Ming Zhou. – Rio de Janeiro: UFRJ/COPPE, 2016. IX, 40 p.: il.; 29; 7cm. Orientadores: Celina Miraglia Herrera de Figueiredo Vinícius Gusmão Pereira de Sá Dissertação (mestrado) – UFRJ/COPPE/Programa de Engenharia de Sistemas e Computação, 2016. Bibliography: p. 38 – 40. 1. graceful labeling. 2. computational proofs. 3. graphs. I. de Figueiredo, Celina Miraglia Herrera et al. II. Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia de Sistemas e Computação. III. Título. iii Resumo da Dissertação apresentada à COPPE/UFRJ como parte dos requisitos necessários para a obtenção do grau de Mestre em Ciências (M.Sc.) GRACEFUL LABELING OF GRAPHS Rodrigo Ming Zhou Dezembro/2016 Orientadores: Celina Miraglia Herrera de Figueiredo Vinícius Gusmão Pereira de Sá Programa: Engenharia de Sistemas e Computação Em 1966, A. -
Improved Lower Bounds for Graph Embedding Problems
Improved Lower Bounds for Graph Embedding Problems Hans L. Bodlaender1;2? and Tom C. van der Zanden1 1 Department of Computer Science, Utrecht University, Utrecht, The Netherlands fH.L.Bodlaender,[email protected] 2 Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands Abstract. In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthog- onal Vector Crafting, and show that these cannot be solved in time 2o(jsj= log jsj), unless the Exponential Time Hypothesis fails. These results are used to obtain simplified hardness results for several graph embedding problems, on more restricted graph classes than previ- ously known: assuming the Exponential Time Hypothesis, there do not exist algorithms that run in 2o(n= log n) time for Subgraph Isomorphism on graphs of pathwidth 1, Induced Subgraph Isomorphism on graphs of pathwidth 1, Graph Minor on graphs of pathwidth 1, Induced Graph Minor on graphs of pathwidth 1, Intervalizing 5-Colored Graphs on trees, and finding a tree or path decomposition with width at most c with a minimum number of bags, for any fixed c ≥ 16. 2Θ(n= log n) appears to be the \correct" running time for many pack- ing and embedding problems on restricted graph classes, and we think String Crafting and Orthogonal Vector Crafting form a useful framework for establishing lower bounds of this form. 1 Introduction Many NP-complete graph problems admit faster algorithms when restricted to planar graphs.