Exact colouring algorithm for weighted graphs applied to timetabling problems with lectures of different lengths

An exact algorithm is presented for determining the interval chromatic number of a weighted graph. The algorithm is based on enumeration and the Branch-and-Bound principle. Computational experiments with the application of the algorithm to random weighted graphs are given. The algorithm and its modifications are used for solving timetabling problems with lectures of different lengths

About equivalent interval colorings of weighted graphs

AbstractGiven a graph G=(V,E) with strictly positive integer weights ωi on the vertices i∈V, a k-interval coloring of G is a function I that assigns an interval I(i)⊆{1,…,k} of ωi consecutive integers (called colors) to each vertex i∈V. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0̸ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi=1 for all vertices i∈V).Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1,…,k such that ℓ∈I1(i) if and only if π(ℓ)∈I2(i) for all vertices i∈V. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions

Relaxing Common Belief for Social Networks

We propose a relaxation of common belief called factional belief that is suitable for the analysis of strategic coordination on social networks. We show how this definition can be used to analyze revolt games on general graphs, including by giving an efficient algorithm that characterizes a structural result about the possible equilibria of such games. This extends prior work on common knowledge and common belief, which has been too restrictive for use in understanding strategic coordination and cooperation in social network settings

Coloration de graphes et attribution d'activités dans des quarts de travail

Revue de littérature -- Organisation de la thèse -- Lower bounds and a tabu search algorithm for the minimum deficiency problem -- On a reduction of the interval coloring problem to a series of bandwidth coloring problems -- About equivalent interval colorings of weighted graphs -- Une approche de programmation en nombres entiers pour la résolution d'un problème d'horaire -- Discussion générale et conclusion

Counting and Sampling Directed Acyclic Graphs for Learning Bayesian Networks

Bayesian networks are probabilistic models that represent dependencies between random variables via directed acyclic graphs (DAGs). They provide a succinct representation for the joint distribution in cases where the dependency structure is sparse. Specifying the network by hand is often unfeasible, and thus it would be desirable to learn the model from observed data over the variables. In this thesis, we study computational problems encountered in different approaches to learning Bayesian networks. All of the problems involve counting or sampling DAGs under various constraints. One important computational problem in the fully Bayesian approach to structure learning is the problem of sampling DAGs from the posterior distribution over all the possible structures for the Bayesian network. From the typical modeling assumptions it follows that the distribution is modular, which means that the probability of each DAG factorizes into per-node weights, each of which depends only on the parent set of the node. For this problem, we give the first exact algorithm with a time complexity bound exponential in the number of nodes, and thus polynomial in the size of the input, which consists of all the possible per-node weights. We also adapt the algorithm such that it outperforms the previous methods in the special case of sampling DAGs from the uniform distribution. We also study the problem of counting the linear extensions of a given partial order, which appears as a subroutine in some importance sampling methods for modular distributions. This problem is a classic example of a #P-complete problem that can be approximately solved in polynomial time by reduction to sampling linear extensions uniformly at random. We present two new randomized approximation algorithms for the problem. The first algorithm extends the applicable range of an exact dynamic programming algorithm by using sampling to reduce the given instance into an easier instance. The second algorithm is obtained by combining a novel, Markov chain-based exact sampler with the Tootsie Pop algorithm, a recent generic scheme for reducing counting into sampling. Together, these two algorithms speed up approximate linear extension counting by multiple orders of magnitude in practice. Finally, we investigate the problem of counting and sampling DAGs that are Markov equivalent to a given DAG. This problem is important in learning causal Bayesian networks, because distinct Markov equivalent DAGs cannot be distinguished only based on observational data, yet they are different from the causal viewpoint. We speed up the state-of-the-art recursive algorithm for the problem by using dynamic programming. We also present a new, tree decomposition-based algorithm, which runs in linear time if the size of the maximum clique is bounded.Bayes-verkot mallintavat satunnaismuuttujien välisiä tilastollisia suhteita suunnattuina syklittöminä verkkoina, joissa solmut vastaavat satunnaismuuttujia ja kaaret niiden välisiä riippuvuuksia. Verkkorakenne havainnollistaa muuttujien kuvaaman ilmiön rakennetta ja mahdollistaa muuttujien yhteisjakauman esittämisen tiiviissä muodossa. Vaikka Bayes-verkko voidaan periaatteessa rakentaa käsin, se on epäkäytännöllistä, mikäli muuttujia on paljon tai mallinnettavaa ilmiötä ei ymmärretä täydellisesti. Tämän takia on hyödyllistä oppia verkon rakenne ilmiöstä kerätyn datan perusteella. Väitöskirjassa tutkitaan laskennallisia ongelmia, jotka liittyvät Bayes-verkon rakenteen oppimiseen. Kaikki nämä ongelmat koskevat suunnattujen syklittömien verkkojen laskemista tai satunnaisotantaa erilaisilla rajoitteilla. Yksi keskeinen ongelma Bayes-verkon rakenteen oppimisessa on rakenteen poiminta posteriorisatunnaisjakaumasta, joka painottaa parhaiten dataa vastaavia rakenteita. Väitöskirjassa esitellään tähän ongelmaan ensimmäinen eksakti algoritmi, joka hyödyntämällä posteriorijakauman erityisominaisuuksia saavuttaa polynomisen aikavaativuuden suhteessa jakauman määrittelevän tietorakenteen kokoon. Algoritmi tarjoaa myös aiempia algoritmeja tehokkaamman tavan suunnattujen syklittömien verkkojen poimintaan tasajakaumasta. Toinen väitöskirjassa tutkittu ongelma on osittaisjärjestyksen lineaariekstensioiden laskenta. Tämä ongelma tiedetään kuuluvaksi vaikeiden laskentaongelmien #P-luokkaan, mutta se voidaan silti ratkaista likimäärisesti polynomisessa ajassa palauttamalla se vastaavaan satunnaisotantaongelmaan. Väitöskirja esittelee kaksi uutta likimääräistä satunnaisalgoritmia lineaariekstensioiden laskentaan. Ensimmäinen algoritmi muuttaa tunnetun eksaktin laskenta-algoritmin likimääräiseksi yhdistämällä siihen satunnaisotokseen perustuvaa arviointia. Toinen algoritmi palauttaa laskentaongelman uuteen Markovin ketjuihin perustuvan satunnaisotantamenetelmään. Yhdessä nämä kaksi algoritmia nopeuttavat käytännön tapauksissa likimääräistä lineaariekstensioiden laskentaa usealla kertaluokalla. Työn loppuosassa tutkitaan tietyssä Markov-ekvivalenssiluokassa olevien suunnattujen syklittömien verkkojen laskenta- ja satunnaisotantaongelmia. Ongelma on tärkeä Bayes-verkkojen käytössä kausaalisten riippuvuuksien mallintamiseen, koska Markov-ekvivalentteja rakenteita ei voi erottaa pelkästään havaintodatan perusteella, vaikka ne ovat kausaalisesta näkökulmasta erilaisia. Työssä esitellään tapa nopeuttaa parasta tunnettua algoritmia dynaamisen ohjelmoinnin avulla. Tämän lisäksi väitöskirja esittelee uuden verkon puuhajotelmaan perustuvan menetelmän, jonka aikavaativuus on lineaarinen, mikäli verkon suurimman klikin koko on rajoitettu

TV-PR: Theme and Variations Planner/Realizer

Due to the character of the original source materials and the nature of batch digitization, quality control issues may be present in this document. Please report any quality issues you encounter to [email protected], referencing the URI of the item.Includes bibliographical references (leaves 51).Computer generated music sounds too random and disconnected. Even when the melody and harmony are very pleasant (a task that computers are increasingly adept at performing), one doesn't have to listen long before realizing that a computer wrote the piece. Leading researchers in the field agree that continuity in computer music deserves more attention at this point. My goal is to improve the overall form and coherence in automated music generation. The main focus of my work will be text planning techniques from the natural language generation field. These text planners guide the text generators and help computers form not only well-formed sentences, but also coherent paragraphs. Currently, there is no computer music system that uses a similar planning scheme. I will adapt the text planner-generator model to computer music, and design a working system that generates a plan for a set of variations on a user-given theme