Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a $(1-\epsilon)$-approximation for the number of perfect matchings of a $\lambda$-dense bipartite graph, using $O(n^{\frac{1-2\lambda}{8\lambda}+\epsilon^{-2}})$ samples. With size $n$ on each side and for $\frac{1}{2}>\lambda>0$, a $\lambda$-dense bipartite graph has all degrees greater than $(\lambda+\frac{1}{2})n$. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes

Packing and Counting Permutations

A permutation class is a set of permutations closed under taking subpermutations. We study two aspects of permutation classes: enumeration and packing. Our work on enumeration consists of two campaigns. First, we enumerate all juxtaposition classes of the form “Av(abc) next to Av(xy)”, where abc and xy are permutations of lengths three and two, respectively. We represent elements from such a juxtaposition class by Dyck paths decorated with sequences of points. Context-free grammars are then used to enumerate these decorated Dyck paths. Second, we classify as algebraic the generating functions of 1×m permutation grid classes where one cell is context-free and the remaining cells are monotone. We rely on properties of combinatorial specifications of context-free classes and use operators to express juxtapositions. Repeated application of operators resolves cases for m > 2. We provide examples to re-prove known results and give new ones. Our methods are algorithmic and could be implemented on a PC. Our work on packing consolidates current knowledge about packing densities of 4-point permutations. We also improve the lower bounds for the packing densities of 1324 and 1342 and provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within 10-4 of the true packing densities. Together with the known bounds, we have a fairly complete picture of 4-point packing densities. Additionally, we obtain several bounds (lower and upper) for permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007. We also present Permpack — a flag algebra package for permutations

Monotonic functions of finite posets

Classes of monotone functions from finite posets to chains are studied. These include order-preserving and strict order-preserving maps. When the maps are required to be bijective they are called linear extensions. Techniques for handling the first two types are closely related; whereas for linear extensions quite distinct methods are often necessary, which may yield results for order-preserving injections. First, many new fundamental properties and inequalities of a combinatorial nature are established for these maps. Quantities considered here include the range, height, depth and cardinalities of subposets. In particular we study convexity in posets and similarly pre-images of intervals in chains. The problem of extending a map defined on a subposet to the whole poset is discussed. We investigate positive correlation inequalities, having implications for the complexity of sorting algorithms. These express monotonicity properties for probabilities concerning sets of relations in posets. New proofs are given for existing inequalities and we obtain corresponding negative correlations, along with a generalization of the xyz inequality. The proofs involve inequalities in distributive lattices, some of which arose in physics. A characterization is given for posets satisfying necessary conditions for correlation properties under linear extensions. A log concavity type inequality is proved for the number of strict or non-strict order-preserving maps of an element. We define an explicit injection whereas the bijective case is proved in the literature using inequalities from the theory of mixed volumes. These results motivate a further group of such inequalities. But now we count numbers of strict or non-strict order-preserving maps of subposets to varying heights in the chain. Lastly we consider the computational cost of producing certain posets which can be associated with classical sorting and selection problems. A lower bound technique is derived for this complexity, incorporating either a new distributive lattice inequality, or the log concavity inequalities

Numerical Linear Algebra applications in Archaeology: the seriation and the photometric stereo problems

The aim of this thesis is to explore the application of Numerical Linear Algebra to Archaeology. An ordering problem called the seriation problem, used for dating findings and/or artifacts deposits, is analysed in terms of graph theory. In particular, a Matlab implementation of an algorithm for spectral seriation, based on the use of the Fiedler vector of the Laplacian matrix associated with the problem, is presented. We consider bipartite graphs for describing the seriation problem, since the interrelationship between the units (i.e. archaeological sites) to be reordered, can be described in terms of these graphs. In our archaeological metaphor of seriation, the two disjoint nodes sets into which the vertices of a bipartite graph can be divided, represent the excavation sites and the artifacts found inside them. Since it is a difficult task to determine the closest bipartite network to a given one, we describe how a starting network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems. Another numerical problem related to Archaeology is the 3D reconstruction of the shape of an object from a set of digital pictures. In particular, the Photometric Stereo (PS) photographic technique is considered

Presence 2005: the eighth annual international workshop on presence, 21-23 September, 2005 University College London (Conference proceedings)

OVERVIEW (taken from the CALL FOR PAPERS) Academics and practitioners with an interest in the concept of (tele)presence are invited to submit their work for presentation at PRESENCE 2005 at University College London in London, England, September 21-23, 2005. The eighth in a series of highly successful international workshops, PRESENCE 2005 will provide an open discussion forum to share ideas regarding concepts and theories, measurement techniques, technology, and applications related to presence, the psychological state or subjective perception in which a person fails to accurately and completely acknowledge the role of technology in an experience, including the sense of 'being there' experienced by users of advanced media such as virtual reality. The concept of presence in virtual environments has been around for at least 15 years, and the earlier idea of telepresence at least since Minsky's seminal paper in 1980. Recently there has been a burst of funded research activity in this area for the first time with the European FET Presence Research initiative. What do we really know about presence and its determinants? How can presence be successfully delivered with today's technology? This conference invites papers that are based on empirical results from studies of presence and related issues and/or which contribute to the technology for the delivery of presence. Papers that make substantial advances in theoretical understanding of presence are also welcome. The interest is not solely in virtual environments but in mixed reality environments. Submissions will be reviewed more rigorously than in previous conferences. High quality papers are therefore sought which make substantial contributions to the field. Approximately 20 papers will be selected for two successive special issues for the journal Presence: Teleoperators and Virtual Environments. PRESENCE 2005 takes place in London and is hosted by University College London. The conference is organized by ISPR, the International Society for Presence Research and is supported by the European Commission's FET Presence Research Initiative through the Presencia and IST OMNIPRES projects and by University College London

Graph labelings and decompositions by partitioning sets of integers

Aquest treball és una contribució a l'estudi de diferents problemes que sorgeixen de dues àrees fortament connexes de la Teoria de Grafs: etiquetaments i descomposicions. Molts etiquetaments de grafs deuen el seu origen als presentats l'any 1967 per Rosa. Un d'aquests etiquetaments, àmpliament conegut com a etiquetament graceful, va ser definit originalment com a eina per atacar la conjectura de Ringel, la qual diu que el graf complet d'ordre 2m+1 pot ser descompost en m copies d'un arbre donat de mida m. Aquí, estudiem etiquetaments relacionats que ens donen certes aproximacions a la conjectura de Ringel, així com també a una altra conjectura de Graham i Häggkvist que, en una forma dèbil, demana la descomposició d'un graf bipartit complet per un arbre donat de mida apropiada. Les principals contribucions que hem fet en aquest tema són la prova de la darrera conjectura per grafs bipartits complets del doble de mida essent descompostos per arbres de gran creixement i un nombre primer d'arestes, i la prova del fet que cada arbre és un subarbre gran de dos arbres pels quals les dues conjectures es compleixen respectivament. Aquests resultats estan principalment basats en una aplicació del mètode polinomial d'Alon. Un altre tipus d'etiquetaments, els etiquetaments magic, també són tractats aquí. Motivats per la noció de quadrats màgics de Teoria de Nombres, en aquest tipus d'etiquetaments volem asignar nombres enters a parts del graf (vèrtexs, arestes, o vèrtexs i arestes) de manera que la suma de les etiquetes assignades a certes subestructures del graf sigui constant. Desenvolupem tècniques basades en particions de certs conjunts d'enters amb algunes condicions additives per construir etiquetaments cycle-magic, un nou tipus d'etiquetament introduït en aquest treball i que estén la noció clàssica d'etiquetament magic. Els etiquetaments magic no donen cap descomposició de grafs, però les tècniques usades per obtenir-los estan al nucli d'un altre problema de descomposició, l'ascending subgraph decomposition (ASD). Alavi, Boals, Chartrand, Erdös i Oellerman, van conjecturar l'any 1987 que tot graf té un ASD. Aquí, estudiem l'ASD per grafs bipartits, una classe de grafs per la qual la conjectura encara no ha estat provada. Donem una condició necessària i una de suficient sobre la seqüència de graus d'un estable del graf bipartit de manera que admeti un ASD en que cada factor sigui un star forest. Les tècniques utilitzades estan basades en l'existència de branca-acoloriments en multigrafs bipartits. També tractem amb el sumset partition problem, motivat per la conjectura ASD, que demana una partició de [n] de manera que la suma dels elements de cada part sigui igual a un valor prescrit. Aquí donem la millor condició possible per la versió modular del problema que ens permet provar els millors resultats ja coneguts en el cas enter per n primer. La prova està de nou basada en el mètode polinomial.This work is a contribution to the study of various problems that arise from two strongly connected areas of the Graph Theory: graph labelings and graph decompositions. Most graph labelings trace their origins to the ones presented in 1967 by Rosa. One of these labelings, widely known as the graceful labeling, originated as a means of attacking the conjecture of Ringel, which states that the complete graph of order 2m+1 can be decomposed into m copies of a given tree of size m. Here, we study related labelings that give some approaches to Ringel's conjecture, as well as to another conjecture by Graham and Häggkvist that, in a weak form, asks for the decomposition of a complete bipartite graph by a given tree of appropriate size. Our main contributions in this topic are the proof of the latter conjecture for double sized complete bipartite graphs being decomposed by trees with large growth and prime number of edges, and the proof of the fact that every tree is a large subtree of two trees for which both conjectures hold respectively. These results are mainly based on a novel application of the so-called polynomial method by Alon. Another kind of labelings, the magic labelings, are also treated. Motivated by the notion of magic squares in Number Theory, in these type of labelings we want to assign integers to the parts of a graph (vertices, edges, or vertices and edges) in such a way that the sums of the labels assigned to certain substructures of the graph remain constant. We develop techniques based on partitions of certain sets of integers with some additive conditions to construct cycle-magic labelings, a new brand introduced in this work that extends the classical magic labelings. Magic labelings do not provide any graph decomposition, but the techniques that we use to obtain them are the core of another decomposition problem, the ascending subgraph decomposition (ASD). In 1987, was conjectured by Alavi, Boals. Chartrand, Erdös and Oellerman that every graph has an ASD. Here, we study ASD of bipartite graphs, a class of graphs for which the conjecture has not been shown to hold. We give a necessary and a sufficient condition on the one sided degree sequence of a bipartite graph in order that it admits an ASD by star forests. Here the techniques are based on the existence of edge-colorings in bipartite multigraphs. Motivated by the ASD conjecture we also deal with the sumset partition problem, which asks for a partition of [n] in such a way that the sum of the elements of each part is equal to a prescribed value. We give a best possible condition for the modular version of the sumset partition problem that allows us to prove the best known results in the integer case for n a prime. The proof is again based on the polynomial method