Gene prioritization involves ranking genes by possible relevance to a disease of interest. This is important in order to narrow down the set of genes to be investigated biologically, and over the years, several computational approaches have been proposed for automat-ically prioritizing genes using some form of gene-related data, mostly using statistical or machine learning methods. Recently, Agarwal and Sengupta (2009) proposed the use of learning-to-rank methods, which have been used extensively in information retrieval and related fields, to learn a ranking of genes from a given data source, and used this approach to successfully identify novel genes related to leukemia and colon cancer using only gene expression data. In this work, we explore the possibility of combining such learning-to-rank methods with rank aggregation techniques to learn a ranking of genes from multiple heterogeneous data sources, such as gene expression data, gene ontology data, protein-protein interaction data, etc. Rank aggregation methods have their origins in voting theory, and have been used successfully in meta-search applications to aggregate webpage rankings from different search engines. Here we use graph-based learning-to-rank methods to learn a ranking of genes from each individual data source represented as a graph, and then apply rank aggregation methods to aggregate these rankings into a single ranking over the genes. The thesis describes our approach, reports experiments with various data sets, and presents our findings and initial conclusions.

Algorithms will be domonstrated for how to embed complete bipartite graphs onto 2xn type grids, where the imimum grid cutwidth is attained.

This thesis consists of two papers. The first paper is a study of the structure of the k-assignment polytope, whose vertices are the m x n (0; 1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. Two equivalent representations of the faces are given, one as (0; 1)-matrices and one as ear decompositions of bipartite graphs. These tools are used to describe properties of the polytope, especially a complete description of the cover relation in the face lattice of the polytope and an exact expression for the diameter. The second paper studies the edge-product space Є(X) for trees on X. This space is generated by the set of edge-weighted finite trees on X, and arises by multiplying the weights of edges on paths in trees. These spaces are closely connected to tree-indexed Markov processes in molecular evolutionary biology. It is known that Є(X) has a natural CW-complex structure, and a combinatorial description of the associated face poset exists which is a poset S(X) of X-forests. In this paper it is shown that the edge-product space is a regular cell complex. One important part in showing that is to conclude that all intervals [Ô, Г], Г Є S(X), have recursive coatom orderings.

Yes / Various inequalities (Boole inequality, Chung–Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra 'quantum correction' term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. A logical approach to CHSH inequalities (which are related to the Frechet inequalities), is studied in this context. It is shown that CHSH inequalities hold for all rank one (factorizable) states, and are violated by some rank two (entangled) states. The reduction of the rank of a pure state by a quantum measurement with both orthogonal and coherent projectors, is studied. Bounds for the average rank reduction are given.

The k–server problem is of significant importance to the theoretical computer science and the operations research community. In this problem, we are given k servers, their initial locations and a sequence of n requests that arrive one at a time. All these locations are points from some metric space and the cost of serving a request is given by the distance between the location of the request and the current location of the server selected to process the request. We must immediately process the request by moving a server to the request location. The objective in this problem is to minimize the total distance traveled by the servers to process all the requests. In this thesis, we present an empirical analysis of a new online algorithm for k-server problem. This algorithm maintains two solutions, online solution, and an approximately optimal offline solution. When a request arrives we update the offline solution and use this update to inform the online assignment. This algorithm is motivated by the Robust-Matching Algorithm [RMAlgorithm, Raghvendra, APPROX 2016] for the closely related online bipartite matching problem. We then give a comprehensive experimental analysis of this algorithm and also provide a graphical user interface which can be used to visualize execution instances of the algorithm. We also consider these problems under stochastic setting and implement a lookahead strategy on top of the new online algorithm. / MS / Motivated by real-time logistics, we study the online versions of the well-known bipartite matching and the k-server problems. In this problem, there are servers (delivery vehicles) located in different parts of the city. When a request for delivery is made, we have to immediately assign a delivery vehicle to this request without any knowledge of the future. Making cost-effective assignments, therefore, becomes incredibly challenging. In this thesis, we implement and empirically evaluate a new algorithm for the k-server and online matching problems.

Matching is one of the most fundamental algorithmic graph problems. Many variants of matching problems have been studied on different classes of graphs, the one of special interest to us being the Maximum Cardinality Bipartite Matching in Planar Graphs. In this work, we present a novel sparsification based approach for computing maximum/perfect bipartite matching in planar graphs. The overall complexity of our algorithm is O(n^6/5 log² n) where n is the number of vertices in the graph, bettering the O(n^3/2) time achieved independently by Hopcroft-Karp algorithm and by Lipton and Tarjan divide and conquer approach using planar separators. Our algorithm combines the best of both these standard algorithms along with our sparsification technique and rich planar graph properties to achieve the speed up. Our algorithm is not the fastest, with the existence of O(n log³ n) algorithm based on max-flow reduction. / MS

The minimum cost bipartite matching problem is a well-studied optimization problem in computer science and operations research, with wide-ranging applications in fields such as machine learning, economics, transportation, logistics and biology. A special instance of this problem is the computation of the p-Wasserstein distance which we define next. Given a complete bipartite graph with two disjoint sets of n points in d-dimensional Euclidean space and an integer p ≥ 1, let the cost of an edge be the p-th power of the Euclidean distance between its endpoints. The objective of this problem is to find a minimum-cost matching in this complete bipartite graph. The Hungarian algorithm is a classical method that solves this problem in O(n^3) time. There are many algorithms that have a run time better than that of the Hungarian algorithm if the graphs have non-negative integer edge costs bounded by C. Since the input points have real-valued coordinates and the Euclidean distances can be irrational, such algorithms only return an approximate matching. Thus, the Hungarian algorithm remains the fastest known algorithm to compute an exact matching. In this thesis, we implement a new algorithm in the divide and conquer framework that computes the exact p-Wasserstein distance and has a run time asymptotically better than the Hungarian algorithm for stochastic point sets. Inspired by the techniques used in the algorithm, we also design an alternate version of the Hungarian algorithm that uses a grid- based approach. Our experimental analysis shows that both of our algorithms significantly outperform the classical Hungarian algorithm. / Master of Science / Suppose we have two sets of equal number of items and a list of compatible pairs of items, where a pair is considered compatible if its items belong to different sets. A perfect matching is a subset of compatible pairs where each item is paired with exactly one other item. When trying to find a perfect matching, there may be multiple options, and minimizing the cost of the perfect matching is often desired. This is referred to as the minimum cost bipartite matching problem, which is extensively studied due to its importance in algorithmic theory and operations research. A special instance of this problem is the calculation of the p- Wasserstein distance. It has many practical applications in fields such as machine learning, economics, transportation, logistics and biology. The Hungarian algorithm is the only known algorithm that can compute the exact p-Wasserstein distance. Therefore, our focus is to develop exact algorithms for this problem that perform better than the Hungarian algorithm and can scale efficiently to large datasets.

Cette thèse étudie les méthodes de regroupement basées sur le principe des plus proches voisins partagés (SNN). Comme la plupart des autres approches de clustering à base de graphe, les méthodes SNN sont effectivement bien adaptées à surmonter la complexité des données, l'hétérogénéité et la haute dimensionnalité. La première contribution de la thèse est de revisiter une méthode existante basée sur les voisins partagés en deux points. Nous présentons d'abord un formalisme basé sur la la théorie de décision à contrario. Cela nous permet de tirer des scores de connectivité plus fiable des groupes et une interprétation plus intuitive des voisinage selectionnés optimalement. Nous proposons également un nouveau algorithme de factorisation pour accélérer le calcul intensif nécessaire des matrices des voisins partagés. La deuxième contribution de cette thèse est une généralisation de la classification SNNau cas multi-source. La principale originalité de notre approche est que nous introduisons une étape de sélection des sources d'information optimales dans le calcul de scores de groupes candidats. Chaque groupe est alors associé à son propre sous-ensemble optimal des modalités. Comme le montre le expériences, cette étape de sélection de source rend notre approche largement robuste à la présence de sources locales aberrantes. Cette nouvelle méthode est appliquée à un large éventail de problèmes, y compris la structuration multimodale des collections d'images et dans le regroupement dans des sous-espaces basés sur les projections aléatoires.La troisième contribution de la thèse est une tentative pour étendre les méthodes SNNdans le contexte des graphes biparites. Nous introduisons de nouvelles mesures de pertinence SNNrevisitées pour ce contexte asymétrique et nous montrons qu'elles peuvent être utiliséespour sélectionner localement des voisinages optimales. En conséquence, nous proposons un nouveau algorithme de clustering bipartite SNN qui est appliqué à la découverte d'objets visuels.Les expériences montrent que cette nouvelle méthode est meilleure par rapport aux méthodes de l'état de l'art. Basé sur les objets découverts, nous introduisons également un paradigme de recherche visuelle, c.-à-d les objet basés sur la suggestion de requêtes visuel les. / This thesis investigates new clustering paradigms and algorithms based on the principle of the shared nearest-neighbors (SNN. As most other graph-based clustering approaches, SNN methods are actually well suited to overcome data complexity, heterogeneity and high-dimensionality.The first contribution of the thesis is to revisit existing shared neighbors methods in two points. We first introduce a new SNN formalism based on the theory of a contrario decision. This allows us to derive more reliable connectivity scores of candidate clusters and a more intuitive interpretation of locally optimum neighborhoods. We also propose a new factorization algorithm for speeding-up the intensive computation of the required sharedneighbors matrices.The second contribution of the thesis is a generalization of the SNN clustering approach to the multi-source case. Whereas SNN methods appear to be ideally suited to sets of heterogeneous information sources, this multi-source problem was surprisingly not addressed in the literature beforehand. The main originality of our approach is that we introduce an information source selection step in the computation of candidate cluster scores. As shown in the experiments, this source selection step makes our approach widely robust to the presence of locally outlier sources. This new method is applied to a wide range of problems including multimodal structuring of image collections and subspace-based clustering based on random projections. The third contribution of the thesis is an attempt to extend SNN methods to the context of bipartite k-nn graphs. We introduce new SNN relevance measures revisited for this asymmetric context and show that they can be used to select locally optimal bi-partite clusters. Accordingly, we propose a new bipartite SNN clustering algorithm that is applied to visual object’s discovery based on a randomly precomputed matching graph. Experiments show that this new method outperformed state-of-the-art object mining results on OxfordBuilding dataset. Based on the discovered objects, we also introduce a new visual search paradigm, i.e. object-based visual query suggestion.

The bin packing problem has been the corner stone of approximation algorithms and has been extensively studied starting from the early seventies. In the classical bin packing problem, we are given a list of real numbers in the range (0, 1], the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. In this thesis we study approximation algorithms for three generalizations of bin packing: geometric bin packing, vector bin packing and weighted bipartite edge coloring. In two-dimensional (2-D) geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. Geometric packing has vast applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. We consider the widely studied orthogonal packing case, where the items must be placed in the bin such that their sides are parallel to the sides of the bin. Here two variants are usually studied, (i) where the items cannot be rotated, and (ii) they can be rotated by 90 degrees. We give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for the versions with and without rotations. We have also shown the limitations of rounding based algorithms, ubiquitous in bin packing algorithms. We have shown that any algorithm that rounds at least one side of each large item to some number in a constant size collection values chosen independent of the problem instance, cannot achieve an asymptotic approximation ratio better than 3/2. In d-dimensional vector bin packing (VBP), each item is a d-dimensional vector that needs to be packed into unit vector bins. The problem is of great significance in resource constrained scheduling and also appears in recent virtual machine placement in cloud computing. Even in two dimensions, it has novel applications in layout design, logistics, loading and scheduling problems. We obtain a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for 2-D VBP. We also obtain a polynomial time algorithm with almost tight (absolute) approximation ratio of $1+\ln(1.5)$ for 2-D VBP. For $d$ dimensions, we give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(d/2) + 1.5 \approx \ln d+0.81$. We also consider vector bin packing under resource augmentation. We give a polynomial time algorithm that packs vectors into $(1+\epsilon)Opt$ bins when we allow augmentation in (d - 1) dimensions and $Opt$ is the minimum number of bins needed to pack the vectors into (1,1) bins. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite graph $G=(V,E)$ with weights $w: E \rightarrow [0,1]$. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. This problem is motivated by rearrangeability of 3-stage Clos networks which is very useful in various applications in interconnected networks and routing. We show a polynomial time approximation algorithm that returns a proper weighted coloring with at most $\lceil 2.2223m \rceil$ colors where $m$ is the minimum number of unit sized bins needed to pack the weight of all edges incident at any vertex. We also show that if all edge weights are $>1/4$ then $\lceil 2.2m \rceil$ colors are sufficient.

While it has been known for some time that connected non-bipartite graphs have unique prime factorizations over the direct product, the same cannot be said of bipartite graphs. This is somewhat vexing, as bipartite graphs do have unique prime factorizations over other graph products (the Cartesian product, for example). However, it is fairly easy to show that a connected bipartite graph has only one prime bipartite factor, which begs the question: is such a prime bipartite factor unique? In other words, although a connected bipartite graph may have multiple prime factorizations over the direct product, do such factorizations contain the same prime bipartite factor? It has previously been shown by Hammack that when the prime bipartite factor is K_2, this is in fact true. The goal of this paper is to prove that this is in fact true for any prime bipartite factor, provided the graph being factored is R-thin. The proof of the main result takes the same initial approach as the proof by Hammack, before moving into new territory in order to prove the final result.