Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

Use of a weighted matching algorithm to sequence clusters in spatial join processing

One of the most expensive operations in a spatial database is spatial join processing. This study focuses on how to improve the performance of such processing. The main objective is to reduce the Input/Output (I/O) cost of the spatial join process by using a technique called cluster-scheduling. Generally, the spatial join is processed in two steps, namely filtering and refinement. The cluster-scheduling technique is performed after the filtering step and before the refinement step and is part of the housekeeping phase. The key point of this technique is to realise order wherein two consecutive clusters in the sequence have maximal overlapping objects. However, finding the maximal overlapping order has been shown to be Nondeterministic Polynomial-time (NP)-complete. This study proposes an algorithm to provide approximate maximal overlapping (AMO) order in a Cluster Overlapping (CO) graph. The study proposes the use of an efficient maximum weighted matching algorithm to solve the problem of finding AMO order. As a result, the I/O cost in spatial join processing can be minimised

Hyperspectral image representation and processing with binary partition trees

The optimal exploitation of the information provided by hyperspectral images requires the development of advanced image processing tools. Therefore, under the title Hyperspectral image representation and Processing with Binary Partition Trees, this PhD thesis proposes the construction and the processing of a new region-based hierarchical hyperspectral image representation: the Binary Partition Tree (BPT). This hierarchical region-based representation can be interpreted as a set of hierarchical regions stored in a tree structure. Hence, the Binary Partition Tree succeeds in presenting: (i) the decomposition of the image in terms of coherent regions and (ii) the inclusion relations of the regions in the scene. Based on region-merging techniques, the construction of BPT is investigated in this work by studying hyperspectral region models and the associated similarity metrics. As a matter of fact, the very high dimensionality and the complexity of the data require the definition of specific region models and similarity measures. Once the BPT is constructed, the fixed tree structure allows implementing efficient and advanced application-dependent techniques on it. The application-dependent processing of BPT is generally implemented through a specific pruning of the tree. Accordingly, some pruning techniques are proposed and discussed according to different applications. This Ph.D is focused in particular on segmentation, object detection and classification of hyperspectral imagery. Experimental results on various hyperspectral data sets demonstrate the interest and the good performances of the BPT representatio

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

Of keyboards and beyond - optimization in human-computer interaction

In this thesis, we present optimization frameworks in the area of Human-Computer Interaction. At first, we discuss keyboard layout problems with a special focus on a project we participated in, which aimed at designing the new French keyboard standard. The special nature of this national-scale project and its optimization ingredients are discussed in detail; we specifically highlight our algorithmic contribution to this project. Exploiting the special structure of this design problem, we propose an optimization framework that was efficiently computes keyboard layouts and provides very good optimality guarantees in form of tight lower bounds. The optimized layout that we showed to be nearly optimal was the basis of the new French keyboard standard recently published in the National Assembly in Paris. Moreover, we propose a relaxation for the quadratic assignment problem (a generalization of keyboard layouts) that is based on semidefinite programming. In a branch-and-bound framework, this relaxation achieves competitive results compared to commonly used linear programming relaxations for this problem. Finally, we introduce a modeling language for mixed integer programs that especially focuses on the challenges and features that appear in participatory optimization problems similar to the French keyboard design process.Diese Arbeit behandelt Ansätze zu Optimierungsproblemen im Bereich Human-Computer Interaction. Zuerst diskutieren wir Tastaturbelegungsprobleme mit einem besonderen Fokus auf einem Projekt, an dem wir teilgenommen haben: die Erstellung eines neuen Standards für die französische Tastatur. Wir gehen auf die besondere Struktur dieses Problems und unseren algorithmischen Beitrag ein: ein Algorithmus, der mit Optimierungsmethoden die Struktur dieses speziellen Problems ausnutzt. Mithilfe dieses Algorithmus konnten wir effizient Tastaturbelegungen berechnen und die Qualität dieser Belegungen effektiv (in Form von unteren Schranken) nachweisen. Das finale optimierte Layout, welches mit unserer Methode bewiesenermaßen nahezu optimal ist, diente als Grundlage für den kürzlich in der französischen Nationalversammlung veröffentlichten neuen französischen Tastaturstandard. Darüberhinaus beschreiben wir eine Relaxierung für das quadratische Zuweisungsproblem (eine Verallgemeinerung des Tastaturbelegungsproblems), die auf semidefinieter Programmierung basiert. Wir zeigen, dass unser Algorithmus im Vergleich zu üblich genutzten linearen Relaxierung gut abschneidet. Abschließend definieren und diskutieren wir eine Modellierungssprache für gemischt integrale Programme. Diese Sprache ist speziell auf die besonderen Herausforderungen abgestimmt, die bei interaktiven Optimierungsproblemen auftreten, welche einen ähnlichen Charakter haben wie der Prozess des Designs der französischen Tastatur

Quantitative Variants of Language Equations and their Applications to Description Logics

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. Together with the special case of matching, they were first investigated in detail for the DL FL0, where these problems can be reduced to solving certain language equations. In this thesis, we extend this service in two directions. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify, in order to account for potential small modelling errors. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL0 can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for particular distance measures. Furthermore, we make a first step towards integrating background knowledge, formulated in so-called TBoxes, by investigating the special case of matching in the presence of TBoxes of different forms. We acquire a tight complexity bound for the general case, while we prove that the problem becomes easier in a restricted setting. To achieve these bounds, we take advantage of an equivalence characterization of FL0 concepts that is based on formal languages. In addition, we incorporate TBoxes in computing concept distances. Even though our results on the approximate setting cannot deal with TBoxes yet, we prepare the framework that future research can build on. Before we journey to the technical details of the above investigations, we showcase our program in the simpler setting of the equational theory ACUI, where we are able to also combine the two extensions. In the course of studying the above problems, we make heavy use of automata theory, where we also derive novel results that could be of independent interest

Set partitioning via inclusion-exclusion

Το παρόν έργο αποτελεί μελέτη του paper των Andreas Bjorklund, Thore Husfeldt και Mikko Koivisto, ”Set partitioning via inclusion-exclusion”. Κύριος στόχος κατά τη συγγραφή ήταν να καταστούν οι έννοιες που παρουσιάζονται όσο το δυνατόν περισσότερο εύληπτες από προπτυχιακούς φοιτητές. Αποδεικνύουμε την αρχή εγκλεισμού-αποκλεισμού και ορίζουμε το z-μετασχηματισμό ενώ δίνουμε και έναν αλγόριθμο που τον υπολογίζει. Δεδομένου ενός συνόλου N, n στοιχείων και μιας οικογένειας F υποσυνόλων του N καθώς και ενός ακεραίου k, παρέχουμε έναν ακριβή αλγόριθμο που υπολογίζει το πλήθος των k-κατατμήσεων σε εκθετικό χρόνο. Επίσης παρέχουμε και άλλους οι οποίοι λύνουν παρόμοια προβλήματα όπως η καταμέτρηση των k-καλυμμάτων, η άθροιση κατατμήσεων με βάρη και η εύρεση της πιο βαριάς κατάτμησης. Στη συνέχεια παρέχουμε παραδείγματα προβλημάτων τα οποία ανάγονται σε αυτά που λύσαμε παραπάνω και για τα οποία οι αναγωγές δεν απαιτούν πολύ χρόνο. Οι προαναφερθέντες αλγόριθμοι στοχεύουν στον ελάχιστο χρόνο, με τη χωρική πολυ- πλοκότητα να είναι εκθετική. Δεδομένου ότι την ευθύνη για αυτό φέρουν αποκλειστικά οι υπολογισμοί του z-μετασχηματισμού, δίνουμε εναλλακτικούς τρόπους επίλυσης των παραπάνω χωρίς τη χρήση του z-μετασχηματισμού σε πολυωνιμικό χώρο. Το μειονέκτημα αυτών είναι ότι χρειάζονται περισσότερο χρόνο. Κλείνουμε με έναν προσεγγιστικό αλγόριθμο πολυωνυμικού χώρου ο οποίος λύνει το Πρόβλημα Χρωματικού Αριθμού Γραφήματος.The present work is a study of the paper by Andreas Bjorklund, Thore Husfeldt and Mikko Koivisto, ”Set partitioning via inclusion-exclusion”. The main aim of the writer was for the ideas presented to be as accessible as possible to undergraduate students. We prove the principle of inclusion-exclusion and define the zeta transform while also giving an algorithm that computes it. Given a n element set N and a family F of subsets of N we provide an exact algorithm that computes the number of k-partitions in time exponential. We also provide others that solve similar problems like k-covers, sum of weighted partitions and max-weighted partition. We then provide examples of problems which are reducible to the ones solved above and for which the reduction does not dominate the time complexity. The aforementioned algorithms are optimized for time with the space complexity being also exponential. Considering that the responsibility for this falls squarely on the calculations for the z-transform, we provide alternate ways of solving the previous problems where we substitute the z-transform by polynomial space tools with the drawback of them being more costly on time. We conclude with an approximation algorithm for the Chromatic Number Problem in polynomial space

Multilevel spectral clustering : graph partitions and image segmentation

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.Includes bibliographical references (p. 145-146).While the spectral graph partitioning method gives high quality segmentation, segmenting large graphs by the spectral method is computationally expensive. Numerous multilevel graph partitioning algorithms are proposed to reduce the segmentation time for the spectral partition of large graphs. However, the greedy local refinement used in these multilevel schemes has the tendency of trapping the partition in poor local minima. In this thesis, I develop a multilevel graph partitioning algorithm that incorporates the inverse powering method with greedy local refinement. The combination of the inverse powering method with greedy local refinement ensures that the partition quality of the multilevel method is as good as, if not better than, segmenting the large graph by the spectral method. In addition, I present a scheme to construct the adjacency matrix, W and degree matrix, D for the coarse graphs. The proposed multilevel graph partitioning algorithm is able to bisect a graph (k = 2) with significantly shorter time than segmenting the original graph without the multilevel implementation, and at the same time achieving the same normalized cut (Ncut) value. The starting eigenvector, obtained by solving a generalized eigenvalue problem on the coarsest graph, is close to the Fiedler vector of the original graph. Hence, the inverse iteration needs only a few iterations to converge the starting vector. In the k-way multilevel graph partition, the larger the graph, the greater the reduction in the time needed for segmenting the graph. For the multilevel image segmentation, the multilevel scheme is able to give better segmentation than segmenting the original image. The multilevel scheme has higher success of preserving the salient part of an object.(cont.) In this work, I also show that the Ncut value is not the ultimate yardstick for the segmentation quality of an image. Finding a partition that has lower Ncut value does not necessary means better segmentation quality. Segmenting large images by the multilevel method offers both speed and quality.by Tian Fook Kong.S.M