7 research outputs found

    Exploiting Dense Structures in Parameterized Complexity

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    Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most ?(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized

    Problems in Extremal Graph Theory

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    This dissertation consists of six chapters concerning a variety of topics in extremal graph theory.Chapter 1 is dedicated to the results in the papers with Antnio Giro, Gbor Mszros, and Richard Snyder. We say that a graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in eachpair. We study the extremal behavior of maximum degree and diameter in some classes of path-pairable graphs. In particular we show that a path-pairable planar graph must have a vertex of linear degree.In Chapter 2 we present a joint work with Antnio Giro and Teeradej Kittipassorn. Given graphs G and H, we say that a graph F is H-saturated in G if F is H-free subgraph of G, but addition of any edge from E(G) to F creates a copy of H. Here we deal with the case when G is a complete k-partite graph with n vertices in each class, and H is a complete graph on r vertices. We prove bounds, which are tight for infinitely many values of k and r, on the minimal number of edges in a H-saturated graph in G, for this choice of G and H, answering questions of Ferrara, Jacobson, Pfender, and Wenger; and generalizing a result of Roberts.Chapter 3 is about a joint paper with Antnio Giro and Teeradej Kittipassorn. A coloring of the vertices of a digraph D is called majority coloring if no vertex of D receives the same color as more than half of its outneighbours. This was introduced by van der Zypen in 2016. Recently, Kreutzer, Oum, Seymour, van der Zypen, and Wood posed a number of problems related to this notion: here we solve several of them.In Chapter 4 we present a joint work with Antnio Giro. We show that given any integer k there exist functions g1(k), g2(k) such that the following holds. For any graph G with maximum degree one can remove fewer than g1(k) ^{1/2} vertices from G so that the remaining graph H has k vertices of the same degree at least (H) g2(k). It is an approximate version of conjecture of Caro and Yuster; and Caro, Lauri, and Zarb, who conjectured that g2(k) = 0.Chapter 5 concerns results obtained together with Kazuhiro Nomoto, Julian Sahasrabudhe, and Richard Snyder. We study a graph parameter, the graph burning number, which is supposed to measure the speed of the spread of contagion in a graph. We prove tight bounds on the graph burning number of some classes of graphs and make a progress towards a conjecture of Bonato, Janssen, and Roshanbin about the upper bound of graph burning number of connected graphs.In Chapter 6 we present a joint work with Teeradej Kittipassorn. We study the set of possible numbers of triangles a graph on a given number of vertices can have. Among other results, we determine the asymptotic behavior of the smallest positive integer m such that there is no graph on n vertices with exactly m copies of a triangle. We also prove similar results when we replace triangle by any fixed connected graph

    Opacity and Structural Resilience in Cyberphysical Systems

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    Cyberphysical systems (CPSs) integrate communication, control, and computation with physical processes. Examples include power systems, water distribution networks, and on a smaller scale, medical devices and home control systems. Since these systems are often controlled over a network, the sharing of information among systems and across geographies makes them vulnerable to attacks carried out (possibly remotely) by malicious adversaries. An attack could be carried out on the physical system, on the computer(s) controlling the system, or on the communication links between the system and the computer. Thus, significant material damage can be caused by an attacker who is able to gain access to the system, and such attacks will often have the consequence of causing widespread disruption to everyday life. Therefore, ensuring the safety of information critical to nominal operation of the system is of utmost importance. This dissertation addresses two problems in the broad area of the Control and Security of Cyberphysical Systems. First, we present a framework for opacity in CPSs modeled as a discrete-time linear time-invariant (DT-LTI) system. The current state-of-the-art in this field studies opacity for discrete event systems (DESs) described by regular languages. However, the states in a DES are discrete; in many practical systems, it is common for states (and other system variables) to take continuous values. We define a notion of opacity called k-initial state opacity (k-ISO) for such systems. A set of secret states is said to be k-ISO with respect to a set of nonsecret states if the outputs at time k of every trajectory starting from the set of secret states is indistinguishable from the output at time k of some trajectory starting from the set of nonsecret states. Necessary and sufficient conditions to establish k-ISO are presented in terms of sets of reachable states. Opacity of a given DT-LTI system is shown to be equivalent to the output controllability of a system obeying the same dynamics, but with different initial conditions. We then study the case where there is more than one adversarial observer, and define several notions of decentralized opacity. These notions of decentralized opacity will depend on whether there is a centralized coordinator or not, and the presence or absence of collusion among the adversaries. We establish conditions for decentralized opacity in terms of sets of reachable states. In the case of colluding adversaries, we present a condition for non-opacity in terms of the structure of the communication graph. We extend this work to formulate notions of opacity for discrete-time switched linear systems. A switched system consists of a finite number of subsystems and a rule that orchestrates switching among them. We distinguish between the cases when the secret is specified as a set of initial modes, a set of initial states, or a combination of the two. The novelty of our schemes is in the fact that we place restrictions on: i) the allowed transitions between modes (specified by a directed graph), ii) the number of allowed changes of modes (specified by lengths of paths in the directed graph), and iii) the dwell times in each mode. Each notion of opacity is characterized in terms of allowed switching sequences and sets of reachable states and/ or modes. Finally we present algorithmic procedures to verify these notions, and provide bounds on their computational complexity. Second, we study the resilience of CPSs to denial-of-service (DoS) and integrity attacks. The CPS is modeled as a linear structured system, and its resilience to an attack is interpreted in a graph-theoretic framework. The structural systems approach presumes knowledge of only the positions of zero and nonzero entries in the system matrices to infer system properties. This approach is attractive due to the fact that these properties will hold for almost every admissible numerical realization of the system. The structural resilience of the system is characterized in terms of unmatched vertices in maximum matchings of the bipartite graph and connected components of directed graph representations of the system under attack. Further, we establish a condition based on the zero structure of an input matrix that will ensure that the system is structurally resilient to a state feedback integrity attack if it is also resilient to a DoS attack. Finally, we formulate an extension to the case of switched structured systems, and derive conditions for such systems to be structurally resilient to a DoS attack

    Finite-State Genericity : on the Diagonalization Strength of Finite Automata

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    Algorithmische Generizit¨atskonzepte spielen eine wichtige Rolle in der Berechenbarkeitsund Komplexit¨atstheorie. Diese Begriffe stehen in engem Zusammenhang mit grundlegenden Diagonalisierungstechniken, und sie wurden zur Erzielung starker Trennungen von Komplexit¨atsklassen verwendet. Da f¨ur jedes Generizit¨atskonzept die zugeh¨origen generischen Mengen eine co-magere Klasse bilden, ist die Analyse generischer Mengen ein wichtiges Hifsmittel f¨ur eine quantitative Analyse struktureller Ph¨anomene. Typischerweise werden Generizit¨atskonzepte mit Hilfe von Erweiterungsfunktionen definiert, wobei die St¨arke eines Konzepts von der Komplexit¨at der zugelassenen Erwiterungsfunktionen abh¨angt. Hierbei erweisen sich die sog. schwachen Generizit¨atskonzepte, bei denen nur totale Erweiterungsfunktionen ber¨ucksichtigt werden, meist als wesentlich schw¨acher als die vergleichbaren allgemeinen Konzepte, bei denen auch partielle Funktionen zugelassen sind. Weiter sind die sog. beschr¨ankten Generizit¨atskonzepte – basierend auf Erweiterungen konstanter L¨ange – besonders interessant, da hier die Klassen der zugeh¨origen generischen Mengen nicht nur co-mager sind sondern zus¨atzlich Maß 1 haben. Generische Mengen diesen Typs sind daher typisch sowohl im topologischen wie im maßtheoretischen Sinn. In dieser Dissertation initiieren wir die Untersuchung von Generizit¨at im Bereich der Theorie der Formalen Sprachen: Wir f¨uhren finite-state-Generizit¨atskonzepte ein und verwenden diese, um die Diagonalisierungsst¨arke endlicher Automaten zu erforschen. Wir konzentrieren uns hierbei auf die beschr¨ankte finite-state-Generizit¨at und Spezialf ¨alle hiervon, die wir durch die Beschr¨ankung auf totale Erweiterungsfunktionen bzw. auf Erweiterungen konstanter L¨ange erhalten. Wir geben eine rein kombinatorische Charakterisierung der beschr¨ankt finite-state-generischen Mengen: Diese sind gerade die Mengen, deren charakteristische Folge saturiert ist, d.h. jedes Bin¨arwort als Teilwort enth¨alt. Mit Hilfe dieser Charakterisierung bestimmen wir die Komplexit¨at der beschr¨ankt finitestate- generischen Mengen und zeigen, dass solch eine generische Menge nicht regul¨ar sein kann es aber kontext-freie Sprachen mit dieser Generizit¨atseigenschaft gibt. Die von uns betrachteten unbeschr¨ankten finite-state-Generizit¨atskonzepte basieren auf Moore-Funktionen und auf Verallgemeinerungen dieser Funktionen. Auch hier vergleichen wir die St¨arke der verschiedenen korrespondierenden Generizit¨atskonzepte und er¨ortern die Frage, inwieweit diese Konzepte m¨achtiger als die beschr¨ankte finite-state-Generizit ¨at sind. Unsere Untersuchungen der finite-state-Generizit¨at beruhen zum Teil auf neuen Ergebnissen ¨uber Bi-Immunit¨at in der Chomsky-Hierarchie, einer neuen Chomsky-Hierarchie f¨ur unendliche Folgen und einer gr¨undlichen Untersuchung der saturierten Folgen. Diese Ergebnisse – die von unabh¨angigem Interesse sind – werden im ersten Teil der Dissertation vorgestellt. Sie k¨onnen unabh¨angig von dem Hauptteil der Arbeit gelesen werden

    Chromatic and structural properties of sparse graph classes

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    A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class
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