Connectedness of graphs and its application to connected matroids through covering-based rough sets

Graph theoretical ideas are highly utilized by computer science fields especially data mining. In this field, a data structure can be designed in the form of tree. Covering is a widely used form of data representation in data mining and covering-based rough sets provide a systematic approach to this type of representation. In this paper, we study the connectedness of graphs through covering-based rough sets and apply it to connected matroids. First, we present an approach to inducing a covering by a graph, and then study the connectedness of the graph from the viewpoint of the covering approximation operators. Second, we construct a graph from a matroid, and find the matroid and the graph have the same connectedness, which makes us to use covering-based rough sets to study connected matroids. In summary, this paper provides a new approach to studying graph theory and matroid theory

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Some Applications of the Weighted Combinatorial Laplacian

The weighted combinatorial Laplacian of a graph is a symmetric matrix which is the discrete analogue of the Laplacian operator. In this thesis, we will study a new application of this matrix to matching theory yielding a new characterization of factor-criticality in graphs and matroids. Other applications are from the area of the physical design of very large scale integrated circuits. The placement of the gates includes the minimization of a quadratic form given by a weighted Laplacian. A method based on the dual constrained subgradient method is proposed to solve the simultaneous placement and gate-sizing problem. A crucial step of this method is the projection to the flow space of an associated graph, which can be performed by minimizing a quadratic form given by the unweighted combinatorial Laplacian.Andwendungen der gewichteten kombinatorischen Laplace-Matrix Die gewichtete kombinatorische Laplace-Matrix ist das diskrete Analogon des Laplace-Operators. In dieser Arbeit stellen wir eine neuartige Charakterisierung von Faktor-Kritikalität von Graphen und Matroiden mit Hilfe dieser Matrix vor. Wir untersuchen andere Anwendungen im Bereich des Entwurfs von höchstintegrierten Schaltkreisen. Die Platzierung basiert auf der Minimierung einer quadratischen Form, die durch eine gewichtete kombinatorische Laplace-Matrix gegeben ist. Wir präsentieren einen Algorithmus für das allgemeine simultane Platzierungs- und Gattergrößen-Optimierungsproblem, der auf der dualen Subgradientenmethode basiert. Ein wichtiger Bestandteil dieses Verfahrens ist eine Projektion auf den Flussraum eines assoziierten Graphen, die als die Minimierung einer durch die Laplace-Matrix gegebenen quadratischen Form aufgefasst werden kann

Tropical Aspects in Geometry, Topology and Physics

The workshop Tropical Aspects in Geometry, Topology and Physics was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject. The development of tropical geometry is based on deep links between problems in real and complex enumerative geometry, symplectic geometry, quantum fields theory, mirror symmetry, dynamical systems and other research areas. On the other hand, new interesting phenomena discovered in the framework of tropical geometry (like refined tropical enumerative invariants) pose the problem of a conceptual understanding of these phenomena in the “classical” geometry and mathematical physics

A system-theoretic approach to multi-agent models

A system-theoretic model for cooperative settings is presented that unifies and ex- tends the models of classical cooperative games and coalition formation processes and their generalizations. The model is based on the notions of system, state and transi- tion graph. The latter describes changes of a system over time in terms of actions governed by individuals or groups of individuals. Contrary to classic models, the pre- sented model is not restricted to acyclic settings and allows the transition graph to have cycles. Time-dependent solutions to allocation problems are proposed and discussed. In par- ticular, Weber’s theory of randomized values is generalized as well as the notion of semi-values. Convergence assertions are made in some cases, and the concept of the Cesàro value of an allocation mechanism is introduced in order to achieve convergence for a wide range of allocation mechanisms. Quantum allocation mechanisms are de- fined, which are induced by quantum random walks on the transition graph and it is shown that they satisfy certain fairness criteria. A concept for Weber sets and two dif- ferent concepts of cores are proposed in the acyclic case, and it is shown under some mild assumptions that both cores are subsets of the Weber set. Moreover, the model of non-cooperative games in extensive form is generalized such that the presented model achieves a mutual framework for cooperative and non-co- operative games. A coherency to welfare economics is made and to each allocation mechanism a social welfare function is proposed

Polütoopide laienditega seotud ülesanded

Väitekirja elektrooniline versioon ei sisalda publikatsiooneLineaarplaneerimine on optimeerimine matemaatilise mudeliga, mille sihi¬funktsioon ja kitsendused on esitatud lineaarsete seostega. Paljusid igapäeva elu väljakutseid võime vaadelda lineaarplaneerimise vormis, näiteks miinimumhinna või maksimaalse tulu leidmist. Sisepunkti meetod saavutab häid tulemusi nii teoorias kui ka praktikas ning lahendite leidmise tööaeg ja lineaarsete seoste arv on polünomiaalses seoses. Sellest tulenevalt eksponentsiaalne arv lineaarseid seoseid väljendub ka ekponentsiaalses tööajas. Iga vajalik lineaarne seos vastab ühele polütoobi P tahule, mis omakorda tähistab lahendite hulka. Üks võimalus tööaja vähendamiseks on suurendada dimensiooni, mille tulemusel väheneks ka polütoobi tahkude arv. Saadud polütoopi Q nimeta¬takse polütoobi P laiendiks kõrgemas dimensioonis ning polütoobi Q minimaalset tahkude arvu nimetakakse polütoobi P laiendi keerukuseks, sellisel juhul optimaalsete lahendite hulk ei muutu. Tekib küsimus, millisel juhul on võimalik leida laiend Q, mille korral tahkude arv on polünomiaalne. Mittedeterministlik suhtluskeerukus mängib olulist rolli tõestamaks polütoopide laiendite keerukuse alampiiri. Polütoobile P vastava suhtluskeerukuse leidmine ning alamtõkke tõestamine väistavad võimalused leida laiend Q, mis ei oleks eksponentsiaalne. Käesolevas töös keskendume me juhuslikele Boole'i funktsioonidele f, mille tihedusfunktsioon on p = p(n). Me pakume välja vähima ülemtõkke ning suurima alamtõkke mittedeterministliku suhtluskeerukuse jaoks. Lisaks uurime me ka pedigree polütoobi graafi. Pedigree polütoop on rändkaupmehe ülesande polütoobi laiend, millel on kombinatoorne struktuur. Polütoobi graafi võib vaadelda kui abstraktset graafi ning see annab informatsiooni polütoobi omaduste kohta.The linear programming (LP for short) is a method for finding an optimal solution, such as minimum cost or maximum profit for a linear function subject to linear constraints. But having an exponential number of inequalities gives the exponential running time in solving linear program. A polytope, let's say P, represents the space of the feasible solution. One idea for decreasing the running time of the problem, is lifting the polytope P tho the higher dimensions with the goal of decresing the number of inequalities. The polytope in higher dimension, let's say Q, is the extension of the original polytope P and the minimum number of facets that Q can have is the extension complexity of P. Then the optimal solution of the problem over Q, gives the optimal solution over P. The natural question may raise is when is it possible to have an extension with a polynomial number of inequalities? Nondeterministic communication complexity is a powerful tool for proving lower bound on the extension complexity of a polytopes. Finding a suitable communication complexity problem corresponded to a polytope P and proving a linear lower bound for the nondeterministic communication complexity of it, will rule out all the attempts for finding sub-exponential size extension Q of P. In this thesis, we focus on the random Boolean functions f, with density p = p(n). We give tight upper and lower bounds for the nondeterministic communication complexity and parameters related to it. Also, we study the rank of fooling set matrix which is an important lower bound for nondeterministic communication complexity. Finally, we investigate the graph of the pedigree polytope. Pedigree polytope is an extension of TSP (traveling salesman problem; the most extensively studied problem in combinatorial optimization) polytopes with a nice combinatorial structure. The graph of a polytope can be regarded as an abstract graph and it reveals meaningful information about the properties of the polytope

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum