Certain graphs can be described by the distribution of the edges in its subgraphs. For example, a cycle C is a graph that satisfies |E(H)| |V (H)| < |E(C)| |V (C)| = 1 for all non-trivial subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)| |V (H)|−1 ≤ |E(T)| |V (T)|−1 = 1 for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such that |E(H)| |V (H)| ≤ |E(G)| |V (G)| and a 1-balanced graph is a graph such that |E(H)| |V (H)|−1 ≤ |E(G)| |V (G)|−1 for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G that satisfy the property |E(H)| ≤ k|V (H)| − l for all non-trivial subgraphs H of G play important roles in defining rigid structures. This dissertation is a formal study of a class of density functions that extends the above mentioned ideas. For a rational number r ≤ 1, a graph G is said to be r-balanced if and only if for each non-trivial subgraph H of G, we have |E(H)| |V (H)|−r ≤ |E(G)| |V (G)|−r . For r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined and the existence of these graphs is discussed. We also define a class of vulnerability measures on graphs similar to the edge-connectivity of graphs and show how it is related to r-balanced graphs. All these definitions are matroidal and the definitions of r-balanced matroids naturally extend the definitions of r-balanced graphs. The vulnerability measures in graphs that we define are ranked and are lesser than the edge-connectivity. Due to the relationship of the r-balanced graphs with the vulnerability measures defined in the dissertation, identifying r-balanced graphs and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced matroids and their weak forms are discussed. For r ∈ {0, 1}, we give a method to construct big r-balanced graphs from small r-balanced graphs. This construction is a generalization of the construction of Cartesian product of two graphs. We present an algorithmic solution of the problem of transforming any given graph into a 1-balanced graph on the same number of vertices and edges as the given graph. This result is extended to a density function defined on the power set of any set E via a pair of matroid rank functions defined on the power set of E. Many interesting results may be derived in the future by choosing suitable pairs of matroid rank functions and applying the above result.

Thesis (M.Sc.). / Title from title page of PDF (viewed 2008/01/30). Written for the School of Computer Science. Includes bibliographical references.

In this thesis we consider two combinatorial optimization problems that relate to the field of persistent monitoring. In the first part, we extend the classic problem of finding the maximum weight Hamiltonian cycle in a graph to the case where the objective is a submodular function of the edges. We consider a greedy algorithm and a 2-matching based algorithm, and we show that they have approximation factors of 1/2+κ and max{2/(3(2+κ)),(2/3)(1-κ)} respectively, where κ is the curvature of the submodular function. Both algorithms require a number of calls to the submodular function that is cubic to the number of vertices in the graph. We then present a method to solve a multi-objective optimization consisting of both additive edge costs and submodular edge rewards. We provide simulation results to empirically evaluate the performance of the algorithms. Finally, we demonstrate an application in monitoring an environment using an autonomous mobile sensor, where the sensing reward is related to the entropy reduction of a given a set of measurements. In the second part, we study the problem of selecting sensors to obtain the most accurate state estimate of a linear system. The estimator is taken to be a Kalman filter and we attempt to optimize the a posteriori error covariance. For a finite time horizon, we show that, under certain restrictive conditions, the problem can be phrased as a submodular function optimization and that a greedy approach yields a 1-1/(e^(1-1/e))-approximation. Next, for an infinite time horizon, we characterize the exact conditions for the existence of a schedule with bounded estimation error covariance. We then present a scheduling algorithm that guarantees that the error covariance will be bounded and that the error will die out exponentially for any detectable LTI system. Simulations are provided to compare the performance of the algorithm against other known techniques.

In this thesis we consider two combinatorial optimization problems that relate to the field of persistent monitoring. In the first part, we extend the classic problem of finding the maximum weight Hamiltonian cycle in a graph to the case where the objective is a submodular function of the edges. We consider a greedy algorithm and a 2-matching based algorithm, and we show that they have approximation factors of 1/2+κ and max{2/(3(2+κ)),(2/3)(1-κ)} respectively, where κ is the curvature of the submodular function. Both algorithms require a number of calls to the submodular function that is cubic to the number of vertices in the graph. We then present a method to solve a multi-objective optimization consisting of both additive edge costs and submodular edge rewards. We provide simulation results to empirically evaluate the performance of the algorithms. Finally, we demonstrate an application in monitoring an environment using an autonomous mobile sensor, where the sensing reward is related to the entropy reduction of a given a set of measurements. In the second part, we study the problem of selecting sensors to obtain the most accurate state estimate of a linear system. The estimator is taken to be a Kalman filter and we attempt to optimize the a posteriori error covariance. For a finite time horizon, we show that, under certain restrictive conditions, the problem can be phrased as a submodular function optimization and that a greedy approach yields a 1-1/(e^(1-1/e))-approximation. Next, for an infinite time horizon, we characterize the exact conditions for the existence of a schedule with bounded estimation error covariance. We then present a scheduling algorithm that guarantees that the error covariance will be bounded and that the error will die out exponentially for any detectable LTI system. Simulations are provided to compare the performance of the algorithm against other known techniques.

Thesis (Ph. D.)--University of Washington, 2007. / Includes bibliographical references (p. 134-142).

Funções submodulares aparecem naturalmente em diversas áreas, tais como probabilidade, geometria e otimização combinatória. Pode-se dizer que o papel desempenhado por essas funções em otimização discreta é similar ao desempenhado por convexidade em otimização contínua. Com efeito, muitos problemas em otimização combinatória podem ser formulados como um problema de minimizar uma função submodular sobre um conjunto apropriado. Além disso, submodularidade está presente em vários teoremas ou problemas combinatórios e freqüentemente desempenha um papel essencial em uma demonstração ou na eficiência de um algoritmo. Nesta dissertação, estudamos aspectos estruturais e algorítmicos de funções submodulares, com ênfase nos recentes avanços em algoritmos combinatórios para minimização dessas funções. Descrevemos com detalhes os primeiros algoritmos combinatórios e fortemente polinomiais para esse propósito, devidos a Schrijver e Iwata, Fleischer e Fujishige, além de algumas outras extensões. Aplicações de submodularidade em otimização combinatória também estão presentes neste trabalho. / Submodular functions arise naturally in various fields, including probability, geometry and combinatorial optimization. The role assumed by these functions in discrete optimization is similar to that played by convexity in continuous optimization. Indeed, we can state many problems in combinatorial optimization as a problem of minimizing a submodular function over an appropriate set. Moreover, submodularity appears in many combinatorial theorems or problems and frequently plays an essencial role in a proof or an algorithm. In this dissertation, we study structural and algorithmic aspects of submodular functions. In particular, we focus on the recent advances in combinatorial algorithms for submodular function minimization. We describe in detail the first combinatorial strongly polynomial-time algorithms for this purpose, due to Schrijver and Iwata, Fleischer, and Fujishige, as well as some extensions. Some applications of submodularity in combinatorial optimization are also included in this work.

A scenario where multiple entities interact with a common environment to achieve individual and common goals either co-operatively or competitively can be classified as a Multi-Agent System. In this thesis, we concentrate on the situations where the agents exhibit selfish, competitive and strategic behaviour, giving rise to interesting game theoretic and optimization problems. From a computational point of view, the presence of multiple agents introduces strategic and temporal issues, apart from enhancing the difficulty of optimization. We study the following natural mathematical models of such multi-agent problems faced in practice: a) combinatorial optimization problems with multi-agent submodular cost functions, b) combinatorial auctions with partially public valuations and c) online vertex-weighted bipartite matching and single bid budgeted allocations. We provide approximation algorithms, online algorithms and hardness of approximation results for these problems.

Funções submodulares aparecem naturalmente em diversas áreas, tais como probabilidade, geometria e otimização combinatória. Pode-se dizer que o papel desempenhado por essas funções em otimização discreta é similar ao desempenhado por convexidade em otimização contínua. Com efeito, muitos problemas em otimização combinatória podem ser formulados como um problema de minimizar uma função submodular sobre um conjunto apropriado. Além disso, submodularidade está presente em vários teoremas ou problemas combinatórios e freqüentemente desempenha um papel essencial em uma demonstração ou na eficiência de um algoritmo. Nesta dissertação, estudamos aspectos estruturais e algorítmicos de funções submodulares, com ênfase nos recentes avanços em algoritmos combinatórios para minimização dessas funções. Descrevemos com detalhes os primeiros algoritmos combinatórios e fortemente polinomiais para esse propósito, devidos a Schrijver e Iwata, Fleischer e Fujishige, além de algumas outras extensões. Aplicações de submodularidade em otimização combinatória também estão presentes neste trabalho. / Submodular functions arise naturally in various fields, including probability, geometry and combinatorial optimization. The role assumed by these functions in discrete optimization is similar to that played by convexity in continuous optimization. Indeed, we can state many problems in combinatorial optimization as a problem of minimizing a submodular function over an appropriate set. Moreover, submodularity appears in many combinatorial theorems or problems and frequently plays an essencial role in a proof or an algorithm. In this dissertation, we study structural and algorithmic aspects of submodular functions. In particular, we focus on the recent advances in combinatorial algorithms for submodular function minimization. We describe in detail the first combinatorial strongly polynomial-time algorithms for this purpose, due to Schrijver and Iwata, Fleischer, and Fujishige, as well as some extensions. Some applications of submodularity in combinatorial optimization are also included in this work.