A linear programming approach to increasing the weight of all minimum spanning trees

Dans cet article nous étudions le problème qui consiste à augmenter au moindre coût le poids de tous les arbres couvrants de poids minimum. Nous considérons le cas où le coût d'augmenter le poids d'une arête du graphe est une fonction linéaire par morceaux, convexe et croissante. Nous formulons ce problème par un programme linéaire et nous donnons un algorithme polynomial pour sa résolution et la résolution du son problème dual.

Wiretapping a hidden network

We consider the problem of maximizing the probability of hitting a strategically chosen hidden virtual network by placing a wiretap on a single link of a communication network. This can be seen as a two-player win-lose (zero-sum) game that we call the wiretap game. The value of this game is the greatest probability that the wiretapper can secure for hitting the virtual network. The value is shown to equal the reciprocal of the strength of the underlying graph. We efficiently compute a unique partition of the edges of the graph, called the prime-partition, and find the set of pure strategies of the hider that are best responses against every maxmin strategy of the wiretapper. Using these special pure strategies of the hider, which we call omni-connected-spanning-subgraphs, we define a partial order on the elements of the prime-partition. From the partial order, we obtain a linear number of simple two-variable inequalities that define the maxmin-polytope, and a characterization of its extreme points. Our definition of the partial order allows us to find all equilibrium strategies of the wiretapper that minimize the number of pure best responses of the hider. Among these strategies, we efficiently compute the unique strategy that maximizes the least punishment that the hider incurs for playing a pure strategy that is not a best response. Finally, we show that this unique strategy is the nucleolus of the recently studied simple cooperative spanning connectivity game

Algorithmic and complexity aspects of simple coalitional games

Simple coalitional games are a fundamental class of cooperative games and voting games which are used to model coalition formation, resource allocation and decision making in computer science, artificial intelligence and multiagent systems. Although simple coalitional games are well studied in the domain of game theory and social choice, their algorithmic and computational complexity aspects have received less attention till recently. The computational aspects of simple coalitional games are of increased importance as these games are used by computer scientists to model distributed settings. This thesis fits in the wider setting of the interplay between economics and computer science which has led to the development of algorithmic game theory and computational social choice. A unified view of the computational aspects of simple coalitional games is presented here for the first time. Certain complexity results also apply to other coalitional games such as skill games and matching games. The following issues are given special consideration: influence of players, limit and complexity of manipulations in the coalitional games and complexity of resource allocation on networks. The complexity of comparison of influence between players in simple games is characterized. The simple games considered are represented by winning coalitions, minimal winning coalitions, weighted voting games or multiple weighted voting games. A comprehensive classification of weighted voting games which can be solved in polynomial time is presented. An efficient algorithm which uses generating functions and interpolation to compute an integer weight vector for target power indices is proposed. Voting theory, especially the Penrose Square Root Law, is used to investigate the fairness of a real life voting model. Computational complexity of manipulation in social choice protocols can determine whether manipulation is computationally feasible or not. The computational complexity and bounds of manipulation are considered from various angles including control, false-name manipulation and bribery. Moreover, the computational complexity of computing various cooperative game solutions of simple games in dierent representations is studied. Certain structural results regarding least core payos extend to the general monotone cooperative game. The thesis also studies a coalitional game called the spanning connectivity game. It is proved that whereas computing the Banzhaf values and Shapley-Shubik indices of such games is #P-complete, there is a polynomial time combinatorial algorithm to compute the nucleolus. The results have interesting significance for optimal strategies for the wiretapping game which is a noncooperative game defined on a network

Densities in graphs and matroids

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

Algorithmic and complexity aspects of simple coalitional games

Simple coalitional games are a fundamental class of cooperative games and voting games which are used to model coalition formation, resource allocation and decision making in computer science, artificial intelligence and multiagent systems. Although simple coalitional games are well studied in the domain of game theory and social choice, their algorithmic and computational complexity aspects have received less attention till recently. The computational aspects of simple coalitional games are of increased importance as these games are used by computer scientists to model distributed settings. This thesis fits in the wider setting of the interplay between economics and computer science which has led to the development of algorithmic game theory and computational social choice. A unified view of the computational aspects of simple coalitional games is presented here for the first time. Certain complexity results also apply to other coalitional games such as skill games and matching games. The following issues are given special consideration: influence of players, limit and complexity of manipulations in the coalitional games and complexity of resource allocation on networks. The complexity of comparison of influence between players in simple games is characterized. The simple games considered are represented by winning coalitions, minimal winning coalitions, weighted voting games or multiple weighted voting games. A comprehensive classification of weighted voting games which can be solved in polynomial time is presented. An efficient algorithm which uses generating functions and interpolation to compute an integer weight vector for target power indices is proposed. Voting theory, especially the Penrose Square Root Law, is used to investigate the fairness of a real life voting model. Computational complexity of manipulation in social choice protocols can determine whether manipulation is computationally feasible or not. The computational complexity and bounds of manipulation are considered from various angles including control, false-name manipulation and bribery. Moreover, the computational complexity of computing various cooperative game solutions of simple games in dierent representations is studied. Certain structural results regarding least core payos extend to the general monotone cooperative game. The thesis also studies a coalitional game called the spanning connectivity game. It is proved that whereas computing the Banzhaf values and Shapley-Shubik indices of such games is #P-complete, there is a polynomial time combinatorial algorithm to compute the nucleolus. The results have interesting significance for optimal strategies for the wiretapping game which is a noncooperative game defined on a network.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Submodularity and Its Applications in Wireless Communications

This monograph studies the submodularity in wireless communications and how to use it to enhance or improve the design of the optimization algorithms. The work is done in three different systems. In a cross-layer adaptive modulation problem, we prove the submodularity of the dynamic programming (DP), which contributes to the monotonicity of the optimal transmission policy. The monotonicity is utilized in a policy iteration algorithm to relieve the curse of dimensionality of DP. In addition, we show that the monotonic optimal policy can be determined by a multivariate minimization problem, which can be solved by a discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm. We show that the DSPSA is able to converge to the optimal policy in real time. For the adaptive modulation problem in a network-coded two-way relay channel, a two-player game model is proposed. We prove the supermodularity of this game, which ensures the existence of pure strategy Nash equilibria (PSNEs). We apply the Cournot tatonnement and show that it converges to the extremal, the largest and smallest, PSNEs within a finite number of iterations. We derive the sufficient conditions for the extremal PSNEs to be symmetric and monotonic in the channel signal-to-noise (SNR) ratio. Based on the submodularity of the entropy function, we study the communication for omniscience (CO) problem: how to let all users obtain all the information in a multiple random source via communications. In particular, we consider the minimum sum-rate problem: how to attain omniscience by the minimum total number of communications. The results cover both asymptotic and non-asymptotic models where the transmission rates are real and integral, respectively. We reveal the submodularity of the minimum sum-rate problem and propose polynomial time algorithms for solving it. We discuss the significance and applications of the fundamental partition, the one that gives rise to the minimum sum-rate in the asymptotic model. We also show how to achieve the omniscience in a successive manner