Node-Weighted Prize Collecting Steiner Tree and Applications

The Steiner Tree problem has appeared in the Karp's list of the first 21 NP-hard problems and is well known as one of the most fundamental problems in Network Design area. We study the Node-Weighted version of the Prize Collecting Steiner Tree problem. In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our goal is to find a subtree T of the graph minimizing the cost of the nodes in T plus penalty of the nodes not in T. By a reduction from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1-o(1))ln n unless NP has quasi-polynomial time algorithms, where n is the number of vertices of the graph. Moss and Rabani claimed an O(log n)-approximation algorithm for the problem using a Primal-Dual approach in their STOC'01 paper \cite{moss2001}. We show that their algorithm is incorrect by providing a counter example in which there is an O(n) gap between the dual solution constructed by their algorithm and the optimal solution. Further, evidence is given that their algorithm probably does not have a simple fix. We propose a new algorithm which is more involved and introduces novel ideas in primal dual approach for network design problems. Also, our algorithm is a Lagrangian Multiplier Preserving algorithm and we show how this property can be utilized to design an O(log n)-approximation algorithm for the Node-Weighted Quota Steiner Tree problem using the Lagrangian Relaxation method. We also show an application of the Node Weighted Quota Steiner Tree problem in designing algorithm with better approximation factor for Technology Diffusion problem, a problem proposed by Goldberg and Liu in \cite{goldberg2012} (SODA 2013). In Technology Diffusion, we are given a graph G and a threshold θ(v) associated with each vertex v and we are seeking a set of initial nodes called the seed set. Technology Diffusion is a dynamic process defined over time in which each vertex is either active or inactive. The vertices in the seed set are initially activated and each other vertex v gets activated whenever there are at least θ(v) active nodes connected to v through other active nodes. The Technology Diffusion problem asks to find the minimum seed set activating all nodes. Goldberg and Liu gave an O(rllog n)-approximation algorithm for the problem where r and l are the diameter of G and the number of distinct threshold values, respectively. We improve the approximation factor to O(min{r,l}log n) by establishing a close connection between the problem and the Node Weighted Quota Steiner Tree problem

Potential-based analysis of social, communication, and distributed networks

In recent years, there has been a wide range of studies on the role of social and distributed networks in various disciplinary areas. In particular, availability of large amounts of data from online social networks and advances in control of distributed systems have drawn the attention of many researchers to exploit the connection between evolutionary behaviors in social, communication and distributed networks. In this thesis, we first revisit several well-known types of social and distributed networks and review some relevant results from the literature. Building on this, we present a set of new results related to four different types of problems, and identify several directions for future research. The study undertaken and the approaches adopted allow us to analyze the evolution of certain types of social and distributed networks and also to identify local and global patterns of their dynamics using some novel potential-theoretic techniques. Following the introduction and preliminaries, we focus on analyzing a specific type of distributed algorithm for quantized consensus known as an unbiased quantized algorithm where a set of agents interact locally in a network in order to reach a consensus. We provide tight expressions for the expected convergence time of such dynamics over general static and time-varying networks. Following this, we introduce new protocols using a special class of Markov chains known as Metropolis chains and obtain the fastest (as of today) randomized quantized consensus protocol. The bounds provided here considerably improve the state of the art over static and dynamic networks. We make a bridge between two classes of problems, namely distributed control problems and game problems. We analyze a class of distributed averaging dynamics known as Hegselmann-Krause opinion dynamics. Modeling such dynamics as a non-cooperative game problem, we elaborate on some of the evolutionary properties of such dynamics. In particular, we answer an open question related to the termination time of such dynamics by connecting the convergence time to the spectral gap of the adjacency matrices of underlying dynamics. This not only allows us to improve the best known upper bound, but also removes the dependency of termination time from the dimension of the ambient space. The approach adopted here can also be leveraged to connect the rate of increase of a so-called kinetic-s-energy associated with multi-agent systems to the spectral gap of their underlying dynamics. We describe a richer class of distributed systems where the agents involved in the network act in a more strategic manner. More specifically, we consider a class of resource allocation games over networks and study their evolution to some final outcomes such as Nash equilibria. We devise some simple distributed algorithms which drive the entire network to a Nash equilibrium in polynomial time for dense and hierarchical networks. In particular, we show that such games benefit from having low price of anarchy, and hence, can be used to model allocation systems which suffer from lack of coordination. This fact allows us to devise a distributed approximation algorithm within a constant gap of any pure-strategy Nash equilibrium over general networks. Subsequently we turn our attention to an important problem related to competition over social networks. We establish a hardness result for searching an equilibrium over a class of games known as competitive diffusion games, and provide some necessary conditions for existence of a pure-strategy Nash equilibrium in such games. In particular, we provide some concentration results related to the expected utility of the players over random graphs. Finally, we discuss some future directions by identifying several interesting problems and justify the importance of the underlying problems