Fundamental domains for proper affine actions of Coxeter groups in three dimensions

We study proper actions of groups $G \cong \Z/2\Z \ast \Z/2\Z \ast \Z/2\Z$ on affine space of three real dimensions. Since $G$ is nonsolvable, work of Fried and Goldman implies that it preserves a Lorentzian metric. A subgroup $\Gamma < G$ of index two acts freely, and $\R^3/\Gamma$ is a Margulis spacetime associated to a hyperbolic surface $\Sigma$. When $\Sigma$ is convex cocompact, work of Danciger, Gu{\'e}ritaud, and Kassel shows that the action of $\Gamma$ admits a polyhedral fundamental domain bounded by crooked planes. We consider under what circumstances the action of $G$ also admits a crooked fundamental domain. We show that it is possible to construct actions of $G$ that fail to admit crooked fundamental domains exactly when the extended mapping class group of $\Sigma$ fails to act transitively on the top-dimensional simplices of the arc complex of $\Sigma$. We also provide explicit descriptions of the moduli space of $G$ actions that admit crooked fundamental domains.

Mathematical Programming Models for Influence Maximization on Social Networks

In this dissertation, we apply mathematical programming techniques (i.e., integer programming and polyhedral combinatorics) to develop exact approaches for influence maximization on social networks. We study four combinatorial optimization problems that deal with maximizing influence at minimum cost over a social network. To our knowl- edge, all previous work to date involving influence maximization problems has focused on heuristics and approximation. We start with the following viral marketing problem that has attracted a significant amount of interest from the computer science literature. Given a social network, find a target set of customers to seed with a product. Then, a cascade will be caused by these initial adopters and other people start to adopt this product due to the influence they re- ceive from earlier adopters. The idea is to find the minimum cost that results in the entire network adopting the product. We first study a problem called the Weighted Target Set Selection (WTSS) Prob- lem. In the WTSS problem, the diffusion can take place over as many time periods as needed and a free product is given out to the individuals in the target set. Restricting the number of time periods that the diffusion takes place over to be one, we obtain a problem called the Positive Influence Dominating Set (PIDS) problem. Next, incorporating partial incentives, we consider a problem called the Least Cost Influence Problem (LCIP). The fourth problem studied is the One Time Period Least Cost Influence Problem (1TPLCIP) which is identical to the LCIP except that we restrict the number of time periods that the diffusion takes place over to be one. We apply a common research paradigm to each of these four problems. First, we work on special graphs: trees and cycles. Based on the insights we obtain from special graphs, we develop efficient methods for general graphs. On trees, first, we propose a polynomial time algorithm. More importantly, we present a tight and compact extended formulation. We also project the extended formulation onto the space of the natural vari- ables that gives the polytope on trees. Next, building upon the result for trees---we derive the polytope on cycles for the WTSS problem; as well as a polynomial time algorithm on cycles. This leads to our contribution on general graphs. For the WTSS problem and the LCIP, using the observation that the influence propagation network must be a directed acyclic graph (DAG), the strong formulation for trees can be embedded into a formulation on general graphs. We use this to design and implement a branch-and-cut approach for the WTSS problem and the LCIP. In our computational study, we are able to obtain high quality solutions for random graph instances with up to 10,000 nodes and 20,000 edges (40,000 arcs) within a reasonable amount of time.