Approximation Algorithms for Faster Communication and Cheaper Networks Using Linear Programming

As we are currently in the information age, people expect access to information to exist by default. In order to facilitate the communication of knowledge, efficient networks must be built. In particular, the networks must be built satisfying some constraints while minimizing cost or time. These constraints often make these problems NP-hard. In this thesis, we investigate two different sets of problems: communicating information quickly and building cheap networks. In the communication problems, the goal is to minimize the number of rounds of communication. In the network design problems the goal is to construct a network of minimum cost. First, we study the communication problem of computing a minimum time schedule to spread rumors in a given graph under the telephone, radio, and wireless (also called edge-star) models. In all of these communication models, the communication happens in discrete rounds. In the telephone model, a matching is given every round and matched nodes exchange all their information with each other. In the radio model, any subset of nodes can broadcast in a round but only nodes with a single broadcasting neighbor get a message. If a node has two or more neighbors broadcasting, they receive no messages due to interference. In the wireless model, any subset of nodes can broadcast in a round and a non-broadcasting node can tune to listen to exactly one of its neighbors. The various rumor spreading problems assume a message at several or all nodes and that each message must reach some target node or set of nodes. The goal is to deliver all messages to their destinations in a minimum number of rounds. The various problems we study include gossip, multicommodity, and asymmetric multicommodity. In the gossip problem, the goal is to communicate messages from every node to every other node. In multicommodity, pairs of nodes are given must exchange their messages with each other. In asymmetric multicommodity, ordered pairs of nodes are given where the source node must send their message to the sink node. Our work exposes relationships across these models and problems and opens up several new avenues for future study. We present a comprehensive study of the approximation algroithms for these problems. We show various reductions from harder to easier models for special demands. In addition, we develop the first sublinear approximation algorithm for the asymmetric multicommodity problem in the edge-star model. Using a reduction from minimium induced matchings, we show the minimum radio gossip time has a hardness of approximation of Ω(n 1 2 − ) in an n-node graph. We continue studying these communication problems in the case where the host graph is planar. To aid us, we first study the poise problem, where the goal is to build a subgraph which minimizes the sum of the diameter of the subgraph and the maximum degree of the subgraph. We show poise has only a O( logn log logn ) integrality gap. This is the first upper bound on the integrality gap of the natural LP; all previous algorithms for poise had yielded only approximations with respect to the integer optimum. Using the poise result and shortest-path separators in planar graphs, we give a poly-logrithmic approximation algorithm for the telephone multicommodity problem. We also use shortest-path separators to show a poly-logarithmic approximation for the radio gossip problem. This is the first result on radio gossip which does not rely on the maximum degree of the graph. Lastly, we show that these results on planar graphs naturally extend to minor-free families of graphs. Next we turn to the problem of building minimum-cost networks. From the publishsubscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and the Internet of Things, new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal builds a physical network in the form of a tree or a star centered at it to push or pull its information from. Some publisher-subscriber pairs of nodes need to coordinate which requires that their corresponding networks overlap at some node. These pairs form the demands which must be satisfied. This simple overlap constraint, along with the requirement that the network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this thesis, we show that with metric costs and a complete demand graph, approximation algorithms with small constant performance ratios exist regardless of whether the networks built must be trees or stars. For the general demand case, we show that a natural LP formulation has a non-constant integrality gap which indicates potential hardness of approximation. Finally, we show that the general demand version has a logarithmic-factor approximation algorithm by adapting previously known methods. Lastly, we study the tree augmentation problem (TAP), where the goal is to augment an existing connected network so that the network becomes 2-edge connected with a minimum cost set of edges. A graph is 2-edge connected if the removal of any one edge does not disconnect the graph. First, we show that in weighted TAP, we can restrict our attention to trees which are binary and all the non-tree edges go between two leaves of the tree. We give two different top-down coloring algorithms. Both of our algorithms differ from known techniques for obtaining a 3 2 -approximation in unweighted TAP and current attempts to reach a 3 2 -approximation in weighted TAP. The first algorithm we describe always gives a 2-approximation starting from any feasible