Graph theoretic results on index coding, causal inference and learning graphical models
Exploiting and learning graph structures is becoming ubiquitous in Network Information Theory and Machine Learning. The former deals with efficient communication schemes in a many-node network. In the latter, inferring graph structured relationships from high dimensional data is important. In this dissertation, some graph theoretic results in these two areas are presented. The first part deals with the problem of optimizing bandwidth resources for a shared broadcast link serving many users each having access to cached content. This problem and its variations are broadly called Index Coding. Index Coding is fundamental to understanding multi-terminal network problems and has applications in networks that deploy caches. The second part deals with the resources required for learning a network structure that encodes distributional and causal relationships among many variables in machine learning. The number of samples needed to learn graphical models that capture crucial distributional information is studied. For learning causal relationships, when passive data acquisition is not sufficient, the number of interventions required is investigated. In the first part, efficient algorithms for placing popular content in a network that deploys a distributed system of caches are provided. Then, the Index Coding problem is considered: every user has its own cache content that is given and transmissions on a shared link are to be optimized. All graph theoretic schemes for Index Coding, known prior to this work, are shown to perform within a constant factor from the one based on graph coloring. Then, `partial' flow-cut gap results for information flow in a multi-terminal network are obtained by leveraging Index Coding ideas. This provides a poly-logarithmic approximation for a known generalization of multi-cut. Finally, optimal cache design in Index Coding for an adversarial demand pattern is considered. Near-optimal algorithms for cache design and delivery within a broad class of schemes are presented. In the second part, sample complexity lower bounds considering average error for learning random Ising Graphical Models, sampled from Erdós-Rényi ensembles, are obtained. Then, the number of bounded interventions required to learn a network of causal relationships under the Pearls model is studied. Upper and lower bounds on the number of size bounded interventions required for various classes of graphs are obtained.