Graphical models use the intuitive and well-studied methods of graph theory to implicitly represent dependencies between variables in large systems. They can model the global behaviour of a complex system by specifying only local factors. This thesis studies inference in discrete graphical models from an algebraic perspective and the ways inference can be used to express and approximate NP-hard combinatorial problems. We investigate the complexity and reducibility of various inference problems, in part by organizing them in an inference hierarchy. We then investigate tractable approximations for a subset of these problems using distributive law in the form of message passing. The quality of the resulting message passing procedure, called Belief Propagation (BP), depends on the influence of loops in the graphical model. We contribute to three classes of approximations that improve BP for loopy graphs A) loop correction techniques; B) survey propagation, another message passing technique that surpasses BP in some settings; and C) hybrid methods that interpolate between deterministic message passing and Markov Chain Monte Carlo inference. We then review the existing message passing solutions and provide novel graphical models and inference techniques for combinatorial problems under three broad classes: A) constraint satisfaction problems such as satisfiability, coloring, packing, set / clique-cover and dominating / independent set and their optimization counterparts; B) clustering problems such as hierarchical clustering, K-median, K-clustering, K-center and modularity optimization; C) problems over permutations including assignment, graph morphisms and alignment, finding symmetries and traveling salesman problem. In many cases we show that message passing is able to find solutions that are either near optimal or favourably compare with today's state-of-the-art approaches.

Many diseases cause significant changes to the concentrations of small molecules (aka metabolites) that appear in a person's biofluids, which means such diseases can often be readily detected from a person's "metabolic profile". This information can be extracted from a biofluid's NMR spectrum. Today, this is often done manually by trained human experts, which means this process is relatively slow, expensive and error-prone. This paper presents a tool, Bayesil, that can quickly, accurately and autonomously produce a complex biofluid's (e.g., serum or CSF) metabolic profile from a 1D1H NMR spectrum. This requires first performing several spectral processing steps then matching the resulting spectrum against a reference compound library, which contains the "signatures" of each relevant metabolite. Many of these steps are novel algorithms and our matching step views spectral matching as an inference problem within a probabilistic graphical model that rapidly approximates the most probable metabolic profile. Our extensive studies on a diverse set of complex mixtures, show that Bayesil can autonomously find the concentration of all NMR-detectable metabolites accurately (~90% correct identification and ~10% quantification error), in <5minutes on a single CPU. These results demonstrate that Bayesil is the first fully-automatic publicly-accessible system that provides quantitative NMR spectral profiling effectively -- with an accuracy that meets or exceeds the performance of trained experts. We anticipate this tool will usher in high-throughput metabolomics and enable a wealth of new applications of NMR in clinical settings. Available at this http URL.