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Combinatorial Optimization via the Sum of Squares Hierarchy

We study the Sum of Squares (SoS) Hierarchy with a view towards combinatorial optimization. We survey the use of the SoS hierarchy to obtain approximation algorithms on graphs using their spectral properties. We present a simplified proof of the result of Feige and Krauthgamer on the performance of the hierarchy for the Maximum Clique problem on random graphs. We also present a result of Guruswami and Sinop that shows how to obtain approximation algorithms for the Minimum Bisection problem on low threshold-rank graphs. We study inapproximability results for the SoS hierarchy for general constraint satisfaction problems and problems involving graph densities such as the Densest $k$-subgraph problem. We improve the existing inapproximability results for general constraint satisfaction problems in the case of large arity, using stronger probabilistic analyses of expansion of random instances. We examine connections between constraint satisfaction problems and density problems on graphs. Using them, we obtain new inapproximability results for the hierarchy for the Densest $k$-subhypergraph problem and the Minimum $p$-Union problem, which are proven via reductions. We also illustrate the relatively new idea of pseudocalibration to construct integrality gaps for the SoS hierarchy for Maximum Clique and Max $K$-CSP. The application to Max $K$-CSP that we present is known in the community but has not been presented before in the literature, to the best of our knowledge.