This thesis investigates the use of semidefinite programming (SDP) for solving (variants of) the well-known stable set and max-cut problems. Chapter 2 considers a generalization of the stable set problem, and a corresponding SDP relaxation. Similarly, Chapter 3 considers a generalization of the max-cut problem that involves complex roots of unity. Various complex SDP relaxations of this generalized max-cut problem are studied. Chapter 4 provides approximation algorithms for the quantum generalization of the max-cut problem. These approximation algorithms employ a hierarchy of SDP relaxations for noncommutative polynomial optimization problems. Chapter 5 provides an SDP algorithm for computing bounds on the stability numbers of graphs. Chapter 6 provides an SDP algorithm for solving the MAX-SAT problem. Chapters 5 and 6 provide extensive numerical results on these algorithms.
Semidefinite relaxations are a powerful tool for approximately solving combinatorial optimization problems such as MAX-CUT and the Grothendieck problem. By exploiting a bounded rank property of extreme points in the semidefinite cone, we make a sub-constant improvement in the approximation ratio of one such problem. Precisely, we describe a polynomial-time algorithm for the positive semidefinite Grothendieck problem -- based on rounding from the standard relaxation -- that achieves a ratio of 2/π+Θ(1/n), whereas the previous best is 2/π+Θ(1/n). We further show a corresponding integrality gap of 2/π+O~(1/n1/3).