In this report, some results on semidefinite programming relaxation of graph coloring are summarized. Two algorithms on semicoloring/coloring are described in detail for 3-colorable graphs. The relation between semidefinite programming relaxation and the Lovász theta function is simply introduced.

in this paper, we study the problem of minimizing the ratio of two quadratic functions subject to a quadratic constraint. first we introduce a parametric equivalent of the problem. then a bisection and a generalized newton-based method algorithms are presented to solve it. in order to solve the quadratically constrained quadratic minimization problem within both algorithms, a semidefinite optim...

This work presents a hybrid approach to solve the maximum stable set problem, using constraint and semidefinite programming. The approach consists of two steps: subproblem generation and subproblem solution. First we rank the variable domain values, based on the solution of a semidefinite relaxation. Using this ranking, we generate the most promising subproblems first, by exploring a search tre...

We investigate the relationships between various sum of squares (SOS) and semidefinite programming (SDP) relaxations for the sensor network localization problem. In particular, we show that Biswas and Ye’s SDP relaxation is equivalent to the degree one SOS relaxation of Kim et al. We also show that Nie’s sparse-SOS relaxation is stronger than the edge-based semidefinite programming (ESDP) relax...

In 2004 Newman [43] suggested a semidefinite programming relaxation for the Linear Ordering Problem (LOP) that is related to the semidefinite program used in the Goemans-Williamson algorithm to approximate the Max Cut problem [22]. Her model is based on the observation that linear orderings can be fully described by a series of cuts. Newman [43] shows that her relaxation seems better suited for...

This paper is concerned with completely positive and semidefinite relaxations of quadratic programs with linear constraints and binary variables as presented in Burer [2]. It observes that all constraints of the relaxation associated with linear constraints of the original problem can be accumulated in a single linear constraint without changing the feasible set of either the completely positiv...

The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order n the number of vertices of the graph that is equivalent to the currently strongest semidefinit...

Semidefinite optimization problems are an expressive family of convex optimization problems that can be solved efficiently. We develop semidefinite optimization-based formulations and approximations for a number of families of optimization problems, including problems arising in spacecraft attitude estimation and in learning tree-structured statistical models. We construct explicit exact reform...

The purpose of this paper is to solve the 0-1 k-item quadratic knapsack problem (kQKP ), a problem of maximizing a quadratic function subject to two linear constraints. We propose an exact method based on semidefinite optimization. The semidefinite relaxation used in our approach includes simple rank one constraints, which can be handled efficiently by interior point methods. Furthermore, we st...