This paper investigates the bounds on the expectation of combinatorial optimization given moment information for each individual random variable. A popular approach to solving this problem, known as the marginal moment model (MMM), is to reformulate it as a semidefinite program (SDP). In this paper, we investigate the structure of MMM with up to fourth-order … Read more
The Nemhauser-Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP … Read more
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1 polynomial programming). For polyhedral relaxations of such problems it is generally not true that variables integer in the relaxed solution will retain the same values in … Read more
Given a discrete optimization problem $Z(\mb{\tilde{c}})=\max\{\mb{\tilde{c}}’\mb{x}:\mb{x}\in \mathcal{X}\}$, with objective coefficients $\mb{\tilde{c}}$ chosen randomly from a distribution ${\mathcal{\theta}}$, we would like to evaluate the expected value $E_\theta(Z(\mb{\tilde{c}}))$ and the probability $P_{\mathcal{\theta}}(x^*_i(\mb{\tilde{c}})=k)$ where $x^*(\mb{\tilde{c}})$ is an optimal solution to $Z(\mb{\tilde{c}})$. We call this the persistency problem for a discrete optimization problem under uncertain objective, and $P_{\mathcal{\theta}}(x^*_i(\mb{\tilde{c}})=k)$, the … Read more