Integer Programming as a Framework for Optimization and Approximability

Structural approximation theory seeks to provide a framework for expressing optimization problems, and isolating structural or syntactic conditions that explain the apparent diierence in the approximation properties of diierent NP-optimization problems. In this paper, we initiate a study of structural approximation using integer programming (an optimization problem in its own right) as a general framework for expressing optimization problems. We isolate three classes of constant-approximable maximization problems, based on restricting appropriately the syntactic form of the integer programs expressing them. The rst of these classes subsumes Max 1, which is the syntactic version of the well-studied class Max NP. Moreover, by allowing variables to take on not just 0/1 values but rather values in a polylogarithmic or polynomial range, we obtain syntactic maximization classes that are polylog-approximable and poly-approximable, respectively. The other two classes contain problems, such as Max Matching, for which no previous structural explanation of ap-proximability has been found. We also investigate structurally-deened classes of integer programs for minimization problems, and show a diier-ence between their maximization counterparts.