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) difference in the approximation properties of different 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 first of these classes subsumes MAX Σ₁, 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 approximability has been found