Integer Programming as a Framework for Optimization and Approximability

Abstract

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 MaxΣ₁, which is the syntactic version of the well-studied class MaxNP. 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 MaxMatching, for which no previous structural explanation of approximability has been found. We also investigate structurally defined classes of integer programs for minimization problems and show a difference between their maximization counterparts.