Integer linear programming formulations of the filter partitioning minimization problem

Combinatorial filters, which take the form of labelled transition graphs, are a general representation for filtering and inference tasks in robotics. They are of particular interest in contexts where the objective is to minimize the computational resources needed to execute the filter. One specific problem is called the filter minimization (FM) problem, in which the goal is to find, for a given original filter, a state-minimal filter equivalent to the original filter. We consider a special case of FM, called the filter partitioning minimization (FPM) problem, in which the reduced filter must partition the state space of the original filter. This problem has been proven to be NP-hard. This paper considers the practical problem of solving FPM in spite of these hardness results. In contrast to the best known algorithm for this problem, a heuristic approach based on graph coloring proposed by O’Kane and Shell, we show how to convert an FPM instance to an instance of the well-known integer linear programming (ILP) problem. We present three distinct formulations of this reduction. Though ILP is itself a challenging problem, reducing FPM to ILP has the advantage that the ILP problem has been studied in great detail, and highly-optimized solvers are readily available. We describe experiments comparing this approach to the heuristic algorithm of O’Kane and Shell. The results show that the proposed ILP technique performs better in computing exact solutions as the filter sizes grow, and that the ILP approach obtains higher-quality feasible solutions, in contexts where time limitations prohibit the computation of exact solutions.