Combinatorial filter reduction: Special cases, approximation, and fixed-parameter tractability

Recent research in algorithmic robotics considers combinatorial filters, which concisely capture the discrete structure underlying many reasoning problems for robots. An important recent result is that the filter minimization problem-Given a filter, find the smallest equivalent filter-is NP-hard. This paper extends that result along several dimensions, including hardness proofs for some natural special cases and for approximation, and new results analyzing the only known algorithm for this problem. We show that this problem is not fixed-parameter tractable for any of the obvious parameters, but it is fixed-parameter tractable for a certain combination of new parameters.