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

Abstract

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 fixedparameter tractable for any of the obvious parameters, but it is fixed-parameter tractable for a certain combination of new parameters.

Cover Combinatorial Filters and their Minimization Problem (Extended Version)

“…(Put another way: the inexactness in arriving at a minimal filter can be traced to the graph coloring giving a suboptimal result.) But this conjecture was later proved to be false by Saberifar et al [8]. They showed there may exist multiple optimal solutions to the graph coloring subproblem, only some of which will lead to the minimal filter (see Theorem 7 in [8]).…”

“…But this conjecture was later proved to be false by Saberifar et al [8]. They showed there may exist multiple optimal solutions to the graph coloring subproblem, only some of which will lead to the minimal filter (see Theorem 7 in [8]). They refined the conjecture, giving the following statement of existence: Idea 1 (see Theorem 8 in [8]).…”

“…They showed there may exist multiple optimal solutions to the graph coloring subproblem, only some of which will lead to the minimal filter (see Theorem 7 in [8]). They refined the conjecture, giving the following statement of existence: Idea 1 (see Theorem 8 in [8]). In O'Kane and Shell's heuristic algorithm [5] for so-fm, there always exists some optimal coloring for each conflict graph in the step-wise conflict refinement process, such that it generates a minimal filter.…”

“…The definition of the reduction problems within [5,8,9] are specified so as to require that the output obtained be deterministic. But this postcondition is never shown formally established.…”

FIR to FIR Model Reduction with Linear Group Delay in Passband by SDP Optimization

“…Many scientists study model reduction filter design and propose many different methods in recent years. An important recent result is that the filter minimization problem-given a filter, find the smallest equivalent filter-is NP-hard [2]. Reference [3] propose a comparative study based on combined special projections and important frequency-weighted model order reduction algorithms to compute optimal approximants for full-order digital filter.…”