A general class of combinatorial filters that can be minimized efficiently

TL;DR

This paper analyzes the complexity of state minimization in combinatorial filters, identifying key factors affecting NP-hardness, and introduces a new algorithm that efficiently minimizes a broad class of such filters.

Contribution

It provides a nuanced complexity analysis, identifies factors influencing hardness, and proposes a new polynomial-time algorithm for a large subclass of combinatorial filters.

Findings

Identifies graph structure and determinism constraints as sources of complexity.

Introduces a new constraint repair algorithm for a broad subclass.

Provides new families of filters that can be minimized efficiently.

Abstract

State minimization of combinatorial filters is a fundamental problem that arises, for example, in building cheap, resource-efficient robots. But exact minimization is known to be NP-hard. This paper conducts a more nuanced analysis of this hardness than up till now, and uncovers two factors which contribute to this complexity. We show each factor is a distinct source of the problem's hardness and are able, thereby, to shed some light on the role played by (1) structure of the graph that encodes compatibility relationships, and (2) determinism-enforcing constraints. Just as a line of prior work has sought to introduce additional assumptions and identify sub-classes that lead to practical state reduction, we next use this new, sharper understanding to explore special cases for which exact minimization is efficient. We introduce a new algorithm for constraint repair that applies to a large sub-class of filters, subsuming three distinct special cases for which the possibility of optimal minimization in polynomial time was known earlier. While the efficiency in each of these three cases previously appeared to stem from seemingly dissimilar properties, when seen through the lens of the present work, their commonality now becomes clear. We also provide entirely new families of filters that are efficiently reducible.