A fixed-parameter tractable algorithm for combinatorial filter reduction

TL;DR

This paper gives a fixed-parameter tractable algorithm for the general reduction problem by exploiting a transformation into clique covering by identifying parameters affecting filter reduction that are based upon inter-constraint couplings, which add to the structural parameters present in the unconstrained problem of minimal clique covering.

Abstract

What is the minimal information that a robot must retain to achieve its task? To design economical robots, the literature dealing with reduction of combinatorial filters approaches this problem algorithmically. As lossless state compression is NP-hard, prior work has examined, along with minimization algorithms, a variety of special cases in which specific properties enable efficient solution. Complementing those findings, this paper refines the present understanding from the perspective of parameterized complexity. We give a fixed-parameter tractable algorithm for the general reduction problem by exploiting a transformation into clique covering. The transformation introduces new constraints that arise from sequential dependencies encoded within the input filter -- some of these constraints can be repaired, others are treated through enumeration. Through this approach, we identify parameters affecting filter reduction that are based upon inter-constraint couplings (expressed as a notion of their height and width), which add to the structural parameters present in the unconstrained problem of minimal clique covering. Compared with existing work, we precisely identify and quantitatively characterize those features that contribute to the problem's hardness: given a problem instance, the combinatorial core may be a fraction of the instance's full size, with a small subset of constraints needing to be considered, and even those may have directly identifiable couplings that collapse degrees of freedom in the enumeration.

Figures (5)

• Figure 1: Diverse examples of combinatorial filters. Sakcak et al. sakcak2024mathematical consider a circular environment like that shown in (a) with two types of break-beam sensors that trigger when crossed by an agent. They derive the 4-state filter, depicted in (b), that outputs grey when it detects that strictly clockwise/anticlockwise motion has been violated. Scenario (c) is re-drawn from rahmani21equivalence, where the robot observes only the cyclic ordering of 4 landmarks. Their 3-state filter, depicted in (d), determines definitively whether the robot's current location is within the cyan region. In (e), the task of orienting a polygon with a squeeze-gripper, based upon Taylor1988SensorbasedMP, is expressed as feedback plan (f), with green encoding the action of rotating the gripper by $65^\circ$, and purple the squeeze action (reproduced from zhang22sso).
• Figure 2: An illustration showing (partially) a filter $\mathscr{F}\xspace$ (left inset) leading to a compatibility graph $G_\mathscr{F}\xspace$ (above). The ${Z^2}$ set is also shown.
• Figure 3: An example downstream-enabled prescription for the pairs in ${\color{domaincolor}D}\xspace = {Z^2}$ corresponding to the instance in Figure \ref{['fig:filter_to_graph']}. Dashed red and solid green outlines represent states in $\mathscr{F}\xspace$ to be split and merged, respectively.
• Figure 4: Partial order of $\mathrlap{\raisebox{1pt}{${}^{\,\ast}$}}{\,\prec\,}$. Here, height $\ell = 4$ and width $\omega=8$.
• Figure 5: Subsets of the set of ${Z^2}$ appearing in Algorithm \ref{['algo:fpt']}. First (on the left), ${Z^2}$ is partitioned into ${\color{domaincolor}D}\xspace$ and ${\color{domaincolor}R}\xspace$. Then (on the right), with some particular prescription given, ${\color{domaincolor}D}\xspace$ is grown to form ${$J@th$} {$${\color{domaincolor}D}$@th$} \hbox{$\overline{ }$} bar{}$ by including those pairs which are upstream of off elements and those pairs downstream of on ones. And ${\color{domaincolor}\underline{R}}\xspace$ is reduced by this same difference so that ${$J@th$} {$${\color{domaincolor}D}$@th$} \hbox{$\overline{ }$} bar{} \cup {\color{domaincolor}\underline{R}}\xspace = {Z^2}$.

Theorems & Definitions (29)

• Definition 1: filter setlabelrss
• Lemma 2: okane17concise
• Definition 3: extensions/compatibility
• Definition 4: determinism-enforcing zipper constraints
• Lemma 5: zhang20cover
• Definition 6: comparable neighborhoods
• Lemma 7: From ullah22structural
• proof
• Definition 8
• Definition 9