On Targeted Complexity of Discrete Motion

TL;DR

This paper introduces targeted simplicial complexity for discrete motion planning, demonstrating its properties, calculating it for specific complexes, and comparing it with relative topological complexity to enhance understanding of configuration spaces.

Contribution

It defines targeted simplicial complexity, proves its homotopy invariance, and compares it with existing topological complexity measures, providing new insights into discrete motion planning.

Findings

Targeted simplicial complexity is strongly homotopy invariant.

It varies between simplicial LS-categories of K and K×K.

In certain cases, it equals relative topological complexity.

Abstract

In this paper, we investigate discrete topological complexity $TC(K)$ introduced for situations where the configuration space possesses a simplicial structure. %Simplicial complexes are well-known and commonly used in programming for robotic motion. Let $K$ be a complex and let $L$ be a subcomplex considered as the target of the motion. We introduce targeted simplicial complexity $TC(K,L)$, which yields smaller values than the discrete version $TC(K)$. We then demonstrate that targeted simplicial complexity is strongly homotopy invariant and it varies between simplicial LS-categories of $K$ and $K \prod K$. Utilizing this information, we calculate targeted simplicial complexity for scenarios such as strongly collapsible complexes. Finally, we compare targeted simplicial complexity with relative topological complexity and we show that $TC(|K|, |L|) \le TC (K,L)$ where $|\cdot|$ denotes the geometric realization functor. Although relative topological complexity is generally lower than targeted simplicial complexity, they are equal in certain cases, such as arbitrary wedges of triangulated circles.