On Targeted Complexity of Discrete Motion

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.