The notion of effective topological complexity, introduced by Błaszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article, we focus on studying several properties of this notion of topological complexity. We introduce a notion of effective LS category which mimics the behaviour the usual LS category has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect to the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal.

We show that well-known invariants like Lusternik–Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

In robotics, a topological theory of motion planning was initiated by M. Farber. We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multi-tasking version of the algorithms.

This work is motivated by the question of whether there are spaces X for which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space ${\open R}\hbox{P}^m$, it is known that $\hbox{TC}^S ({\open R}\hbox{P}^{m})$ captures, with a few potential exceptional cases, the Euclidean embedding dimension of ${\open R}\hbox{P}^{m}$. We now show that, for all m≥1, $\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$ is characterized as the smallest positive integer n for which there is a symmetric ${\open Z}_{2}$-biequivariant map Sm×Sm→Sn with a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of ${\open R}\hbox{P}^{2^{e}}$ for e≥1. In particular, this leaves the torus S1×S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) invariant is currently unknown.

We study a partial differential equation on a bounded domain $\Omega\subset\mathbb{R}^N$ with a $p(x)$-growth condition in the divergence operator and we establish the existence of at least two nontrivial weak solutions in the generalized Sobolev space $W_0^{1,p(x)}(\Omega)$. Such equations have been derived as models of several physical phenomena. Our proofs rely essentially on critical point theory combined with corresponding variational techniques.

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable Xn,q: the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for Xn,q. If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.