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We study the computational power that oblivious robots operating in the plane have under sequential schedulers. We show that this power is much stronger than the obvious capacity these schedulers offer of breaking symmetry, and thus to create a leader. In fact, we prove that, under any sequential scheduler, the robots are capable of solving problems that are unsolvable even with a leader under the fully synchronous scheduler FSYNC. More precisely, we consider the class of pattern formation problems, and focus on the most general problem in this class, Universal Pattern Formation (UPF), which requires the robots to form every pattern given in input, starting from any initial configurations (where some robots may occupy the same point). We first show that UPF is unsolvable under FSYNC, even if the robots are endowed with additional strong capabilities (multiplicity detection, rigid movement, agreement on coordinate systems, presence of a unique leader). On the other hand, we prove that, except for point formation (Gathering), UPF is solvable under any sequential scheduler without any additional assumptions. We then turn our attention to the Gathering problem, and prove that weak multiplicity detection is necessary and sufficient for solvability under sequential schedulers. The obtained results show that the computational power of the robots under FSYNC (where Gathering is solvable without any multiplicity detection) and that under sequential schedulers are orthogonal.

Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies., Comment: 19 pages, 8 figures

The \textsc{Mutual Visibility} is a well-known problem in the context of mobile robots. For a set of $n$ robots disposed in the Euclidean plane, it asks for moving the robots without collisions so as to achieve a placement ensuring that no three robots are collinear. For robots moving on graphs, we consider the \textsc{Geodesic Mutual Visibility} ($\GMV$) problem. Robots move along the edges of the graph, without collisions, so as to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means that there is a shortest path (i.e., a "geodesic") between each pair of robots along which no other robots reside. We study this problem in the context of finite and infinite square grids, for robots operating under the standard Look-Compute-Move model. In both scenarios, we provide resolution algorithms along with formal correctness proofs, highlighting the most relevant peculiarities arising within the different contexts, while optimizing the time complexity., Comment: 28 pages, 9 figures. To appear in the proceedings of the 25th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2023), Institute for Future Technologies, Jersey City, New Jersey, USA, October 2-4, 2023

Programmable Matter (PM) has been widely investigated in recent years. It refers to some kind of matter with the ability to change its physical properties (e.g., shape or color) in a programmable way. One reference model is certainly Amoebot, with its recent canonical version (DISC 2021). Along this line, with the aim of simplification and to better address concurrency, the SILBOT model has been introduced (AAMAS 2020), which heavily reduces the available capabilities of the particles composing the PM. In SILBOT, in fact, particles are asynchronous, without any direct means of communication (silent) and without memory of past events (oblivious). Within SILBOT, we consider the Line Formation primitive in which particles are required to end up in a configuration where they are all aligned and connected. We propose a simple and elegant distributed algorithm - optimal in terms of number of movements, along with its correctness proof., Comment: The paper appears in the Proceedings of the 25th International Symposium 19 on Stabilization, Safety, and Security of Distributed Systems (SSS), 2023. A brief announcement appears in the proceedings of the 37th International Symposium on Distributed Computing (DISC) 2023

Among fundamental problems in the context of distributed computing by autonomous mobile entities, one of the most representative and well studied is {\sc Point Convergence}: given an arbitrary initial configuration of identical entities, disposed in the Euclidean plane, move in such a way that, for all $\eps>0$, a configuration in which the separation between all entities is at most $\eps$ is eventually reached and maintained. The problem has been previously studied in a variety of settings, including full visibility, exact measurements (like distances and angles), and synchronous activation of entities. Our study concerns the minimal assumptions under which entities, moving asynchronously with limited and unknown visibility range and subject to limited imprecision in measurements, can be guaranteed to converge in this way. We present an algorithm that solves {\sc Point Convergence}, for entities in the plane, in such a setting, provided the degree of asynchrony is bounded: while any one entity is active, any other entity can be activated at most $k$ times, for some arbitrarily large but fixed $k$. This provides a strong positive answer to a decade old open question posed by Katreniak. We also prove that in a comparable setting that permits unbounded asynchrony, {\sc Point Convergence} in the plane is impossible, contingent on the natural assumption that algorithms maintain the (visible) connectivity among entities present in the initial configuration. This variant, that we call {\sc Cohesive Convergence}, serves to distinguish the power of bounded and unbounded asynchrony in the control of autonomous mobile entities, settling at the same time a long-standing question whether in the Euclidean plane synchronously scheduled entities are more powerful than asynchronously scheduled entities.

In this paper, we study the Orienteering Aisle-graphs Single-access Problem (OASP), a variant of the orienteering problem for a robot moving in a so-called single-access aisle-graph, i.e., a graph consisting of a set of rows that can be accessed from one side only. Aisle-graphs model, among others, vineyards or warehouses. Each aisle-graph vertex is associated with a reward that a robot obtains when visits the vertex itself. As the robot's energy is limited, only a subset of vertices can be visited with a fully charged battery. The objective is to maximize the total reward collected by the robot with a battery charge. We first propose an optimal algorithm that solves OASP in O(m^2 n^2) time for aisle-graphs with a single access consisting of m rows, each with n vertices. With the goal of designing faster solutions, we propose four greedy sub-optimal algorithms that run in at most O(mn (m+n)) time. For two of them, we guarantee an approximation ratio of 1/2(1-1/e), where e is the base of the natural logarithm, on the total reward by exploiting the well-known submodularity property. Experimentally, we show that these algorithms collect more than 80% of the optimal reward., Comment: re-submitted revised version to IEEE Transactions on Robotics (T-RO) after a conditionally accepted response

Given a set R of robots, each one located at different vertices of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that |R|=|F|, APF asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections. So far, as possible graph discretizing the Euclidean plane only the standard square grid has been considered in the context of the classical Look-Compute-Move model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph.

Following the wide investigation in distributed computing issues by mobile entities of the last two decades, we consider the need of a structured methodology to tackle the arisen problems. The aim is to simplify both the design of the resolution algorithms and the writing of the required correctness proofs. We would encourage the usage of a common framework in order to help both algorithm designer and reviewers in the intricate work of analyzing the proposed resolution strategies. In order to better understand the potentials of our methodology, we consider the Pattern Formation (PF) problem approached in [Fujinaga et al. SIAM J. Comput., 2015] as case study. Since the proposed resolution algorithm has turned out to be inaccurate and also of difficult fixing, we design a new algorithm guided by the proposed methodology, hence fully characterizing the problem.

In this paper, we consider the Constant-cost Orienteering Problem (COP) where a robot, constrained by a limited travel budget, aims at selecting a path with the largest reward in an aisle-graph. The aisle-graph consists of a set of loosely connected rows where the robot can change lane only at either end, but not in the middle. Even when considering this special type of graphs, the orienteering problem is known to be NP-hard. We optimally solve in polynomial time two special cases, COP-FR where the robot can only traverse full rows, and COP-SC where the robot can access the rows only from one side. To solve the general COP, we then apply our special case algorithms as well as a new heuristic that suitably combines them. Despite its light computational complexity and being confined into a very limited class of paths, the optimal solutions for COP-FR turn out to be competitive even for COP in both real and synthetic scenarios. Furthermore, our new heuristic for the general case outperforms state-of-art algorithms, especially for input with highly unbalanced rewards.

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ is a simple $3$-plane graph, while it can be solved in linear time, for any $h$, when $G$ is $1$-plane., Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require $\Omega(n^\frac{2}{3})$ edges each having $\Omega(n^\frac{1}{3})$ bends in the worst case. The lower bound holds even when the function that maps vertices to points is not a bijection but it is defined by a 3-coloring. In contrast, a constant number of bends per edge can be obtained for 3-colored paths and for 3-colored caterpillars whose leaves all have the same color. Such results answer to a long standing open problem., Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017)

Given any multiset F of points in the Euclidean plane and a set R of robots such that |R|=|F|, the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard LCM cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left-right orientation. We are interested in removing such a restriction. While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the validity as well as the relevance of strategies proposed in the literature. Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results. In fact, it comes out that well-established results like [Fujinaga et al, SIAM J. Comp. 44(3) 2015], more recent approaches like [Bramas et al, PODC and SSS 2016] and 'unofficial' results like [Dieudonne et al, arXiv:0902.2851] revealed to be not correct., Comment: To appear in Distributed Computing