Collaborative delivery with energy-constrained mobile robots

The basic primitive for a mobile agent is the ability to visit all the nodes of the graph in a systematic manner. This chapter considers the exploration of unknown graphs in full detail, for the specific mobile agent model considered in this book. The graph is considered to be finite, undirected and connected. Other than this fact, no prior knowledge of the graph is assumed. Several exploration techniques are introduced and explained for either a single agent, or multiple agents, exploring either labelled or unlabelled graphs. We focus on the efficiency of exploration and consider three different complexity measures, the time taken, the amount of memory used by the agents and the storage needed at each node of the graph. For exploration by multiple agents, we consider collaborative exploration by a team of colocated agents as well as distributed exploration by agents scattered in a graph. The concluding section presents some brief ideas and references on more advanced topics on graph exploration that are not covered in this chapter.

/pdf/graph-explorations-with-mobile-agents-1ounf00q6s.pdf

Graph Explorations with Mobile Agents

We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n-vertex graph using O(log log n) pebbles, even when n is not known a priori. The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O(log log n) agents with constant memory can explore any n-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n) when we allow each agent to have at most O((log n)1-e) bits of memory for any e > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n) pebbles to explore any n-vertex graph.

Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents

We study the problem of exploration of a tree by mobile agents (robots) that have limited energy. The energy constraint bounds the number of edges that can be traversed by a single agent. We use a team of agents to collectively explore the tree and the objective is to minimize the size of this team. The agents start at a single node, the designated root of the tree and the height of the tree is assumed to be less than the energy bound B of the agents. The agents have local vision and communication capabilities; two agents can exchange information only when they are collocated at the same node. We provide an exploration algorithm for visiting all nodes of the unknown tree and we compare our algorithm with the optimal offline algorithm that has complete knowledge of the tree. Our algorithm has a competitive ratio of O(log B), independent of the number of nodes in the tree. We also show that this is the best possible competitive ratio for exploration of unknown trees.

Collaborative Exploration of Trees by Energy-Constrained Mobile Robots

We consider the problem of collectively delivering some message from a specified source to a designated target location in a graph, using multiple mobile agents. Each agent has a limited energy which constrains the distance it can move. Hence multiple agents need to collaborate to move the message, each agent handing over the message to the next agent to carry it forward. Given the positions of the agents in the graph and their respective budgets, the problem of finding a feasible movement schedule for the agents can be challenging. We consider two variants of the problem: in non-returning delivery, the agents can stop anywhere; whereas in returning delivery, each agent needs to return to its starting location, a variant which has not been studied before. We first provide a polynomial-time algorithm for returning delivery on trees, which is in contrast to the known (weak) $\mathrm {NP}$-hardness of the non-returning version. In addition, we give resource-augmented algorithms for returning delivery in general graphs. Finally, we give tight lower bounds on the required resource augmentation for both variants of the problem. In this sense, our results close the gap left by previous research.

Collaborative Delivery with Energy-Constrained Mobile Robots

We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm, thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04.Additionally, we consider the online Dial-A-Ride problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41.Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O(n2) for closed offline TSP on the line with release dates and show that both variants of offline Dial-A-Ride on the line are NP-hard for any capacity c ≥ 2 of the server.

https://hal.inria.fr/hal-01549685/document

Tight bounds for online TSP on the line

We consider the fundamental problem of exploring an undirected and initially unknown graph by an agent with little memory. The vertices of the graph are unlabeled, and the edges incident to a vertex have locally distinct labels. In this setting, it is known that Θ(log n) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We show that this memory requirement can be decreased significantly by making a part of the memory distributable in the form of pebbles. A pebble is a device that can be dropped to mark a vertex and can be collected when the agent returns to the vertex. We show that for an agent O(log log n) distinguishable pebbles and bits of memory are sufficient to explore any bounded-degree graph with at most n vertices. We match this result with a lower bound exhibiting that for any agent with sub-logarithmic memory, Ω(log log n) distinguishable pebbles are necessary for exploration.

Undirected graph exploration with Θ(log log n) pebbles

We consider the problem of delivering m messages between specified source-target pairs in an undirected graph, by k mobile agents initially located at distinct nodes of the graph. Each edge has a designated length and each agent consumes energy proportional to the distance it travels in the graph. We are interested in optimizing the total energy consumption for the team of agents. Unlike previous related work, we consider heterogeneous agents with different rates of energy consumption (weights w_i). To solve the delivery problem, agents face three major challenges: Collaboration (how to work together on each message), Planning (which route to take) and Coordination (how to assign agents to messages).

We first show that the delivery problem can be 2-approximated without collaborating and that this is best possible, i.e., we show that the benefit of collaboration is 2 in general. We also show that the benefit of collaboration for a single message is 1 / log 2 which is approximately 1.44. Planning turns out to be NP-hard to approximate even for a single agent, but can be 2-approximated in polynomial time if agents have unit capacities and do not collaborate. We further show that coordination is NP-hard even for agents with unit capacity, but can be efficiently solved exactly if they additionally have uniform weights. Finally, we give a polynomial-time c-approximation for message delivery with unit capacities.

https://drops.dagstuhl.de/opus/volltexte/2017/7023/pdf/LIPIcs-STACS-2017-10.pdf/

Energy-Efficient Delivery by Heterogeneous Mobile Agents

We consider the problem of delivering $m$ messages between specified source-target pairs in a weighted undirected graph, by $k$ mobile agents initially located at distinct nodes of the graph. Each agent consumes energy proportional to the distance it travels in the graph and we are interested in optimizing the total energy consumption for the team of agents. Unlike previous related work, we consider heterogeneous agents with different rates of energy consumption (weights~$w_i$). To solve the delivery problem, agents face three major challenges: \emph{Collaboration} (how to work together on each message), \emph{Planning} (which route to take) and \emph{Coordination} (how to assign agents to messages). 
We first show that the delivery problem can be 2-approximated \emph{without} collaborating and that this is best possible, i.e., we show that the \emph{benefit of collaboration} is 2 in general. We also show that the benefit of collaboration for a single message is~$1/\ln 2 \approx 1.44$. Planning turns out to be \NP-hard to approximate even for a single agent, but can be 2-approximated in polynomial time if agents have unit capacities and do not collaborate. We further show that coordination is \NP-hard even for agents with unit capacity, but can be efficiently solved exactly if they have uniform weights. Finally, we give a polynomial-time $(4\max\tfrac{w_i}{w_j})$-approximation for message delivery with unit capacities.

Jan Hackfeld

Papers

Tight bounds for online TSP on the line

Undirected graph exploration with Θ(log log n) pebbles

Energy-Efficient Delivery by Heterogeneous Mobile Agents

Tight Bounds for Undirected Graph Exploration with Pebbles and Multiple Agents

Energy-efficient Delivery by Heterogeneous Mobile Agents