A graph environment must be explored by a collection of mobile robots. Some of the robots, a priori unknown, may turn out to be unreliable. The graph is weighted and each node is assigned a deadline. The exploration is successful if each node of the graph is visited before its deadline by a reliable robot. The edge weight corresponds to the time needed by a robot to traverse the edge. Given the number of robots which may crash, is it possible to design an algorithm, which will always guarantee the exploration, independently of the choice of the subset of unreliable robots by the adversary? We find the optimal time, during which the graph may be explored. Our approach permits to find the maximal number of robots, which may turn out to be unreliable, and the graph is still guaranteed to be explored. We concentrate on line graphs and rings, for which we give positive results. We start with the case of the collections involving only reliable robots. We give algorithms finding optimal times needed for exploration when the robots are assigned to fixed initial positions as well as when such starting positions may be determined by the algorithm. We extend our consideration to the case when some number of robots may be unreliable. Our most surprising result is that solving the line exploration problem with robots at given positions, which may involve crash-faulty ones, is NP-hard. The same problem has polynomial solutions for a ring and for the case when the initial robots' positions on the line are arbitrary. The exploration problem is shown to be NP-hard for star graphs, even when the team consists of only two reliable robots.
This paper deals with nding a lower and an upper bound for the minimum number of vehicles needed to serve all locations of the multiple traveling salesman problem with time windows. The two types of precedence graphs are introduced the start-time precedence graph and the end-time precedence graph. The bounds are generated by covering the precedence graph with minimum number of paths. Instances for which bounds are tight are presented, as well as instances for which bounds can be arbitrary bad. The closeness of such instances is discussed. Some satellites transmit a piece of information for a set duration, then proceed with another piece of information. A ground station receiving from several such satellites and wishing to avoid data loss faces a real-time scheduling problem. The pinwheel is a formalization of this problem. Given a multiset A of integers=(a sub 1 , a sub 2 , . . ., a sub n ), a successful schedule S is an infinite sequence over (1, 2, . . ., n) such that any subsequence of a sub i (1 > This paper resolves the complexity status of the well-known Traveling Repairman Problem on a line (Line-TRP) with general processing times at the request locations and deadline restrictions. It has long remained an open research question whether an exact solution procedure with pseudo-polynomial running time can be developed for this version of the Traveling Repairman Problem that was known to be at least binary NP-hard. The presented proof of strong NP-completeness of the problem is provided by a reduction from 3-PARTITION. Since recent literature provides significant new results for further variants of the Line-TRP and the Line-TSP, a brief updated overview of the complexity status of the different variants is given. Another major contribution is that a practically applicable exact best-first search Branch&Bound approach that optimally solves instances of real-world size in reasonable time is proposed. By applying sophisticated dominance rules as well as lower bounds, the number of enumerated partial solutions is kept limited. The efficiency of the new approach and the applied instruments is validated by a computational study.