Towards an Iterative Algorithm for the Optimal Boundary Coverage of a 3D Environment

This paper presents a new optimal algorithm for locating a set of sensors in 3D able to see the boundaries of a polyhedral environment. Our approach is iterative and is based on a lower bound on the sensors' number and on a restriction of the original problem requiring each face to be observed in its entirety by at least one sensor. The lower bound allows evaluating the quality of the solution obtained at each step, and halting the algorithm if the solution is satisfactory. The algorithm asymptotically converges to the optimal solution of the unrestricted problem if the faces are subdivided into smaller part

On the phase transitions of graph coloring and independent sets

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number

Network conduciveness with application to the graph-coloring and independent-set optimization transitions

We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another. We exemplify its use through an application to the two problems in combinatorial optimization that, given an undirected graph, ask that its so-called chromatic and independence numbers be found. Though NP-hard, when solved on sequences of expanding random graphs there appear marked transitions at which optimal solutions can be obtained substantially more easily than right before them. We demonstrate that these phenomena can be understood by resorting to the network that represents the solution space of the problems for each graph and examining its conduciveness between the non-optimal solutions and the optimal ones. At the said transitions, this network becomes strikingly more conducive in the direction of the optimal solutions than it was just before them, while at the same time becoming less conducive in the opposite direction. We believe that, besides becoming useful also in other areas in which network theory has a role to play, network conduciveness may become instrumental in helping clarify further issues related to NP-hardness that remain poorly understood

On Maximum Weight Clique Algorithms, and How They Are Evaluated

Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments

Comparison of Multiple Spacecraft Configuration Designs for Coordinated Flight Missions

Coordinated flight allows the replacement of a single monolithic spacecraft with multiple smaller ones, based on the idea of distributed systems. According to the mission objectives and in order to ensure a safe relative motion, constraints on the relative distances need to be satisfied. At first a proper orbit design can limit the differential perturbations, then through corrective maneuvers their induced differential drifts can be canceled. In this work several designs are surveyed, defining the initial configuration of a group of spacecraft while counteracting the differential perturbations. For each of the investigated designs the focus is on the number of deployable spacecraft and on the possibility to ensure safe relative motion through station keeping of the initial configuration, with particular attention to the required $\Delta V$ budget and the constraints violations