We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a 1 2-approximation algorithm based on local improvements and show that the problem is APXhard. For the minimization version, we show that it is not approximable within n 1−ǫ for any fixed ǫ > 0. When every color appears in the graph at most r times and r is an increasing function of n, the problem is shown not to be approximable within factor O(r 1−ǫ). For fixed constant r we analyze a polynomial-time (r + H r)/2-approximation algorithm, where H r is the r-th harmonic number, and prove APX-hardness for r = 2. For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances.

In this paper we consider the problem of optimizing the ecological connectivity of a landscape under a budget constraint by improving habitat areas and ecological corridors between them. We consider a formulation of this problem in terms of graphs in which vertices represent the habitat areas and arcs represent a probability of connection between two areas that depend on the quality of the respective corridor. We propose a new generalized flow model that improves existing models for this problem and an efficient preprocessing algorithm that reduces the size of the graphs on which generalized flows is computed.Reported numerical experiments highlight the benefice of these two contributions on computation times and show that larger problems can be solved using them. Our experiments also show that several variants of greedy algorithms perform relatively well on practical instances while they return arbitrary bad solutions in the worst case.

The automatic detection of vessel behaviours from Automatic Identification System (AIS) data is a challenging aspect of designing intelligent systems and aiding maritime situational awareness. The development of such systems remains limited to some activities like fishing, and by geographical biases that prevent systems to generalise to other areas than that used for training.To contribute to these questions, we investigate how to treat raw data or engineered features so that they do not convey such biases at training time and we propose methods for point-wise behaviour detection in the context of container vessels with four target behaviours. Several systems are studied, with raw data or engineered features as inputs, followed by shallow or deep learning classifiers. While good performances are obtained by several of them, we observe that a decision tree classifier with engineered features outperforms an LSTM in areas where no labelled data is available for training.

a b s t r a c tGiven an undirected graph on n vertices with weights on its edges, Min WCF (p) consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices. It has been proved that Min WCF (p) is NP-hard for any p ≥ 4 (Imielinska et al. (1993) [10]) but (2 − 1 n )-approximable (Goemans and Williamson (1995) [9]). While Min WCF(2) is polynomial-time solvable, already the unweighted version of Min WCF (3) is NP-hard even on planar bipartite graphs of maximum degree 3. We prove here that for any p ≥ 4, the unweighted version is NP-hard, even for planar bipartite graphs of maximum degree 3; moreover, the unweighted version for any p ≥ 3 has no ptas for bipartite graphs of maximum degree 3. The latter theorem is the first-ever APX-hardness result on this problem. On the other hand, we show that Min WCF (p) is polynomial-time solvable on graphs with bounded treewidth, and for any p bounded by O( log n log log n ) it has a ptas on planar graphs.