Labeled Traveling Salesman Problems: Complexity and approximation

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 APX-hard. For the minimization version, we show that it is not approximable within $n^{1-\epsilon}$ for any fixed $\epsilon>0$. 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 $O(r^{1-\epsilon})$. 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

On the complexity of color-avoiding site and bond percolation

The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation. In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.Comment: 14 page

Trophic transfer of copper decreases the condition index in Crassostrea gigas spat in concomitance with a change in the microalgal fatty acid profile and enhanced oyster energy demand

Due to new usages and sources, copper (Cu) concentrations are increasing in the Arcachon Basin, an important shellfish production area in France. In the present paper, the trophic transfer of Cu was studied between a microalga, Tetraselmis suecica, and Crassostrea gigas (Pacific oyster) spat. An experimental approach was developed to assess Cu exposure, transfer and toxicity on both phytoplankton and spat. Exposure of microalgal cultures to Cu for 7–8 days (3.1 ± 0.1, 15.7 ± 0.2 and 50.4 ± 1.0 μg Cu·L−1 for the control, Cu15 and Cu50 conditions, respectively) led to concentrations in microalgae (28.3 ± 0.9 and 110.7 ± 11.9 mg Cu·kg dry weight−1 for Cu15 and Cu50, respectively) close to those measured in the field. Despite Cu accumulation, the physiology of the microalgae remained poorly affected. Exposed cultures could only be discriminated from controls by a higher relative content in intracellular reactive oxygen species, and a lower relative content in lipids together with a reduced metabolic activity. By contrast, the fatty acid profile of microalgae was modified, with a particularly relevant lower content of the essential polyunsaturated fatty acid 22:6n-3 (docosahexaenoic acid [DHA]). Following 21 days of spat feeding with Cu15 and Cu50 microalgal cultures, trophic transfer of Cu was observed with a high initial Cu concentration in spat tissues. No effect was observed on oxidative stress endpoints. Cu exposure was responsible for a decrease in the spat condition index, an outcome that could be related to an insufficient DHA supply and extra energy demand as suggested by the overexpression of genes involved in energy metabolism, ATP synthesis and glycogen catabolism. © 202

Strategic Multiway Cut and Multicut Games

We consider cut games where players want to cut themselves off from different parts of a network. These games arise when players want to secure themselves from areas of potential infection. For the game-theoretic version of Multiway Cut, we prove that the price of stability is 1, i.e., there exists a Nash equilibrium as good as the centralized optimum. For the game-theoretic version of Multicut, we show that there exists a 2-approximate equilibrium as good as the centralized optimum. We also give poly-time algorithms for finding exact and approximate equilibria in these games. 1