Topological Price of Anarchy bounds for clustering games on networks

We consider clustering games in which the players are embedded in a network and want to coordinate (or anti-coordinate) their choices with their neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding (worst-case) Price of Anarchy bounds. As one of our main results, we derive (tight) bounds on the Price of Anarchy for clustering games on Erdős-Rényi random graphs, which, depending on the graph density, stand in stark contrast to the known Price of Anarchy bounds

Parameterized complexity of the MINCCA problem on graphs of bounded decomposability

In an edge-colored graph, the cost incurred at a vertex on a path when two incident edges with different colors are traversed is called reload or changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem consists in finding an arborescence with a given root vertex such that the total changeover cost of the internal vertices is minimized. It has been recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when parameterized by the treewidth and the maximum degree of the input graph. In this article we present the following results for the MINCCA problem: - the problem is W[1]-hard parameterized by the treedepth of the input graph, even on graphs of average degree at most 8. In particular, it is W[1]-hard parameterized by the treewidth of the input graph, which answers the main open problem of G\"oz\"upek et al. [TCS 2016]; - it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the input multigraph; - it is FPT parameterized by the star tree-cutwidth of the input graph, which is a slightly restricted version of tree-cutwidth. This result strictly generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016]; - it remains NP-hard on planar graphs even when restricted to instances with at most 6 colors and 0/1 symmetric costs, or when restricted to instances with at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure

Stellar spectral-type (mass) dependence of the dearth of close-in planets around fast-rotating stars. Architecture of Kepler confirmed single-exoplanet systems compared to star-planet evolution models

In 2013 a dearth of close-in planets around fast-rotating host stars was found using statistical tests on Kepler data. The addition of more Kepler and Transiting Exoplanet Survey Satellite (TESS) systems in 2022 filled this region of the diagram of stellar rotation period (Prot) versus the planet orbital period (Porb). We revisited the Prot extraction of Kepler planet-host stars, we classify the stars by their spectral type, and we studied their Prot-Porb relations. We only used confirmed exoplanet systems to minimize biases. In order to learn about the physical processes at work, we used the star-planet evolution code ESPEM (French acronym for Evolution of Planetary Systems and Magnetism) to compute a realistic population synthesis of exoplanet systems and compared them with observations. Because ESPEM works with a single planet orbiting around a single main-sequence star, we limit our study to this population of Kepler observed systems filtering out binaries, evolved stars, and multi-planets. We find in both, observations and simulations, the existence of a dearth in close-in planets orbiting around fast-rotating stars, with a dependence on the stellar spectral type (F, G, and K), which is a proxy of the mass in our sample of stars. There is a change in the edge of the dearth as a function of the spectral type (and mass). It moves towards shorter Prot as temperature (and mass) increases, making the dearth look smaller. Realistic formation hypotheses included in the model and the proper treatment of tidal and magnetic migration are enough to qualitatively explain the dearth of hot planets around fast-rotating stars and the uncovered trend with spectral type.Comment: Accepted in A&A. 13 pages, 8 figure

Scheduling Games with Machine-Dependent Priority Lists

We consider a scheduling game in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We characterize four classes of instances in which a pure Nash equilibrium (NE) is guaranteed to exist, and show, by means of an example, that none of these characterizations can be relaxed. We then bound the performance of Nash equilibria for each of these classes with respect to the makespan of the schedule and the sum of completion times. We also analyze the computational complexity of several problems arising in this model. For instance, we prove that it is NP-hard to decide whether a NE exists, and that even for instances with identical machines, for which a NE is guaranteed to exist, it is NP-hard to approximate the best NE within a factor of $2-\frac{1}{m}-\epsilon$ for all $\epsilon>0$. In addition, we study a generalized model in which players' strategies are subsets of resources, each having its own priority list over the players. We show that in this general model, even unweighted symmetric games may not have a pure NE, and we bound the price of anarchy with respect to the total players' costs.Comment: 19 pages, 2 figure

The k-Color Shortest Path Problem

This paper proposes a mathematical model and an exact algorithm for a novel problem, the k-Color Shortest Path Problem. This problem is defined on a edge-colored weighted graph, and its aim is to find a shortest path that uses at most k different edge-colors. The main support and motivation for this problem arise in the field of transmission networks design, where two crucial matters, reliability and cost, can be addressed using both colors and arc distances in the solution of a constrained shortest path problem. In this work, we describe a first mathematical formulation of the problem of interest and present an exact solution approach based on a branch and bound technique

Anti-coordination Games and Stable Graph Colorings

Abstract. Motivated by understanding non-strict and strict pure strat-egy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We charac-terize the cases when such colorings exist and when the decision problem is NP-hard. These correspond to finding pure strategy equilibria in the anti-coordination games, whose price of anarchy we also analyze. We further consider the directed case, a generalization that captures both coordination and anti-coordination. We prove the decision problem for non-strict equilibria in directed graphs is NP-hard. Our notions also have multiple connections to other combinatorial questions, and our results re-solve some open problems in these areas, most notably the complexity of the strictly unfriendly partition problem.