Nash Equilibria in Reverse Temporal Voronoi Games

We study Voronoi games on temporal graphs as introduced by Boehmer et al. (IJCAI 2021) where two players each select a vertex in a temporal graph with the goal of reaching the other vertices earlier than the other player. In this work, we consider the reverse temporal Voronoi game, that is, a player wants to maximize the number of vertices reaching her earlier than the other player. Since temporal distances in temporal graphs are not symmetric in general, this yields a different game. We investigate the difference between the two games with respect to the existence of Nash equilibria in various temporal graph classes including temporal trees, cycles, grids, cliques and split graphs. Our extensive results show that the two games indeed behave quite differently depending on the considered temporal graph class

Training Neural Networks is NP-Hard in Fixed Dimension

We study the parameterized complexity of training two-layer neural networks with respect to the dimension of the input data and the number of hidden neurons, considering ReLU and linear threshold activation functions. Albeit the computational complexity of these problems has been studied numerous times in recent years, several questions are still open. We answer questions by Arora et al. [ICLR '18] and Khalife and Basu [IPCO '22] showing that both problems are NP-hard for two dimensions, which excludes any polynomial-time algorithm for constant dimension. We also answer a question by Froese et al. [JAIR '22] proving W[1]-hardness for four ReLUs (or two linear threshold neurons) with zero training error. Finally, in the ReLU case, we show fixed-parameter tractability for the combined parameter number of dimensions and number of ReLUs if the network is assumed to compute a convex map. Our results settle the complexity status regarding these parameters almost completely.Comment: Paper accepted at NeurIPS 202