Recent years have witnessed significant interest in convex relaxations of the power flows, with several papers showing that the second-order cone relaxation is tight for tree networks under various conditions on loads or voltages. This paper shows that ac-feasibility, i.e., to find whether some generator dispatch can satisfy a given demand, is NP-hard for tree networks.

Pathfinding in uniform-cost grid environments is a problem commonly found in application areas such as robotics and video games. The state-of-the-art is dominated by hierarchical pathfinding algorithms which are fast and have small memory overheads but usually return suboptimal paths. In this paper we present a novel search strategy, specific to grids, which is fast, optimal and requires no memory overhead. Our algorithm can be described as a macro operator which identifies and selectively expands only certain nodes in a grid map which we call jump points. Intermediate nodes on a path connecting two jump points are never expanded. We prove that this approach always computes optimal solutions and then undertake a thorough empirical analysis, comparing our method with related works from the literature. We find that searching with jump points can speed up A* by an order of magnitude and more and report significant improvement over the current state of the art.

Any-angle pathfinding is a fundamental problem in robotics and computer games. The goal is to find a shortest path between a pair of points on a grid map such that the path is not artificially constrained to the points of the grid. Prior research has focused on approximate online solutions. A number of exact methods exist but they all require super-linear space and pre-processing time. In this study, we describe Anya: a new and optimal any-angle pathfinding algorithm. Where other works find approximate any-angle paths by searching over individual points from the grid, Anya finds optimal paths by searching over sets of states represented as intervals. Each interval is identified on-thefly. From each interval Anya selects a single representative point that it uses to compute an admissible cost estimate for the entire set. Anya always returns an optimal path if one exists. Moreover it does so without any offline pre-processing or the introduction of additional memory overheads. In a range of empirical comparisons we show that Anya is competitive with several recent (sub-optimal) online and pre-processing based techniques and is up to an order of magnitude faster than the most common benchmark algorithm, a grid-based implementation of A*.