Online Two-Dimensional Vector Packing with Advice

We consider the online two-dimensional vector packing problem, showing a lower bound of $11/5$ on the competitive ratio of any {\sc AnyFit} strategy for the problem. We provide strategies with competitive ratio $\max\!\left\{2,6\big/\big(1+3\tan(\pi/4-\gamma/2)\big)+\epsilon\right\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi/4-\gamma/2,\pi/4+\gamma/2]$, for $0\leq\gamma<\pi/3$ and $\max\left\{5/2,4\big/\big(1+2\tan(\pi/4-\gamma/2)\big)+\epsilon\right\}$ and logarithmic advice, for any instance where all the input vectors are restricted to have angles in the range $[\pi/4-\gamma/2,\pi/4+\gamma/2]$, for $0\leq\gamma\leq\pi/3$. In addition, we give a $5/2$-competitive strategy also using logarithmic advice for the unrestricted vectors case. These results should be contrasted to the currently best competitive strategy, FirstFit, having competitive ratio~$27/10$.Comment: 15 pages, 4 figures. This an extended version of an article published in "Algorithms and Complexity. CIAC 2021." Lecture Notes in Computer Science, vol 12701. Springer, https://doi.org/10.100

Approximation Algorithms for the Two-Watchman Route in a Simple Polygon

The two-watchman route problem is that of computing a pair of closed tours in an environment so that the two tours together see the whole environment and some length measure on the two tours is minimized. Two standard measures are: the minmax measure, where we want the tours where the longest of them has smallest length, and the minsum measure, where we want the tours for which the sum of their lengths is the smallest. It is known that computing a minmax two-watchman route is NP-hard for simple rectilinear polygons and thus also for simple polygons. Also, any c-approximation algorithm for the minmax two-watchman route is automatically a 2c-approximation algorithm for the minsum two-watchman route. We exhibit two constant factor approximation algorithms for computing minmax two-watchman routes in simple polygons with approximation factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively, where n is the number of vertices of the polygon. We also use the same techniques to obtain a 6.922-approximation for the fixed two-watchman route problem running in O(n^2) time, i.e., when two starting points of the two tours are given as input.Comment: 36 pages, 14 figure

Parallel searching on m rays☆☆This research is supported by the DFG-Project “Diskrete Probleme”, No. Ot 64/8-3.

AbstractWe investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio defined as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance.If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9—independent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if m⩾2.If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1+2(k+1)k+1/kk where k=⌈logm⌉ where log is used to denote the base-2 logarithm. We also give a strategy that obtains this ratio

Online Bin Covering with Exact Parameter Advice

We show an asymptotic 2/3-competitive strategy for the bin covering problem using O(b+log n) bits of advice, where b is the number of bits used to encode a rational value and n is the length of the input sequence.Comment: 10 pages, 2 figure, submitted to Informatic

Assessing energy security: An overview of commonly used methodologies

This paper provides an overview of methodologies used for quantitative evaluations of security of supply. The studied material is mainly based on peer-reviewed articles and the methodologies are classified according to which stage in the supply chain their main focus is directed to, as well as their scientific background. Our overview shows that a broad variety of approaches is used, but that there are still some important gaps, especially if the aim is to study energy security in a future-oriented way. First, there is a need to better understand how sources of insecurity can develop over time and how they are affected by the development of the energy system. Second, the current tendency to study the security of supply for each energy carrier separately needs to be complemented by comparisons of different energy carrier's supply chains. Finally, the mainly static perspective on system structure should be complemented with perspectives that to a greater extent take the systems' adaptive capacity and transformability into account, as factors with a potential to reduce the systems vulnerabilities. Furthermore, it may be beneficial to use methodological combinations, conduct more thorough sensitivity analysis and alter the mind-set from securing energy flows to securing energy services

Trajectory-Following and Off-Tracking Minimization of Long Combination Vehicles: A Comparison Between Nonlinear and Linear Model Predictive Control

In this paper, we compared the linear and nonlinear motion prediction models of a long combination vehicle (LCV). We designed a nonlinear model predictive control (NMPC) for trajectory-following and off-tracking minimisation of the LCV. The used prediction model allowed coupled longitudinal and lateral dynamics together with the possibility of a combined steering, propulsion and braking control of those vehicles in long prediction horizons and in all ranges of forward velocity. For LCVs where the vehicle model is highly nonlinear, we showed that the control actions calculated by a linear time-varying model predictive control (LTV-MPC) are relatively close to those obtained by the NMPC if the guess linearisation trajectory is sufficiently close to the nonlinear solution, in contrast to linearising for specific operating conditions that limit the generality of the designed function. We discussed how those guess trajectories can be obtained allowing off-line fixed time-varying model linearisation that is beneficial for real-time implementation of MPC in LCVs with long prediction horizons. The long prediction horizons are necessary for motion planning and trajectory-following of LCVs to maintain stability and tracking quality, e.g. by optimally reducing the speed prior to reaching a curve, and by generating control actions within the actuators limits

The Complexity of Guarding Monotone Polygons

Abstract A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations

Real-time Predictive Energy Management of Hybrid Electric Heavy Vehicles by Sequential Programming

With the objective of reducing fuel consumption, this paper presents real-time predictive energy management of hybrid electric heavy vehicles. We propose an optimal control strategy that determines the power split between different vehicle power sources and brakes. Based on the model predictive control (MPC) and sequential programming, the optimal trajectories of the vehicle velocity and battery state of charge are found for upcoming horizons with a length of 5-20 km. Then, acceleration and brake pedal positions together with the battery usage are regulated to follow the requested speed and state of charge that is verified using a vehicle plant model. The main contribution of this paper is the development of a sequential linear program for predictive energy management that is faster and simpler than sequential quadratic programming in tested solvers and gives trajectories that are very close to the best trajectories found by nonlinear programming. The performance of the method is also compared to two different sequential quadratic programs

Illuminating the x-Axis by ?-Floodlights

Given a set S of regions with piece-wise linear boundary and a positive angle α &lt; 90°, we consider the problem of computing the locations and orientations of the minimum number of α-floodlights positioned at points in S which suffice to illuminate the entire x-axis. We show that the problem can be solved in O(n log n) time and O(n) space, where n is the number of vertices of the set S

Three-dimensional topologically gauged N=6 ABJM type theories

In this paper we construct the $\mathcal N=6$ conformal supergravity in three dimensions from a set of Chern-Simons-like terms one for each of the graviton, gravitino, and R-symmetry gauge field and then couple this theory to the $\mathcal N=6$ superconformal ABJM theory. In a first step part of the coupled Lagrangian for this topologically gauged ABJM theory is derived by demanding that all terms of third and second order in covariant derivatives cancel in the supersymmtry variation of the Lagrangian. To achieve this the transformation rules of the two separate sectors must be augmented by new terms. In a second step we analyze all terms in $\delta L$ that are of first order in covariant derivatives. The cancelation of these terms require additional terms in the transformation rules as well as a number of new terms in the Lagrangian. As a final step we check that all remaining terms in $\delta L$ which are bilinear in fermions cancel which means that the presented Lagrangian and transformation rules constitute the complete answer. In particular we find in the last step new terms in the scalar potential containing either one or no structure constant. The non-derivative higher fermion terms in $\delta L$ that have not yet been completely analyzed are briefly discussed.Comment: 26 pages, v.2 minor corrections, comment on relation to chiral gravity added