A Linear Time Algorithm for the 33-neighbour Traveling Salesman Problem on Halin graphs and extensions

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamilton cycle in an edge-weighted graph, where costs are defined on all pairs of edges contained in the Hamilton cycle. This is a more general version than the commonly studied QTSP which only considers pairs of adjacent edges. We define a restricted version of QTSP, the $k$-neighbour TSP (TSP($k$)), and give a linear time algorithm to solve TSP($k$) on a Halin graph for $k\leq 3$. This algorithm can be extended to solve TSP($k$) on any fully reducible class of graphs for any fixed $k$ in polynomial time. This result generalizes corresponding results for the standard TSP. TSP($k$) can be used to model various machine scheduling problems as well as an optimal routing problem for unmanned aerial vehicles (UAVs)

A Survey of Motion Planning and Control Techniques for Self-driving Urban Vehicles

Self-driving vehicles are a maturing technology with the potential to reshape mobility by enhancing the safety, accessibility, efficiency, and convenience of automotive transportation. Safety-critical tasks that must be executed by a self-driving vehicle include planning of motions through a dynamic environment shared with other vehicles and pedestrians, and their robust executions via feedback control. The objective of this paper is to survey the current state of the art on planning and control algorithms with particular regard to the urban setting. A selection of proposed techniques is reviewed along with a discussion of their effectiveness. The surveyed approaches differ in the vehicle mobility model used, in assumptions on the structure of the environment, and in computational requirements. The side-by-side comparison presented in this survey helps to gain insight into the strengths and limitations of the reviewed approaches and assists with system level design choices

Improved Dynamic Graph Coloring

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within $n^{1-\epsilon}$ for any $\epsilon > 0$, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs. Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for $C$-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. In WADS'17, Barba et al. devised two complementary algorithms: For any $\beta > 0$ the first (respectively, second) maintains an $O(C \beta n^{1/\beta})$ (resp., $O(C \beta)$)-coloring while recoloring $O(\beta)$ (resp., $O(\beta n^{1/\beta})$) vertices per update. Our contribution is two-fold: - We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any $\beta > 0$, we get a $\tilde{O}(\frac{C}{\beta}\log^2 n)$-coloring with $O(\beta)$ recolorings per update, where the $\tilde{O}$ notation supresses polyloglog$(n)$ factors. In particular, for $\beta=O(1)$ we get constant recolorings with polylog$(n)$ colors; this is an exponential improvement over the previous bound. - For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest.Comment: Appeared in ESA 201

Egalitarian Graph Orientations

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic

Density decompositions of networks

We introduce a new topological descriptor of a network called the density decomposition which is a partition of the nodes of a network into regions of uniform density. The decomposition we define is unique in the sense that a given network has exactly one density decomposition. The number of nodes in each partition defines a density distribution which we find is measurably similar to the degree distribution of given real networks (social, internet, etc.) and measurably dissimilar in synthetic networks (preferential attachment, small world, etc.)

Incorporating Type II Error Probabilities from Independence Tests into Score-Based Learning of Bayesian Network Structure

We give a new consistent scoring function for structure learning of Bayesian networks. In contrast to traditional approaches to score-based structure learning, such as BDeu or MDL, the complexity penalty that we propose is data-dependent and is given by the probability that a conditional independence test correctly shows that an edge cannot exist. What really distinguishes this new scoring function from earlier work is that it has the property of becoming computationally easier to maximize as the amount of data increases. We prove a polynomial sample complexity result, showing that maximizing this score is guaranteed to correctly learn a structure with no false edges and a distribution close to the generating distribution, whenever there exists a Bayesian network which is a perfect map for the data generating distribution. Although the new score can be used with any search algorithm, in our related UAI 2013 paper [BS13], we have given empirical results showing that it is particularly effective when used together with a linear programming relaxation approach to Bayesian network structure learning. The present paper contains all details of the proofs of the finite-sample complexity results in [BS13] as well as detailed explanation of the computation of the certain error probabilities called beta-values, whose precomputation and tabulation is necessary for the implementation of the algorithm in [BS13].Comment: 118 pages, 13 Figure

Material Optimization in Transverse Electromagnetic Scattering Applications

A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration a model is established on the basis of an appropriately parametrized material tensor. The resulting nonlinear parametrization is treated on the level of the sub-problem, for which, globally optimal solutions can be computed due to the block separability of the model. Although global optimization of non-convex design problems is generally prohibitive, a well chosen combination of analytic solutions along with standard global optimization techniques leads to a very efficient algorithm for most relevant material parametrizations. A global convergence result for the overall algorithm is established. The effectiveness of the approach in terms of both computation time and solution quality is demonstrated by numerical examples, including the optimal design of cloaking layers for a nano-particle and the identification of multiple materials with different optical properties in a matrix

Distributed Discrete-time Optimization in Multi-agent Networks Using only Sign of Relative State

This paper proposes distributed discrete-time algorithms to cooperatively solve an additive cost optimization problem in multi-agent networks. The striking feature lies in the use of only the sign of relative state information between neighbors, which substantially differentiates our algorithms from others in the existing literature. We first interpret the proposed algorithms in terms of the penalty method in optimization theory and then perform non-asymptotic analysis to study convergence for static network graphs. Compared with the celebrated distributed subgradient algorithms, which however use the exact relative state information, the convergence speed is essentially not affected by the loss of information. We also study how introducing noise into the relative state information and randomly activated graphs affect the performance of our algorithms. Finally, we validate the theoretical results on a class of distributed quantile regression problems.Comment: Part of this work has been presented in American Control Conference (ACC) 2018, first version posted on arxiv on Sep. 2017, IEEE Transactions on Automatic Control, 201

Runtime Guarantees for Regression Problems

We study theoretical runtime guarantees for a class of optimization problems that occur in a wide variety of inference problems. these problems are motivated by the lasso framework and have applications in machine learning and computer vision. Our work shows a close connection between these problems and core questions in algorithmic graph theory. While this connection demonstrates the difficulties of obtaining runtime guarantees, it also suggests an approach of using techniques originally developed for graph algorithms. We then show that most of these problems can be formulated as a grouped least squares problem, and give efficient algorithms for this formulation. Our algorithms rely on routines for solving quadratic minimization problems, which in turn are equivalent to solving linear systems. Finally we present some experimental results on applying our approximation algorithm to image processing problems

Incorporating prior knowledge in medical image segmentation: a survey

Medical image segmentation, the task of partitioning an image into meaningful parts, is an important step toward automating medical image analysis and is at the crux of a variety of medical imaging applications, such as computer aided diagnosis, therapy planning and delivery, and computer aided interventions. However, the existence of noise, low contrast and objects' complexity in medical images are critical obstacles that stand in the way of achieving an ideal segmentation system. Incorporating prior knowledge into image segmentation algorithms has proven useful for obtaining more accurate and plausible results. This paper surveys the different types of prior knowledge that have been utilized in different segmentation frameworks. We focus our survey on optimization-based methods that incorporate prior information into their frameworks. We review and compare these methods in terms of the types of prior employed, the domain of formulation (continuous vs. discrete), and the optimization techniques (global vs. local). We also created an interactive online database of existing works and categorized them based on the type of prior knowledge they use. Our website is interactive so that researchers can contribute to keep the database up to date. We conclude the survey by discussing different aspects of designing an energy functional for image segmentation, open problems, and future perspectives.Comment: Survey paper, 30 page