Revisiting the Minimum Constraint Removal Problem in Mobile Robotics

The minimum constraint removal problem seeks to find the minimum number of constraints, i.e., obstacles, that need to be removed to connect a start to a goal location with a collision-free path. This problem is NP-hard and has been studied in robotics, wireless sensing, and computational geometry. This work contributes to the existing literature by presenting and discussing two results. The first result shows that the minimum constraint removal is NP-hard for simply connected obstacles where each obstacle intersects a constant number of other obstacles. The second result demonstrates that for $n$ simply connected obstacles in the plane, instances of the minimum constraint removal problem with minimum removable obstacles lower than $(n+1)/3$ can be solved in polynomial time. This result is also empirically validated using several instances of randomly sampled axis-parallel rectangles.Comment: Accepted for presentation at the 18th international conference on Intelligent Autonomous System 202

How to Navigate Through Obstacles?

Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t in V(G), and k in N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph. We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by k. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem. We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex v, there exists a set of paths whose cardinality is upper bounded by a function of k, that "represents" the valid s-t paths containing subsets of colors from v. We employ these structural results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions

Improved Approximation Bounds for the Minimum Constraint Removal Problem

In the minimum constraint removal problem, we are given a set of geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable, and (perhaps surprisingly) no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is a new approximation technique that gives O(sqrt{n})-approximation for rectangles, disks as well as rectilinear polygons. The technique also gives O(sqrt{n})-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem

On the complexity of barrier resilience for fat regions and bounded ply

In the barrier resilience problem (introduced by Kumar et al., Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, we show that the problem is NP-hard when the collection only contains fat regions with bounded ply ¿ (even when they are axis-aligned rectangles of aspect ratio ). We also show that the problem is fixed-parameter tractable (FPT) for unit disks and for similarly-sized ß-fat regions with bounded ply ¿ and pairwise boundary intersections. We then use our FPT algorithm to construct an -approximation algorithm that runs in time, where .Peer ReviewedPostprint (author's final draft

Implications of Motion Planning: Optimality and k-survivability

We study motion planning problems, finding trajectories that connect two configurations of a system, from two different perspectives: optimality and survivability. For the problem of finding optimal trajectories, we provide a model in which the existence of optimal trajectories is guaranteed, and design an algorithm to find approximately optimal trajectories for a kinematic planar robot within this model. We also design an algorithm to build data structures to represent the configuration space, supporting optimal trajectory queries for any given pair of configurations in an obstructed environment. We are also interested in planning paths for expendable robots moving in a threat environment. Since robots are expendable, our goal is to ensure a certain number of robots reaching the goal. We consider a new motion planning problem, maximum k-survivability: given two points in a stochastic threat environment, find n paths connecting two given points while maximizing the probability that at least k paths reach the goal. Intuitively, a good solution should be diverse to avoid several paths being blocked simultaneously, and paths should be short so that robots can quickly pass through dangerous areas. Finding sets of paths with maximum k-survivability is NP-hard. We design two algorithms: an algorithm that is guaranteed to find an optimal list of paths, and a set of heuristic methods that finds paths with high k-survivability

Computational and Near-Optimal Trade-Offs in Renewable Electricity System Modelling

In the decades to come, the European electricity system must undergo an unprecedented transformation to avert the devastating impacts of climate change. To devise various possibilities for achieving a sustainable yet cost-efficient system, in the thesis at hand, we solve large optimisation problems that coordinate the siting of generation, storage and transmission capacities. Thereby, it is critical to capture the weather-dependent variability of wind and solar power as well as transmission bottlenecks. In addition to modelling at high spatial and temporal resolution, this requires a detailed representation of the electricity grid. However, since the resulting computational challenges limit what can be investigated, compromises on model accuracy must be made, and methods from informatics become increasingly relevant to formulate models efficiently and to compute many scenarios. The first part of the thesis is concerned with justifying such trade-offs between model detail and solving times. The main research question is how to circumvent some of the challenging non-convexities introduced by transmission network representations in joint capacity expansion models while still capturing the core grid physics. We first examine tractable linear approximations of power flow and transmission losses. Subsequently, we develop an efficient reformulation of the discrete transmission expansion planning (TEP) problem based on a cycle decomposition of the network graph, which conveniently also accommodates grid synchronisation options. Because discrete investment decisions aggravate the problem\u27s complexity, we also cover simplifying heuristics that make use of sequential linear programming (SLP) and retrospective discretisation techniques. In the second half, we investigate other trade-offs, namely between least-cost and near-optimal solutions. We systematically explore broad ranges of technologically diverse system configurations that are viable without compromising the system\u27s overall cost-effectiveness. For example, we present solutions that avoid installing onshore wind turbines, bypass new overhead transmission lines, or feature a more regionally balanced distribution of generation capacities. Such alternative designs may be more widely socially accepted, and, thus, knowing about these degrees of freedom is highly policy-relevant. The method we employ to span the space of near-optimal solutions is related to modelling-to-generate-alternatives, a variant of multi-objective optimisation. The robustness of our results is further strengthened by considering technology cost uncertainties. To efficiently sweep the cost parameter space, we leverage multi-fidelity surrogate modelling techniques using sparse polynomial chaos expansion in combination with low-discrepancy sampling and extensive parallelisation on high-performance computing infrastructure

Recursive marginal quantization: extensions and applications in finance

Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum