PTAS for geometric hitting set problems via local search

The increasing demand for the use of database systems as an integrating factor in CAD/CAM applications has necessitated the development of database systems with appropriate modelling and retrieval capabilities. One essential problem is the treatment of geometric data which has led to the development of spatial databases. Unfortunately, most proposals only deal with simple geometric objects like multidimensional points and rectangles. On the other hand, there has been a rapid development in the field of representing geometric objects with free form curves or surfaces, initiated by engineering applications such as mechanical engineering, aviation or astronautics. Therefore, we propose a concept for the realization of spatial retrieval operations on geometric objects with free form boundaries, such as B-spline or Bezier curves, which can easily be integrated in a database management system. The key concept is the encapsulation of geometric operations in a so-called query processor. First, this enables the definition of an interface allowing the integration into the data model and the definition of the query language of a database system for complex objects. Second, the approach allows the use of an arbitrary representation of the geometric objects. After a short description of the query processor, we propose some representations for free form objects determined by B-spline or Bezier curves. The goal of efficient query processing in a database environment is achieved using a combination of decomposition techniques and spatial access methods. Finally, we present some experimental results indicating that the performance of decomposition techniques is clearly superior to traditional query processing strategies for geometric objects with free form boundaries.

We present the first study on the approximability of geometric versions of the unique coverage problem and the minimum membership set cover problem. In the former problem, one is given a family of sets of elements from some universe and aims to select sets that maximize the number of elements contained in precisely one selected set. Unique Coverage has important applications in wireless networks and it is thus natural to consider it in a geometric setting. We use the well-known (unit) disk model for wireless networks and show that Unique Coverage remains NP-hard in this case and present a polynomial-time 1/18-approximation algorithm. This algorithm is extended to the budgeted low-coverage problem, where covering can element multiple times yields less profit and we have a fixed budget to 'buy' sets. We give an asymptotic FPTAS in case the disks have arbitrary size, but bounded ply. For the case that the geometric objects are arbitrary fat objects, we show that these problems are as hard to approximate as in the general case. In the minimum membership set cover problem, the goal is to cover all elements while minimizing the maximum number of sets in which any element is contained. For unit squares and unit disks, we show that the problem remains NP-hard and does not admit a polynomial-time approximation algorithm with ratio smaller than 2 unless P=NP. For unit squares, we give a 5-approximation algorithm for instances where the optimum objective value is bounded by a constant.

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (WUDC) problem asks for a subset of disks of minimum total weight that covers all given points. WUDC is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted WUDC problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted WUDC problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for WUDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.

Minimum Membership Covering and Hitting

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Matching two geometric objects in two-dimensional (2D) and three-dimensional (3D) spaces is a central problem in computer vision, pattern recognition, and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various distance measures. It is well known that the Hausdorff distance is useful for matching two point sets, and that the Fréchet distance is a superior measure for matching two polygonal chains. The discrete Fréchet distance closely approximates the (continuous) Fréchet distance, and is a natural measure for the geometric similarity of the folded 3D structures of biomolecules such as proteins. In this paper, we present new algorithms for matching two polygonal chains in two dimensions to minimize their discrete Fréchet distance under translation and rotation, and an effective heuristic for matching two polygonal chains in three dimensions. We also describe our empirical results on the application of the discrete Fréchet distance to protein structure-structure alignment.

Training a neural network is a difficult optimization problem because of the nonconvex objective function. Therefore, as an alternative to local search algorithms, many global search algorithms have been used to train neural networks. However, local search algorithms are more efficient with computational resources, and therefore numerous random restarts with a local algorithm may be more effective than a global algorithm at obtaining a low value of the objective function. This study examines, through Monte-Carlo simulations, the relative efficiency of a local search algorithm to 8 stochastic global algorithms: 2 simulated annealing algorithms, 1 simple random stochastic algorithm, 1 genetic algorithm and 4 evolutionary strategy algorithms. The results show that even ignoring the computational requirements of the global algorithms, there is little evidence to support the use of the global algorithms examined for training neural networks.

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1 and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm. A simple polynomial-time local search (β/(β + 1) − ε)-approximation algorithm, for any ε > 0. Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 − 1/e − ε)-approximation and a local search algorithm with (½ - ε)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a ½-approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1/e unless NP ⫅ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem. We extend the (1 − 1/e)-approximation algorithm to a constant-factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a ½ - ε approximation algorithm for the maximum k-median problem with hard capacities.

In this paper, we propose a generalisation of the bin packing problem, obtained by adding precedences between items that can assume heterogeneous non-negative integer values. Such generalisation also models the well-known Simple Assembly Line Balancing Problem of type I. To solve the problem, we propose a simple and effective iterated local search algorithm that integrates in an innovative way of constructive procedures and neighbourhood structures to guide the search to local optimal solutions. Moreover, we apply some preprocessing procedures and adapt classical lower bounds from the literature. Extensive computational experiments on benchmark instances suggest that the developed algorithm is able to generate good quality solutions in a reasonable computational time.

We study the predictability of geometric concepts, in particular those defined by boolean combinations of simple geometric objects. First, we give a negative results, showing that the problem of predicting the class of convex polytopes encoded by listing their vertices is prediction complete for P . Thus, an efficient solution to this prediction problem implies the existence of efficient solutions to all prediction problems whose associated evaluation problems are in P . Assuming the existence of a one-way function that is hard on iterates, there are such prediction problems which do not admit efficient solutions. Thus we show under minimal cryptographic assumptions that the class of convex polytopes encoded by listing their vertices is not predictable. As a side effect, we show that determining membership in the convex hull of a given set of points is complete for P with respect to log space reductions. Next, we establish the predictability of the class consisting of unions of a fixed number of flats by reducing its prediction problem to that of the class of flats, which has previously been shown to be predictable. Finally, we give an Occam algorithm for predicting fixed finite unions of boxes. Both constructive results for flats and boxes hold if the dimension is variable.

In this study, a recently introduced feature-based quality measure called SAL, which provides information about the structure, amplitude, and location of a quantitative precipitation forecast (QPF) in a prespecified domain, is applied to different sets of synthetic and realistic QPFs in the United States. The focus is on a detailed discussion of selected cases and on the comparison of the verification results obtained with SAL and some classical gridpoint-based error measures. For simple geometric precipitation objects it is shown that SAL adequately captures errors in the size and location of the objects, however, not in their orientation. The artificially modified (so-called fake) cases illustrate that SAL has the potential to distinguish between forecasts where intense precipitation objects are either isolated or embedded in a larger-scale low-intensity precipitation area. The real cases highlight that a quality assessment with SAL can lead to contrasting results compared to the application of classical error measures and that, overall, SAL provides useful guidance for identifying the specific shortcomings of a particular QPF. It is also discussed that verification results with SAL and other error measures should be interpreted with care if considering large domains, which may contain meteorologically distinct precipitation systems.

We present a memetic approach designed to tackle the 2020 Computational Geometry Challenge on the Minimum Convex Partition problem. It is based on a simple local search algorithm hybridized with a genetic approach. The population is brought down to its smallest possible size - only 2 individuals - for a very simple implementation. This algorithm was applied to all the instances, without any specific parameterization or adaptation.

Search algorithms play a crucial role in systematic design optimization of optical structures. Though many sophisticated methods appear in the literature, physically motivated guiding principles for the design and enhancement of such methods are few. We introduce such a guiding principle—an objective-trait-bias metaheuristic—and demonstrate its value in practice. Specifically, we present a case study in which application of an instance of this metaheuristic—a transmission-bias metaheuristic—leads to significantly better performing variants of a simple stochastic local search algorithm—restarted iterative best improvement—on a challenging design optimization problem—combinatorial design optimization of a multi-spatial-mode photonic crystal waveguide bend that preserves modal content.

Search algorithms play a crucial role in systematic design optimization of optical structures. We demonstrate how search landscape analysis sheds light on the behavior of search algorithms in this context and thereby leads to valuable insight. Specifically, we present a case study in which search landscape analysis demystifies the surprisingly good performance of a simple stochastic local search algorithm—restarted iterative best improvement—on a challenging design optimization problem—combinatorial design optimization of a multi-spatial-mode photonic crystal waveguide bend that preserves modal content. Moreover, the search landscape analysis leads to insight potentially valuable in design optimization of a broad class of optical structures.

Finding the optimal hyper-parameters values for a given problem is essential for most machine learning algorithms. In this paper, we propose a novel hyper-parameter optimization algorithm that is very simple to implement and still competitive with the state-of-the-art L-SHADE variant of Differential Evolution. While the most common method for hyper-parameter optimization is a combination of grid and manual search, random search has recently shown itself to be more effective and has been proposed as a baseline against which to measure other methods. In this paper, we compare three optimization algorithms, namely, the state-of-the-art L-SHADE algorithm, the random search algorithm, and our novel and simple adaptive random search algorithm. We find that our simple adaptive random search strategy is capable of finding parameters that achieve results comparable to the state-of-the-art L-SHADE algorithm, both of which achieve significantly better performance than random search when optimizing the hyper-parameters of the state-of-the-art XGBoost algorithm for 11 datasets. Because of the significant performance increase of our simple algorithm when compared to random search, we propose this as the new go-to method for tuning the hyper-parameters of machine learning algorithms when desiring a simple-to-implement algorithm, and also propose to use this algorithm as a new baseline against which other strategies should be measured.