Let $P$ be a crossing-free polygon and $mathcal C$ a set of shortcuts, where each shortcut is a directed straight-line segment connecting two vertices of $P$. A shortcut hull of $P$ is another crossing-free polygon that encloses $P$ and whose oriente
d boundary is composed of elements from $mathcal C$. Shortcut hulls find their application in geo-related problems such as the simplification of contour lines. We aim at a shortcut hull that linearly balances the enclosed area and perimeter. If no holes in the shortcut hull are allowed, the problem admits a straight-forward solution via shortest paths. For the more challenging case that the shortcut hull may contain holes, we present a polynomial-time algorithm that is based on computing a constrained, weighted triangulation of the input polygons exterior. We use this problem as a starting point for investigating further variants, e.g., restricting the number of edges or bends. We demonstrate that shortcut hulls can be used for drawing the rough extent of point sets as well as for the schematization of polygons.
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(loglog n)$. The results
are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.
We consider dynamic loading and unloading problems for heavy geometric objects. The challenge is to maintain balanced configurations at all times: minimize the maximal motion of the overall center of gravity. While this problem has been studied from
an algorithmic point of view, previous work only focuses on balancing the final center of gravity; we give a variety of results for computing balanced loading and unloading schemes that minimize the maximal motion of the center of gravity during the entire process. In particular, we consider the one-dimensional case and distinguish between loading and unloading. In the unloading variant, the positions of the intervals are given, and we search for an optimal unloading order of the intervals. We prove that the unloading variant is NP-complete and give a 2.7-approximation algorithm. In the loading variant, we have to compute both the positions of the intervals and their loading order. We give optimal approaches for several variants that model different loading scenarios that may arise, e.g., in the loading of a transport ship with containers.
In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the {em adversarial TSP} problem (ATSP). Given a metric space $(X, d)$ and a set of subsets $R = {R_1, R_2, ...
, R_n} : R_i subseteq X$, the goal is to devise an ordering of the regions, $sigma_R$, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by $sigma_R$ is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the {em adversarial model} in which once the choice of $sigma_R$ is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions $R$ that is optimal with respect to the worst point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a $3$-approximation when $R$ is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which $R$ is a set of disjoint unit disks in the plane.
The Euclidean TSP with neighborhoods (TSPN) problem seeks a shortest tour that visits a given collection of $n$ regions ({em neighborhoods}). We present the first polynomial-time approximation scheme for TSPN for a set of regions given by arbitrary d
isjoint fat regions in the plane. This improves substantially upon the known approximation algorithms, and is the first PTAS for TSPN on regions of non-comparable sizes. Our result is based on a novel extension of the $m$-guillotine method. The result applies to regions that are fat in a very weak sense: each region $P_i$ has area $Omega([diam(P_i)]^2)$, but is otherwise arbitrary.
In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we pr
esent new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.
We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points (guards) that are universal in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower b
ounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.
We study several natural instances of the geometric hitting set problem for input consisting of sets of line segments (and rays, lines) having a small number of distinct slopes. These problems model path monitoring (e.g., on road networks) using the
fewest sensors (the hitting points). We give approximation algorithms for cases including (i) lines of 3 slopes in the plane, (ii) vertical lines and horizontal segments, (iii) pairs of horizontal/vertical segments. We give hardness and hardness of approximation results for these problems. We prove that the hitting set problem for vertical lines and horizontal rays is polynomially solvable.
In the relay placement problem the input is a set of sensors and a number $r ge 1$, the communication range of a relay. In the one-tier version of the problem the objective is to place a minimum number of relays so that between every pair of sensors
there is a path through sensors and/or relays such that the consecutive vertices of the path are within distance $r$ if both vertices are relays and within distance 1 otherwise. The two-tier version adds the restrictions that the path must go through relays, and not through sensors. We present a 3.11-approximation algorithm for the one-tier version and a PTAS for the two-tier version. We also show that the one-tier version admits no PTAS, assuming P $ e$ NP.
A path or a polygonal domain is C-oriented if the orientations of its edges belong to a set of C given orientations; this is a generalization of the notable rectilinear case (C = 2). We study exact and approximation algorithms for minimum-link C-orie
nted paths and paths with unrestricted orientations, both in C-oriented and in general domains. Our two main algorithms are as follows: A subquadratic-time algorithm with a non-trivial approximation guarantee for general (unrestricted-orientation) minimum-link paths in general domains. An algorithm to find a minimum-link C-oriented path in a C-oriented domain. Our algorithm is simpler and more time-space efficient than the prior algorithm. We also obtain several related results: - 3SUM-hardness of determining the link distance with unrestricted orientations (even in a rectilinear domain). - An optimal algorithm for finding a minimum-link rectilinear path in a rectilinear domain. The algorithm and its analysis are simpler than the existing ones. - An extension of our methods to find a C-oriented minimum-link path in a general (not necessarily C-oriented) domain. - A more efficient algorithm to compute a 2-approximate C-oriented minimum-link path. - A notion of robust paths. We show how minimum-link C-oriented paths approximate the robust paths with unrestricted orientations to within an additive error of 1.
