QPTAS for Weighted Geometric Set Cover on Pseudodisks and Halfspaces

International audienceWeighted geometric set-cover problems arise naturally in several geometric and non-geometric settings (e.g. the breakthrough of Bansal and Pruhs (FOCS 2010) reduces a wide class of machine scheduling problems to weighted geometric set-cover). More than two decades of research has succeeded in settling the (1 + status for most geometric set-cover problems, except for some basic scenarios which are still lacking. One is that of weighted disks in the plane for which, after a series of papers, Varadarajan (STOC 2010) presented a clever quasi-sampling technique, which together with improvements by Chan et al. (SODA 2012), yielded an O(1)-approximation algorithm. Even for the unweighted case, a PTAS for a fundamental class of objects called pseudodisks (which includes half-spaces, disks, unit-height rectangles, translates of convex sets etc.) is currently unknown. Another fundamental case is weighted halfspaces in R 3 , for which a PTAS is currently lacking. In this paper, we present a QPTAS for all of these remaining problems. Our results are based on the separator framework of Adamaszek and Wiese (FOCS 2013, SODA 2014), who recently obtained a QPTAS for weighted independent set of polygonal regions. This rules out the possibility that these problems are APX-hard, assuming NP DTIME(2 polylog(n)). Together with the recent work of Chan and Grant (CGTA 2014), this settles the APX-hardness status for all natural geometric set-cover problems

ON PARAMETERIZED COMPLEXITY OF HITTING SET PROBLEM FOR AXIS–PARALLEL SQUARES INTERSECTING A STRAIGHT LINE

The Hitting Set Problem (HSP) is the well known extremal problem adopting research interest in the fields of combinatorial optimization, computational geometry, and statistical learning theory for decades. In the general setting, the problem is NP-hard and hardly approximable. Also, the HSP remains intractable even in very specific geometric settings, e.g. for axis-parallel rectangles intersecting a given straight line. Recently, for the special case of the problem, where all the rectangles are unit squares, a polynomial but very time consuming optimal algorithm was proposed. We improve this algorithm to decrease its complexity bound more than 100 degrees of magnitude. Also, we extend it to the more general case of the problem  and show that the geometric HSP for axis-parallel (not necessarily unit) squares intersected by a line is polynomially solvable for any fixed range of squares to hit

Packing and covering with balls on Busemann surfaces

In this note we prove that for any compact subset $S$ of a Busemann surface $({\mathcal S},d)$ (in particular, for any simple polygon with geodesic metric) and any positive number $\delta$, the minimum number of closed balls of radius $\delta$ with centers at $\mathcal S$ and covering the set $S$ is at most 19 times the maximum number of disjoint closed balls of radius $\delta$ centered at points of $S$: $\nu(S) \le \rho(S) \le 19\nu(S)$, where $\rho(S)$ and $\nu(S)$ are the covering and the packing numbers of $S$ by ${\delta}$-balls.Comment: 27 page

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D

Algorithms for Geometric Covering and Piercing Problems

This thesis involves the study of a range of geometric covering and piercing problems, where the unifying thread is approximation using disks. While some of the problems addressed in this work are solved exactly with polynomial time algorithms, many problems are shown to be at least NP-hard. For the latter, approximation algorithms are the best that we can do in polynomial time assuming that P is not equal to NP. One of the best known problems involving unit disks is the Discrete Unit Disk Cover (DUDC) problem, in which the input consists of a set of points P and a set of unit disks in the plane D, and the objective is to compute a subset of the disks of minimum cardinality which covers all of the points. Another perspective on the problem is to consider the centre points (denoted Q) of the disks D as an approximating set of points for P. An optimal solution to DUDC provides a minimal cardinality subset Q*, a subset of Q, so that each point in P is within unit distance of a point in Q*. In order to approximate the general DUDC problem, we also examine several restricted variants. In the Line-Separable Discrete Unit Disk Cover (LSDUDC) problem, P and Q are separated by a line in the plane. We write that l^- is the half-plane defined by l containing P, and l^+ is the half-plane containing Q. LSDUDC may be solved exactly in O(m^2n) time using a greedy algorithm. We augment this result by describing a 2-approximate solution for the Assisted LSDUDC problem, where the union of all disks centred in l^+ covers all points in P, but we consider using disks centred in l^- as well to try to improve the solution. Next, we describe the Within-Strip Discrete Unit Disk Cover (WSDUDC) problem, where P and Q are confined to a strip of the plane of height h. We show that this problem is NP-complete, and we provide a range of approximation algorithms for the problem with trade-offs between the approximation factor and running time. We outline approximation algorithms for the general DUDC problem which make use of the algorithms for LSDUDC and WSDUDC. These results provide the fastest known approximation algorithms for DUDC. As with the WSDUDC results, we present a set of algorithms in which better approximation factors may be had at the expense of greater running time, ranging from a 15-approximate algorithm which runs in O(mn + m log m + n log n) time to a 18-approximate algorithm which runs in O(m^6n+n log n) time. The next problems that we study are Hausdorff Core problems. These problems accept an input polygon P, and we seek a convex polygon Q which is fully contained in P and minimizes the Hausdorff distance between P and Q. Interestingly, we show that this problem may be reduced to that of computing the minimum radius of disk, call it k_opt, so that a convex polygon Q contained in P intersects all disks of radius k_opt centred on the vertices of P. We begin by describing a polynomial time algorithm for the simple case where P has only a single reflex vertex. On general polygons, we provide a parameterized algorithm which performs a parametric search on the possible values of k_opt. The solution to the decision version of the problem, i.e. determining whether there exists a Hausdorff Core for P given k_opt, requires some novel insights. We also describe an FPTAS for the decision version of the Hausdorff Core problem. Finally, we study Generalized Minimum Spanning Tree (GMST) problems, where the input consists of imprecise vertices, and the objective is to select a single point from each imprecise vertex in order to optimize the weight of the MST over the points. In keeping with one of the themes of the thesis, we begin by using disks as the imprecise vertices. We show that the minimization and maximization versions of this problem are NP-hard, and we describe some parameterized and approximation algorithms. Finally, we look at the case where the imprecise vertices consist of just two vertices each, and we show that the minimization version of the problem (which we call 2-GMST) remains NP-hard, even in the plane. We also provide an algorithm to solve the 2-GMST problem exactly if the combinatorial structure of the optimal solution is known. We identify a number of open problems in this thesis that are worthy of further study. Among them: Is the Assisted LSDUDC problem NP-complete? Can the WSDUDC results be used to obtain an improved PTAS for DUDC? Are there classes of polygons for which the determination of the Hausdorff Core is easy? Is there a PTAS for the maximum weight GMST problem on (unit) disks? Is there a combinatorial approximation algorithm for the 2-GMST problem (particularly with an approximation factor under 4)