We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far.
Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. A constrained variant of Stabbing turns out to be even APX-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity.
Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.
We consider the problem of finding a planar embedding of a graph at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard. We give polynomial time algorithms achieving an O(n^{1/2} log n) approximation for paths and matchings, and an O(n) approximation for general graphs.
We study a geometric version of the Red-Blue Set Cover problem originally proposed by Carr, Doddi, Konjevod, and Marathe (SODA 2000): given a red point set, a blue point set, and a set of objects, we want to use objects to cover all the blue points, while minimizing the number of red points covered. We prove that the problem is NP-hard even when the objects are unit squares in 2D, and we give the first PTAS for this case. The technique we use simplifies and unifies previous PTASes for the weighted geometric set cover problem and the unique maximum coverage problem for 2D unit squares.
One advantage of smart grids is that they can reduce the peak load by distributing electricity-demands over multiple short intervals. Finding a schedule that minimizes the peak load corresponds to a variant of a strip packing problem. Normally, for strip packing problems, a given set of axis-aligned rectangles must be packed into a fixed-width strip, and the goal is to minimize the height of the strip. The electricity-allocation application can be modelled as strip packing with slicing: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions in which a vertical line intersects two slices of the same rectangle.
We give a fully polynomial time approximation scheme for this problem, as well as a practical polynomial time algorithm that slices each rectangle at most once and yields a solution of height at most 5/3 times the optimal height.
In the conflict-free coloring problem, for a given range space, we want to bound the minimum value F(n) such that every set P of n points can be colored with F(n) colors with the property that every nonempty range contains a unique color. We prove a new upper bound O(n^{0.368}) with respect to orthogonal ranges in two dimensions (i.e., axis-parallel rectangles), which is the first improvement over the previous bound O(n^{0.382}) by Ajwani, Elbassioni, Govindarajan, and Ray [SPAA'07]. This result leads to an O(n^{1-0.632/2^{d-2}}) upper bound with respect to orthogonal ranges (boxes) in dimension d, and also an O(n^{1-0.632/(2^{d-3}-0.368)}) upper bound with respect to dominance ranges (orthants) in dimension d >= 4.
We also observe that combinatorial results on conflict-free coloring can be applied to the analysis of approximation algorithms for discrete versions of geometric independent set problems. Here, given a set P of (weighted) points and a set S of ranges, we want to select the largest(-weight) subset Q of P with the property that every range of S contains at most one point of Q. We obtain, for example, a randomized O(n^{0.368})-approximation algorithm for this problem with respect to orthogonal ranges in the plane.
The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC.
Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2^{O(log* n)}-approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant. We also obtain an O(log log* n)-approximation for weighted fat triangle cover.
We study several geometric set cover problems in which the goal is to compute a minimum cover of a given set of points in Euclidean space by a family of geometric objects. We give a short proof that this problem is APX-hard when the objects are axis-aligned fat rectangles, even when each rectangle is an epsilon-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest.
In contrast, we give a polynomial-time dynamic programming algorithm for 2-dimensional set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required.
We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points {s_1,s_2,...,s_n} in d-dimensional Euclidean space, where each point s_i is active with some independent and arbitrary but known probability p_i. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems has not been investigated before in computational geometry to our knowledge, and is motivated by uncertainty inherent in many sources of geometric data.
We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, that leads to a constant-factor approximation.
Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.
The piercing problem seeks the minimum number of points for a set of objects such that each object contains at least one of the points. We present a polynomial-time approximation scheme (PTAS) for the piercing problem for a set of axis-parallel unit-height rectangles. We also examine the problem in a dynamic setting and show how to maintain a factor-2 approximation under insertions in logarithmic amortized time, by solving an incremental version of the maximum independent set problem for interval graphs.
We study two problems for a given n-point set in 3-space: finding a largest subset with diameter at most one, and finding a subset of k points with minimum diameter. For the former problem we suggest several polynomial-time algorithms with constant approximation factors, the best of which has factor pi / arccos(1/3) < 2.553. For the latter problem we observe that there is a polynomial-time approximation scheme.
Finding the maximum independent set in the intersection graph of n axis-parallel rectangles is NP-hard. We re-examine two known approximation results for this problem. For the case of rectangles of unit height, Agarwal, van Kreveld, and Suri (1997) gave a (1+1/k)-factor algorithm with an O(n log n + n^{2k-1}) time bound for any integer constant k >= 1; we describe a similar algorithm running in only O(n log n + nD^{k-1}) time, where D <= n denotes the maximum number of rectangles a point can be in. For the general case, Berman, DasGupta, Muthukrishnan, and Ramaswami (2001) gave a log_k n-factor algorithm with an O(n^{k+1}) time bound for any integer constant k >= 2; we describe similar algorithms running in O(n log n + nD^{k-2}) and n^{O(k/log k)} time.
Let tau_K^{(d)} be the analogous ratio in d-dimensional space. Khuller et al. showed that tau_3^{(d)} < 1.667 for any d. We observe that tau_3^{(d)} < 1.633.
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