scholarly journals MAXIMIZING THE AREA OF OVERLAP OF TWO UNIONS OF DISKS UNDER RIGID MOTION

2009 ◽  
Vol 19 (06) ◽  
pp. 533-556 ◽  
Author(s):  
SERGIO CABELLO ◽  
MARK DE BERG ◽  
PANOS GIANNOPOULOS ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
High Probability ◽  
Rigid Motion ◽  
Maximum Area ◽  
Empty Intersection ◽  
Rigid Motions ◽  
Unit Disks

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log (m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log 2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.

Minimum Dominating Set Problem for Unit Disks Revisited

2015 ◽  
Vol 25 (03) ◽  
pp. 227-244 ◽  
Author(s):  
Paz Carmi ◽  
Gautam K. Das ◽  
Ramesh K. Jallu ◽  
Subhas C. Nandy ◽  
Prajwal R. Prasad ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Time Complexity ◽  
Dominating Set ◽  
Minimum Dominating Set ◽  
Shifting Lemma ◽  
Dominating Set Problem ◽  
Unit Disks ◽  
A Minor ◽  
Text Unit

In this article, we study approximation algorithms for the problem of computing minimum dominating set for a given set [Formula: see text] of [Formula: see text] unit disks in [Formula: see text]. We first present a simple [Formula: see text] time 5-factor approximation algorithm for this problem, where [Formula: see text] is the size of the output. The best known 4-factor and 3-factor approximation algorithms for the same problem run in time [Formula: see text] and [Formula: see text] respectively [M. De, G. K. Das, P. Carmi and S. C. Nandy, Approximation algorithms for a variant of discrete piercing set problem for unit disks, Int. J. of Computational Geometry and Appl., 22(6):461–477, 2013]. We show that the time complexity of the in-place 4-factor approximation algorithm for this problem can be improved to [Formula: see text]. A minor modification of this algorithm produces a [Formula: see text]-factor approximation algorithm in [Formula: see text] time. The same techniques can be applied to have a 3-factor and a [Formula: see text]-factor approximation algorithms in time [Formula: see text] and [Formula: see text] respectively. Finally, we propose a very important shifting lemma, which is of independent interest, and it helps to present [Formula: see text]-factor approximation algorithm for the same problem. It also helps to improve the time complexity of the proposed PTAS for the problem.

Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

APPROXIMATION ALGORITHMS FOR SCHEDULING MALLEABLE TASKS UNDER PRECEDENCE CONSTRAINTS

2002 ◽  
Vol 13 (04) ◽  
pp. 613-627 ◽  
Author(s):  
RENAUD LEPÈRE ◽  
DENIS TRYSTRAM ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Precedence Constraints ◽  
Performance Guarantee ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Special Cases ◽  
Special Case ◽  
Polynomial Time Approximation

This work presents approximation algorithms for scheduling the tasks of a parallel application that are subject to precedence constraints. The considered tasks are malleable which means that they may be executed on a varying number of processors in parallel. The considered objective criterion is the makespan, i.e., the largest task completion time. We demonstrate a close relationship between this scheduling problem and one of its subproblems, the allotment problem. By exploiting this relationship, we design a polynomial time approximation algorithm with performance guarantee arbitrarily close to [Formula: see text] for the special case of series parallel precedence constraints and for the special case of precedence constraints of bounded width. These special cases cover the important situation of tree structured precedence constraints. For arbitrary precedence constraints, we give a polynomial time approximation algorithm with performance guarantee [Formula: see text].

Online Conflict-Free Colouring for Hypergraphs

2009 ◽  
Vol 19 (4) ◽  
pp. 493-516 ◽  
Author(s):  
A. BAR-NOY ◽  
P. CHEILARIS ◽  
S. OLONETSKY ◽  
S. SMORODINSKY
Keyword(s):  
High Probability ◽  
Online Algorithm ◽  
Asymptotically Optimal ◽  
Unit Disks

We provide a framework for online conflict-free colouring of any hypergraph. We introduce the notion of a degenerate hypergraph, which characterizes hypergraphs that arise in geometry. We use our framework to obtain an efficient randomized online algorithm for conflict-free colouring of any k-degenerate hypergraph with n vertices. Our algorithm uses O(k log n) colours with high probability and this bound is asymptotically optimal. Moreover, our algorithm uses O(k log k log n) random bits with high probability. We introduce algorithms that are allowed to perform a few recolourings of already coloured points. We provide deterministic online conflict-free colouring algorithms for points on the line with respect to intervals and for points on the plane with respect to half-planes (or unit disks) that use O(log n) colours and perform a total of at most O(n) recolourings.

MAXIMUM AREA INDEPENDENT SETS IN DISK INTERSECTION GRAPHS

2010 ◽  
Vol 20 (02) ◽  
pp. 105-118 ◽  
Author(s):  
SERGEY BEREG ◽  
ADRIAN DUMITRESCU ◽  
MINGHUI JIANG
Keyword(s):  
Combinatorial Optimization ◽  
Approximation Algorithms ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Constant Ratio ◽  
Maximum Weight ◽  
Intersection Graphs ◽  
Maximum Area ◽  
Geometric Setting ◽  
Disk Area

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

Planar and Spatial Rigid Motion as Special Cases of Spherical and 3-Spherical Motion

1983 ◽  
Vol 105 (3) ◽  
pp. 569-575 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rigid Motion ◽  
Arc Length ◽  
Spatial Motion ◽  
Special Cases ◽  
Curvature Theory ◽  
Rigid Motions ◽  
Spherical Motion ◽  
Smooth Transformation

This paper examines spherical and 3-spherical rigid motions with instantaneous invariants approaching zero. It is shown that these motions may be identified with planar and spatial motion, respectively. The instantaneous invariants are ratios of arc-length along the surface of the sphere to its radius, thus the process of shrinking their value may be viewed as expanding the sphere while bounding the instantaneous displacements on the sphere. This allows a smooth transformation of the results of the curvature theory of spherical and 3-spherical motion into their planar and spatial counterparts.

CUTTING OUT POLYGONS WITH LINES AND RAYS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 227-248 ◽  
Author(s):  
OVIDIU DAESCU ◽  
JUN LUO
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Convex Polygon ◽  
Linear Time ◽  
Constant Factor ◽  
Open Problems ◽  
Cutting Out ◽  
Cutting Sequence

We present approximation algorithms for cutting out a polygon P with n vertices from another convex polygon Q with m vertices by line cuts and ray cuts. For line cuts we require both P and Q are convex while for ray cuts we require Q is convex and P is ray cuttable. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O( log 2 n)-factor approximation of an optimal cutting sequence. No algorithms were previously known for the ray cutting version.

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