Constant Ratio Approximation Algorithms for the Rectangle Stabbing Problem and the Rectilinear Partitioning Problem

2000 ◽  
pp. 211-219 ◽  
Author(s):  
Daya Ram Gaur ◽  
Toshihide Ibaraki ◽  
Ramesh Krishnamurti
Keyword(s):  
Approximation Algorithms ◽  
Constant Ratio ◽  
Partitioning Problem

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.

scholarly journals On the Performance of a Simple Approximation Algorithm for the Longest Path Problem

2019 ◽  
Vol 35 (1) ◽  
pp. 57-68
Author(s):  
Nguyen Thi Phuong ◽  
Tran Vinh Duc ◽  
Le Cong Thanh
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Undirected Graph ◽  
Simple Algorithm ◽  
Performance Ratio ◽  
Constant Ratio ◽  
Time Approximation ◽  
Longest Path ◽  
Longest Path Problem

The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\sqrt{|V| \log |V|}$ and so has performance ratio less than $1+4\sqrt{\log |V|/|V|}$.\

scholarly journals Approximating Smallest Containers for Packing Three-Dimensional Convex Objects

2018 ◽  
Vol 28 (02) ◽  
pp. 111-128 ◽  
Author(s):  
Helmut Alt ◽  
Nadja Scharf
Keyword(s):  
Exact Solution ◽  
Approximation Algorithms ◽  
Three Dimensional ◽  
Convex Polyhedra ◽  
Minimum Volume ◽  
Constant Ratio ◽  
Polynomial Time Algorithms ◽  
Axis Parallel ◽  
Arbitrary Convex ◽  
Rigid Motions

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are [Formula: see text]-hard so that we cannot expect to find polynomial time algorithms to determine the exact solution. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for packing cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimum volume containers for the objects described.

The constant-ratio rule and visual displays.

1961 ◽  
Author(s):  
Milton H. Hodge ◽  
Morris J. Crawford ◽  
Mary L. Piercy
Keyword(s):  
Constant Ratio ◽  
Visual Displays ◽  
Ratio Rule

Analysis of Myocardial Time-Activity Curves of 123l-Heptadecanoic Acid II. The Acquisition Time

1987 ◽  
Vol 26 (06) ◽  
pp. 248-252 ◽  
Author(s):  
M. J. van Eenige ◽  
F. C. Visser ◽  
A. J. P. Karreman ◽  
C. M. B. Duwel ◽  
G. Westera ◽  
...  
Keyword(s):  
Standard Deviation ◽  
Acquisition Time ◽  
Half Time ◽  
Constant Ratio ◽  
Time Activity ◽  
Time Activity Curve ◽  
Time Value ◽  
Activity Curve ◽  
Low Amplitude ◽  
Parameter Values

Optimal fitting of a myocardial time-activity curve is accomplished with a monoexponential plus a constant, resulting in three parameters: amplitude and half-time of the monoexponential and the constant. The aim of this study was to estimate the precision of the calculated parameters. The variability of the parameter values as a function of the acquisition time was studied in 11 patients with cardiac complaints. Of the three parameters the half-time value varied most strongly with the acquisition time. An acquisition time of 80 min was needed to keep the standard deviation of the half-time value within ±10%. To estimate the standard deviation of the half-time value as a function of the parameter values, of the noise content of the time-activity curve and of the acquisition time, a model experiment was used. In most cases the SD decreased by 50% if the acquisition time was increased from 60 to 90 min. A low amplitude/constant ratio and a high half-time value result in a high SD of the half-time value. Tables are presented to estimate the SD in a particular case.

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