scholarly journals A POLYNOMIAL-TIME APPROXIMATION ALGORITHM FOR A GEOMETRIC DISPERSION PROBLEM

2009 ◽  
Vol 19 (03) ◽  
pp. 267-288 ◽  
Author(s):  
MARC BENKERT ◽  
JOACHIM GUDMUNDSSON ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
ALEXANDER WOLFF
Keyword(s):  
Approximation Algorithm ◽  
Polynomial Time ◽  
Packing Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Dispersion Problem ◽  
Unit Disks ◽  
Polynomial Time Approximation

We consider the following packing problem. Let α be a fixed real in (0, 1]. We are given a bounding rectangle ρ and a set [Formula: see text] of n possibly intersecting unit disks whose centers lie in ρ. The task is to pack a set [Formula: see text] of m disjoint disks of radius α into ρ such that no disk in B intersects a disk in [Formula: see text], where m is the maximum number of unit disks that can be packed. In this paper we present a polynomial-time algorithm for α = 2/3. So far only the case of packing squares has been considered. For that case, Baur and Fekete have given a polynomial-time algorithm for α = 2/3 and have shown that the problem cannot be solved in polynomial time for any α > 13/14 unless [Formula: see text].

scholarly journals Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
M. Bouznif ◽  
R. Giroudeau
Keyword(s):  
Approximation Algorithm ◽  
Large Class ◽  
Bipartite Graph ◽  
Polynomial Time ◽  
Performance Guarantee ◽  
Makespan Minimization ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Precedence Graph ◽  
Polynomial Time Approximation

We investigate complexity and approximation results on a processor networks where the communication delay depends on the distance between the processors performing tasks. We then prove that there is no heuristic with a performance guarantee smaller than 4/3 for makespan minimization for precedence graph on a large class of processor networks like hypercube, grid, torus, and so forth, with a fixed diameter . We extend complexity results when the precedence graph is a bipartite graph. We also design an efficient polynomial-time -approximation algorithm for the makespan minimization on processor networks with diameter .

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].

ON TWO APPROXIMATION ALGORITHMS FOR THE CLIQUE PROBLEM

1993 ◽  
Vol 04 (02) ◽  
pp. 117-133
Author(s):  
IAIN A. STEWART
Keyword(s):  
Approximation Algorithms ◽  
Polynomial Time ◽  
Optimization Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Deterministic Algorithm ◽  
Time Approximation ◽  
Nondeterministic Algorithm ◽  
Polynomial Time Approximation ◽  
Clique Problem

We look at well-known polynomial-time approximation algorithms for the optimization problem MAX-CLIQUE (“find the size of the largest clique in a graph”) with regard to how easy it is to compute the actual cliques yielded by these approximation algorithms. We show that even for two “pretty useless” deterministic polynomial-time approximation algorithms, it is unlikely that the resulting clique can be computed efficiently in parallel. We also show that for each non-deterministic algorithm, it is unlikely that there is some deterministic polynomial-time algorithm that decides whether any given vertex appears in some clique yielded by that nondeterministic algorithm.

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