Parallel approximation schemes for Subset Sum and Knapsack problems

1987 ◽  
Vol 24 (4) ◽  
pp. 417-432 ◽  
Author(s):  
Joseph G. Peters ◽  
Larry Rudolph
Keyword(s):  
Knapsack Problems ◽  
Approximation Schemes ◽  
Subset Sum ◽  
Parallel Approximation

scholarly journals Trivial geometric heuristic for subset sum problem

2017 ◽  
Author(s):  
R U
Keyword(s):  
Greedy Algorithm ◽  
Dimensional Space ◽  
Approximate Solutions ◽  
Exact Algorithms ◽  
Approximation Schemes ◽  
Brute Force ◽  
Discrete Structure ◽  
Subset Sum Problem ◽  
Subset Sum ◽  
Hyper Plane

All exact algorithms for solving subset sum problem (SUBSET\_SUM) are exponential (brute force, branch and bound search, dynamic programming which is pseudo-polynomial). To find the approximate solutions both a classical greedy algorithm and its improved variety, and different approximation schemes are used.This paper is an attempt to build another greedy algorithm by transferring representation of analytic geometry to such an object of discrete structure as a set. Set of size $n$ is identified with $n$-dimensional space with Euclidean metric, the subset-sum is identified with (hyper)plane.

scholarly journals Approximation Schemes for Subset Sum Ratio Problems

2020 ◽  
pp. 96-107
Author(s):  
Nikolaos Melissinos ◽  
Aris Pagourtzis ◽  
Theofilos Triommatis
Keyword(s):  
Approximation Schemes ◽  
Subset Sum

scholarly journals Parallel Approximation Schemes for problems on planar graphs

1996 ◽  
Vol 33 (4) ◽  
pp. 387-408 ◽  
Author(s):  
J. Díaz ◽  
M. J. Serna ◽  
J. Torán
Keyword(s):  
Planar Graphs ◽  
Approximation Schemes ◽  
Parallel Approximation

Cellular Solutions to Some Numerical NP-Complete Problems

2011 ◽  
pp. 115-149 ◽  
Author(s):  
Andrés Cordón-Franco ◽  
Miguel A. Gutiérrez-Naranjo ◽  
Mario J. Pérez-Jiménez ◽  
Agustín Riscos-Núñez
Keyword(s):  
Polynomial Time ◽  
Efficient Solutions ◽  
Cellular Systems ◽  
P Systems ◽  
Knapsack Problems ◽  
Simulation Tool ◽  
Working Space ◽  
Complete Problems ◽  
Subset Sum ◽  
Np Complete

This chapter is devoted to the study of numerical NP-complete problems in the framework of cellular systems with membranes, also called P systems (Pun, 1998). The chapter presents efficient solutions to the subset sum and the knapsack problems. These solutions are obtained via families of P systems with the capability of generating an exponential working space in polynomial time. A simulation tool for P systems, written in Prolog, is also described. As an illustration of the use of this tool, the chapter includes a session in the Prolog simulator implementing an algorithm to solve one of the above problems.

Approximation schemes for r-weighted Minimization Knapsack problems

2018 ◽  
Vol 279 (1-2) ◽  
pp. 367-386
Author(s):  
Khaled Elbassioni ◽  
Areg Karapetyan ◽  
Trung Thanh Nguyen
Keyword(s):  
Knapsack Problems ◽  
Approximation Schemes

scholarly journals Approximation Schemes for Two Non-Linear Knapsack Problems and Their Applications in Alternating Current Electric Systems

2017 ◽  
Vol 33 (2) ◽  
pp. 165-179
Author(s):  
Thanh Nguyen
Keyword(s):  
Objective Function ◽  
Polynomial Time ◽  
Alternating Current ◽  
Knapsack Problems ◽  
Approximation Schemes ◽  
Programming Approach ◽  
Approximation Factor ◽  
Electrical Systems ◽  
Non Linear ◽  
Efficient Approximation

The purpose of this paper is to study the approximability of two non-linear Knapsack problems, which are motivated by important applications in alternating current electrical systems. The first problem is to maximize a nonnegative linear objective function subject to a quadratic constraint, whilst the second problem is a dual version of the first one, where an objective function is minimized. Both problems are $\np$-hard since they generalize the unbounded Knapsack problem, and it is unlikely to obtain polynomial-time algorithms for them, unless $\p=\np$. It is therefore of great interest to know whether or not there exist efficient approximation algorithms which can return an approximate solution in polynomial time with a reasonable approximation factor. Our contribution of this paper is to present polynomial-time approximation schemes (PTASs) and this is the best possible result one can hope for the studied problems. Our technique is based on the linear-programming approach which seems to be more simple and efficient than the previous one.

scholarly journals Approximation schemes for knapsack problems with shelf divisions

2006 ◽  
Vol 352 (1-3) ◽  
pp. 71-84 ◽  
Author(s):  
E.C. Xavier ◽  
F.K. Miyazawa
Keyword(s):  
Knapsack Problems ◽  
Approximation Schemes
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