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### Optimal Scheduling of File Transfers with Divisible Sizes on Multiple Disjoint Paths [article]

Mugurel Ionut Andreica
2012 arXiv   pre-print
multiple knapsack problem with divisible item sizes).  ...  In the offline case, I present a novel, heuristic, algorithm for scheduling files with divisible sizes on multiple disjoint paths, in order to maximize the total profit (the problem is equivalent to the  ...  Even for this particular case with divisible item sizes, we present only a pseudopolynomial O(n•S•min{n,S•log(S)}) time algorithm, where S is the maximum size of an item.  ...

### On Strong NP-Completeness of Rational Problems [chapter]

Dominik Wojtczak
2018 Lecture Notes in Computer Science
We show that all of these problems in this setting become strongly NP-complete and, as a result, no pseudo-polynomial algorithm can exist for solving them unless P=NP.  ...  Despite this result we show that they all still admit a fully polynomial-time approximation scheme.  ...  A polynomial-time approximation scheme (PTAS) is an algorithm that, for every ρ > 0, runs in polynomial-time and has relative performance ρ.  ...

### Approximation schemes for knapsack problems with shelf divisions

E.C. Xavier, F.K. Miyazawa
2006 Theoretical Computer Science
Given a knapsack of size K, non-negative values d and , and a set S of items, each item e ∈ S with size s e and value v e , we define a shelf as a subset of items packed inside a bin with total items size  ...  The SHELF-KNAPSACK Problem (SK) is to find a subset S ⊆ S partitioned into shelves with total shelves size at most K and maximum value.  ...  If I is an instance for the problem CCSK and SS ε is an ε-relaxed polynomial time algorithm for the problem SMALL then the algorithm G ε with subroutine SS ε is a polynomial time approximation scheme for  ...

### Page 7814 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews
(NL-PRL; Eindhoven) A polynomial-time algorithm for knapsack with divisible item sizes. (English summary) Inform. Process. Lett. 62 (1997), no. 4, 217-221.  ...  Summary: “We consider the special case of the bounded knapsack problem with divisible item sizes, and present an algorithm that solves it in polynomial time.  ...

### Where are the hard knapsack problems?

David Pisinger
2005 Computers & Operations Research
The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems  ...  Not only can it be solved in pseudo-polynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature.  ...  Also for some large-sized subset sum problems MThard takes unproportionally long time. The best performance is obtained with the Combo algorithm.  ...

### Approximation Algorithms for Traffic Grooming in WDM Rings

K. Corcoran, S. Flaxman, M. Neyer, P. Scherpelz, C. Weidert, R. Libeskind-Hadas
2009 2009 IEEE International Conference on Communications
Although this problem is NP-complete, we give polynomial time approximation algorithms with excellent theoretical performance validated with experimental results.  ...  Given a set of requests at the destination nodes, where each request comprises a bandwidth demand and a profit for fulfilling the request, our objective is to select a subset of the requests and pack (  ...  Recall that a polynomial time approximation scheme (PTAS) is an infinite family of approximation algorithms such that for any > 0 there exists a (1 − )-approximation algorithm that runs in time polynomial  ...

### Where are the hard knapsack problems?

D PISINGER
2004 Computers & Operations Research
The purpose of this paper is to give an overview of all recent exact solution approaches, and to show that the knapsack problem still is hard to solve for these algorithms for a variety of new test problems  ...  Not only can it be solved in pseudo-polynomial time, but also decades of algorithmic improvements have made it possible to solve nearly all standard instances from the literature.  ...  Also for some large-sized subset sum problems MThard takes unproportionally long time. The best performance is obtained with the Combo algorithm.  ...

### Fast Algorithms for Knapsack via Convolution and Prediction [article]

2018 arXiv   pre-print
Our main results are algorithms with near-linear running times (in terms of the size of the knapsack and the number of items) for the knapsack problem, if either the values or sizes of items are small  ...  The goal is to pack a knapsack of size t with the maximum value from a collection of n items with given sizes and values.  ...  D Strongly Polynomial Time Algorithms for Knapsack with Multiplicities In this section, we study the knapsack problem where items have multiplicities.  ...

### Approximate and online multi-issue negotiation

Shaheen S. Fatima, Michael Wooldridge, Nicholas R. Jennings
2007 Proceedings of the 6th international joint conference on Autonomous agents and multiagent systems - AAMAS '07
The agents have time constraints in the form of both deadlines and discount factors. There are m > 1 issues for negotiation where each issue is viewed as a pie of size one.  ...  These approximate strategies also have polynomial time complexity.  ...  An approximation algorithm is said to be fully polynomial if for any > 0 it finds a solution satisfying Equation 5 in time polynomially bounded by size of the problem (for the 0-1 knapsack problem, the  ...

### Page 4075 of Mathematical Reviews Vol. , Issue 82i [page]

1982 Mathematical Reviews
Associated with each period are a number of types of items, each with a value and weight.  ...  Author’s summary: “We consider a class of algorithms which use the combined powers of branch-and-bound, dynamic programming and rudimentary divisibility arguments for solving the zero-one knapsack problem  ...

### Page 9754 of Mathematical Reviews Vol. , Issue 2003m [page]

2003 Mathematical Reviews
When the sizes of items and the sizes of bins are divisible, the algorithms give optimal solutions.  ...  Moreover, a polynomial-time approximation with performance ratio 2+ e for ¢ > 0 is found by using a local search technique.  ...

### Inapproximability of power allocation with inelastic demands in AC electric systems and networks

Majid Khonji, Chi-Kin Chau, Khaled Elbassioni
2014 2014 23rd International Conference on Computer Communication and Networks (ICCCN)
Moreover, for the case when φ is arbitrarily close to π, neither a PTAS nor any bi-criteria approximation algorithm with polynomial guarantees can exist.  ...  We show that there is no bi-criteria approximation algorithm with polynomial guarantees for this networked setting, even all power demands are real (non-complex) numbers.  ...  For one-dimensional knapsack problem (1DKP), there is a pseudo-polynomial time algorithm using dynamic programming achieving exact optimization when all item values are integers.  ...

### Evolution of hyperheuristics for the biobjective 0/1 knapsack problem by multiobjective genetic programming

Rajeev Kumar, Ashwin H. Joshi, Krishna K. Banka, Peter I. Rockett
2008 Proceedings of the 10th annual conference on Genetic and evolutionary computation - GECCO '08
Many different approaches have been taken to obtain an approximate solution to the problem in polynomial time. Here we consider the biobjective 0/1 knapsack problem.  ...  The genetic programming (GP) system outlined here evolves a heuristic which decides whether or not to add an item to the knapsack in such a way that the final solution is one of the Pareto optimal solutions  ...  For a single objective m-dimensional knapsack problem, a polynomial time approximation scheme (PTAS) was presented by Frieze & Clarke  . Erlebach et al.  ...

### Smoothed Analysis: Analysis of Algorithms Beyond Worst Case

Bodo Manthey, Heiko Röglin
2011 it - Information Technology
We give a gentle, not too formal introduction to smoothed analysis by means of two examples: the k-means method for clustering and the Nemhauser/Ullmann algorithm for the knapsack problem.  ...  The reason for this discrepancy is that worst-case analysis is often a way too pessimistic measure for the performance of an algorithm.  ...  Let S n denote the set of all instances of size n for a specific problem, and let t(X) be the running time of the algorithm on instance X.  ...

### Computational Analysis and Efficient Algorithms for Micro and Macro OFDMA Downlink Scheduling

R. Cohen, L. Katzir
2010 IEEE/ACM Transactions on Networking
them, and how to construct the complex OFDMA frame matrix as a collection of rectangles that fit into a single matrix with fixed dimensions.  ...  Before transmitting a frame on the downlink, an OFDMA base station has to invoke an algorithm that determines which of the pending packets will be transmitted, what modulation should be used for each of  ...  In contrast, a polynomial time algorithm is polynomial in the size of the input (the number of digits).  ...
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