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Knapsack and Subset Sum with Small Items

2021

*Knapsack*

*and*

*Subset*

*Sum*are fundamental NP-hard problems in combinatorial optimization. ... We give algorithms that run in time O(n + s³)

*and*O(n + v³) for the

*Knapsack*problem,

*and*in time Õ(n + s^{5/3}) for the

*Subset*

*Sum*problem. ... In time O(|I| + λ I ) we can find all

*subset*

*sums*in S(I) that are smaller than Θ(λ I ) using Bringmann's algorithm. ...

##
###
Worst-Case Execution Time Test Generation for Solutions of the Knapsack Problem Using a Genetic Algorithm
[chapter]

2014
*
Communications in Computer and Information Science
*

It is evaluated on five algorithms, including one simple branch-

doi:10.1007/978-3-662-45049-9_1
fatcat:65yma2qsrfd7pjwq4l6nmqwrva
*and*-bound algorithm, two algorithms by David Pisinger*and*their partial implementations. ... This is especially true in the case of the*knapsack*problem, which is often called "the easiest NP-complete problem". ... N*items*are given, each characterized by weight w i*and*profit p i ,*and*a*knapsack**with*a weight capacity of W . ...##
###
A Subquadratic Approximation Scheme for Partition
[article]

2018
*
arXiv
*
pre-print

The subject of this paper is the time complexity of approximating

arXiv:1804.02269v1
fatcat:4oapl5qvxzbcro2ylgkmixzm4u
*Knapsack*,*Subset**Sum*, Partition,*and*some other related problems. ... Our main contribution lies in designing a mechanism that reduces an instance of*Subset**Sum*to several simpler instances, each*with*some special structure,*and*keeps track of interactions between them. ... Since Partition is a special case of*Subset**Sum*,*and**Subset**Sum*is a special case of*Knapsack*, an algorithm for*Knapsack*also works for*Subset**Sum**and*Partition. ...##
###
A Study on the Computational Complexity of the Bilevel Knapsack Problem

2014
*
SIAM Journal on Optimization
*

We also discuss the somewhat easier situation where the weight

doi:10.1137/130906593
fatcat:aw7nh7iagngvdpglkyoxv5xquy
*and*profit coefficients in the*knapsack*problem are encoded in unary: two of the considered bilevel variants become solvable in polynomial ... Bilevel optimization is a special case of the general multilevel optimization problem, which deals*with*a hierarchy of decision makers at an arbitrary number of levels. ... . , r − 1 we create a padding*item*p j*with*a(p j ) = 1*and*b(p j ) = Q + 2 j . (Proof of if.) Assume that the integer S*with*R ≤ S < R + 2 r cannot be represented as a*subset**sum*of the q i . ...##
###
Hardness of Approximation for Knapsack Problems

2014
*
Theory of Computing Systems
*

Furthermore, we give a simple new algorithm for approximating

doi:10.1007/s00224-014-9550-z
fatcat:qwgirc5sifa6flvmb7imrkcmpi
*knapsack**and**subset*-*sum*, that can be adapted to work for*small*space, or in*small*parallel time. ... than*with*an FPTAS. ... give the full contradiction*with*the ETH. ...##
###
There is no EPTAS for two-dimensional knapsack

2010
*
Information Processing Letters
*

The goal is to select a

doi:10.1016/j.ipl.2010.05.031
fatcat:i5lhqvgr4jhqfinwy7xmu2sze4
*subset*of the*items*of maximum total profit such that the*sum*of all vectors is bounded by the bin capacity in each dimension. ... In the d-dimensional (vector)*knapsack*problem given is a set of*items*, each having a d-dimensional size vector*and*a profit,*and*a d-dimensional bin. ... Furthermore, a reduction given in [4] , from perfect code*with*a parameter k to sized*subset**sum**with*the same parameter k, implies that there is no algorithm for sized*subset**sum**with*running time f ...##
###
Mathematical models and decomposition methods for the multiple knapsack problem

2019
*
European Journal of Operational Research
*

We consider the multiple

doi:10.1016/j.ejor.2018.10.043
fatcat:mwwhzwgmd5hb7oekjgxpx7rkcm
*knapsack*problem, that calls for the optimal assignment of a set of*items*, each having a profit*and*a weight, to a set of*knapsacks*, each having a maximum capacity. ... We then present two new pseudo-polynomial formulations, together*with*specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. ... Acknowledgments Research supported by MIUR-Italy (Grant PRIN 2015, Nonlinear*and*Combinatorial Aspects of Complex Networks )*and*by Air Force Office of Scientific Research (under award number FA9550- ...##
###
On exponential time lower bound of Knapsack under backtracking

2010
*
Theoretical Computer Science
*

Alekhovich et al.

doi:10.1016/j.tcs.2009.12.004
fatcat:tcmtg3gzirbiflqy736r74awoq
*and*the Ω(2 0.66n / √ n) lower bound of Li et al. , Improved exponential time lower bound of*Knapsack*problem under BT model, in: Proc 4th TAMC 2007, in: LNCS, vol. 4484, 2007, pp. 624 ... Keywords: Model of algorithms The pBT model*Knapsack*problem Exponential time lower bounds Backtracking a b s t r a c t We prove an Ω(2 0.69n / √ n) time lower bound of*Knapsack*problem under the adaptive ... Acknowledgements We thank Professor Ke Xu for joining our discussion*and*providing helpful comments. We thank Professor Kaile Su for his encouragement*and*support. ...##
###
Polynomial Kernels for Weighted Problems
[article]

2015
*
arXiv
*
pre-print

Furthermore, when parameterized by the different

arXiv:1507.03439v1
fatcat:35iefk6kyzgnhaeqh3b2z3z5ni
*item*sizes we obtain a polynomial kernelization for*Subset**Sum**and*an exponential kernelization for*Knapsack*. ... Among open problems in kernelization it has been asked many times whether there are deterministic polynomial kernelizations for*Subset**Sum**and**Knapsack*when parameterized by the number n of*items*. ... Next, we consider the*Knapsack*problem*and*its special case*Subset**Sum*, in Sect. 5. For*Subset**Sum*instances*with*only k*item*sizes, we derive a kernel of size polynomial in k. ...##
###
Volume Constraint Model and Algorithm for the 0-1 Knapsack Problem

2015
*
Research Journal of Applied Sciences Engineering and Technology
*

In this study, we have proposed a new model formulation

doi:10.19026/rjaset.9.2584
fatcat:ohhl2xnw6ncxjfko27pzzudkye
*and*algorithm design for the 0-1*knapsack*problem. ... of the*knapsack*since every*knapsack*also has a volume, there would be no need to force it into the*knapsack*. ... INTRODUCTION The classical*Knapsack*Problem is the problem of choosing a*subset*of the n*items*such that the corresponding profit*sum*is maximized without having the weight*sum*to exceed the capacity c ...##
###
Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem

1999
*
Management science
*

(

doi:10.1287/mnsc.45.3.414
fatcat:jmxmjvttw5bvtpejkg23mrozly
*Knapsack*Problem; Dynamic Programming; Branch-*and*-Bound; Surrogate Relaxation) ... The algorithm is able to solve all classical test instances,*with*up to 10,000 variables, in less than 0.2 seconds on a HP9000-735/99 computer. ... They also thank MURST*and*CNR, Italy,*and*are grateful to two anonymous referees for helpful comments that considerably improved the presentation. ...##
###
The subset sum game revisited

2021
*
Theory of Computing Systems
*

AbstractWe discuss a game theoretic variant of the

doi:10.1007/s00224-021-10034-z
fatcat:bphbxgk6gzazvaalmfjlcvwj6q
*subset**sum*problem, in which two players compete for a common resource represented by a*knapsack*. ... Each player owns a private set of*items*, players pack*items*alternately,*and*each player either wants to maximize the total weight of his own*items*packed into the*knapsack*or to minimize the total weight ... Example 2.2 Consider the*subset**sum*game*with*B-*items*4,4,4,7,7,*with*A-*items*6,6,11,*and**with*a*knapsack*of capacity c = 24. The first move belongs to the adversary B. ...##
###
Polynomial Time Approximation Schemes for Class-Constrained Packing Problems
[chapter]

2000
*
Lecture Notes in Computer Science
*

We consider variants of the classic bin packing

doi:10.1007/3-540-44436-x_24
fatcat:7qmt2omuffdfxb72oegyjtkj7a
*and*multiple*knapsack*problems, in which sets of*items*of di erent classes (colours) need to be placed in bins; the*items*may have di erent sizes*and*values ... We show that GCCP is APX-hard, already for the case of a single*knapsack*, where all*items*have the same size*and*the same value. ... Fund -Smoler Research Fund,*and*by the Fund for the Promotion of Research at the Technion. ...##
###
Group Fairness for Knapsack Problems
[article]

2021
*
arXiv
*
pre-print

We study the

arXiv:2006.07832v3
fatcat:ekvbhgzfyrddbaynmkokwsuz7a
*knapsack*problem*with*group fairness constraints. ... The goal of this problem is to select a*subset*of*items*such that all categories are fairly represented, the total weight of the selected*items*does not exceed the capacity of the*knapsack*,*and*the total ... AL was supported in part by SERB Award ECR/2017/003296*and*a Pratiksha Trust Young Investigator Award. ...##
###
Uniqueness in quadratic and hyperbolic 0–1 programming problems

2013
*
Operations Research Letters
*

Acknowledgements Vladimir Deineko acknowledges support by Warwick University's Centre for Discrete Mathematics

doi:10.1016/j.orl.2013.08.013
fatcat:2fmbgitgmrcs7abzfoyqt2ylxq
*and*Its Applications (DIMAP)*and*by EPSRC fund EP/F017871. ... Gerhard Woeginger acknowledges support by the Netherlands Organization for Scientific Research (NWO), grant 639.033.403,*and*by DIAMANT (an NWO mathematics cluster). ... For every*item*i in the*knapsack*instance, we create a corresponding*item*i in the*subset**sum*instance*with*weight q i = 3w i P + p i . Furthermore we define the goal value as Q = (3W + 1)P . ...
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