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Knapsack and Subset Sum with Small Items

Adam Polak, Lars Rohwedder, Karol Węgrzycki, Nikhil Bansal, Emanuela Merelli, James Worrell
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.  ... 
doi:10.4230/lipics.icalp.2021.106 fatcat:juxt4agrivhoxpceopainjo6ka

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

Maxim Buzdalov, Anatoly Shalyto
2014 Communications in Computer and Information Science  
It is evaluated on five algorithms, including one simple branch-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 .  ... 
doi:10.1007/978-3-662-45049-9_1 fatcat:65yma2qsrfd7pjwq4l6nmqwrva

A Subquadratic Approximation Scheme for Partition [article]

Marcin Mucha, Karol Węgrzycki, Michał Włodarczyk
2018 arXiv   pre-print
The subject of this paper is the time complexity of approximating 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.  ... 
arXiv:1804.02269v1 fatcat:4oapl5qvxzbcro2ylgkmixzm4u

A Study on the Computational Complexity of the Bilevel Knapsack Problem

Alberto Caprara, Margarida Carvalho, Andrea Lodi, Gerhard J. Woeginger
2014 SIAM Journal on Optimization  
We also discuss the somewhat easier situation where the weight 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 .  ... 
doi:10.1137/130906593 fatcat:aw7nh7iagngvdpglkyoxv5xquy

Hardness of Approximation for Knapsack Problems

Harry Buhrman, Bruno Loff, Leen Torenvliet
2014 Theory of Computing Systems  
Furthermore, we give a simple new algorithm for approximating 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.  ... 
doi:10.1007/s00224-014-9550-z fatcat:qwgirc5sifa6flvmb7imrkcmpi

There is no EPTAS for two-dimensional knapsack

Ariel Kulik, Hadas Shachnai
2010 Information Processing Letters  
The goal is to select a 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  ... 
doi:10.1016/j.ipl.2010.05.031 fatcat:i5lhqvgr4jhqfinwy7xmu2sze4

Mathematical models and decomposition methods for the multiple knapsack problem

Mauro Dell'Amico, Maxence Delorme, Manuel Iori, Silvano Martello
2019 European Journal of Operational Research  
We consider the multiple 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-  ... 
doi:10.1016/j.ejor.2018.10.043 fatcat:mwwhzwgmd5hb7oekjgxpx7rkcm

On exponential time lower bound of Knapsack under backtracking

Xin Li, Tian Liu
2010 Theoretical Computer Science  
Alekhovich et al. 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.  ... 
doi:10.1016/j.tcs.2009.12.004 fatcat:tcmtg3gzirbiflqy736r74awoq

Polynomial Kernels for Weighted Problems [article]

Michael Etscheid, Stefan Kratsch, Matthias Mnich, Heiko Röglin
2015 arXiv   pre-print
Furthermore, when parameterized by the different 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.  ... 
arXiv:1507.03439v1 fatcat:35iefk6kyzgnhaeqh3b2z3z5ni

Volume Constraint Model and Algorithm for the 0-1 Knapsack Problem

M.A. Ofosu, S.K. Amponsah, F. Appau-Yeboah
2015 Research Journal of Applied Sciences Engineering and Technology  
In this study, we have proposed a new model formulation 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  ... 
doi:10.19026/rjaset.9.2584 fatcat:ohhl2xnw6ncxjfko27pzzudkye

Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem

Silvano Martello, David Pisinger, Paolo Toth
1999 Management science  
(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.  ... 
doi:10.1287/mnsc.45.3.414 fatcat:jmxmjvttw5bvtpejkg23mrozly

The subset sum game revisited

Astrid Pieterse, Gerhard J. Woeginger
2021 Theory of Computing Systems  
AbstractWe discuss a game theoretic variant of the 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.  ... 
doi:10.1007/s00224-021-10034-z fatcat:bphbxgk6gzazvaalmfjlcvwj6q

Polynomial Time Approximation Schemes for Class-Constrained Packing Problems [chapter]

Hadas Shachnai, Tami Tamir
2000 Lecture Notes in Computer Science  
We consider variants of the classic bin packing 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.  ... 
doi:10.1007/3-540-44436-x_24 fatcat:7qmt2omuffdfxb72oegyjtkj7a

Group Fairness for Knapsack Problems [article]

Deval Patel, Arindam Khan, Anand Louis
2021 arXiv   pre-print
We study the 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.  ... 
arXiv:2006.07832v3 fatcat:ekvbhgzfyrddbaynmkokwsuz7a

Uniqueness in quadratic and hyperbolic 0–1 programming problems

Vladimir G. Deineko, Bettina Klinz, Gerhard J. Woeginger
2013 Operations Research Letters  
Acknowledgements Vladimir Deineko acknowledges support by Warwick University's Centre for Discrete Mathematics 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 .  ... 
doi:10.1016/j.orl.2013.08.013 fatcat:2fmbgitgmrcs7abzfoyqt2ylxq
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