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Problem in combinatorial optimization
The knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items
Knapsack_problem
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special
List_of_knapsack_problems
Algorithmic problem in computer science
computer science, the continuous knapsack problem (also known as the fractional knapsack problem) is an algorithmic problem in combinatorial optimization
Continuous_knapsack_problem
The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective
Quadratic_knapsack_problem
Mathematical optimization problem
multiple knapsack problem (MKP) is a generalization of both the max-sum MSSP and the knapsack problem. In this problem, there are m knapsacks and n items
Multiple_subset_sum
Decision problem in computer science
quickly in practice. SSP is a special case of the knapsack problem and of the multiple subset sum problem. The time complexity of SSP depends on two parameters:
Subset_sum_problem
Subfield of mathematical optimization
optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such
Combinatorial_optimization
Mathematical problem in operations research
NP-hard problem reducible to the knapsack problem. The problem can be formulated as an integer linear programming problem. A paper machine can produce an
Cutting_stock_problem
Concept in complexity theory
time O ( ( log n ) 6 ) {\displaystyle O((\log {n})^{6})} . In the knapsack problem, we are given n {\displaystyle n} items with weight w i {\displaystyle
Pseudo-polynomial_time
Form of public key cryptography
for decryption. It is based on the subset sum problem (a special case of the knapsack problem). The problem is as follows: given a set of integers A {\displaystyle
Merkle–Hellman knapsack cryptosystem
Merkle–Hellman_knapsack_cryptosystem
Combinatorial optimization problem
problem reduces to the multiple knapsack problem. If there is a single agent, then, this problem reduces to the knapsack problem. In the following, we have
Generalized assignment problem
Generalized_assignment_problem
Probability of shared birthdays
confident about their estimates. A related problem is the partition problem, a variant of the knapsack problem from operations research. Some weights are
Birthday_problem
Sequence of locally optimal choices
below a fixed constraint. Unlike the knapsack problem, which is known to be NP-hard, the fractional knapsack problem admits a polynomial time greedy algorithm
Greedy_algorithm
Unsolved problem in computer science
for many NP-complete problems, such as the knapsack problem, the traveling salesman problem, and the Boolean satisfiability problem, that can solve to optimality
P_versus_NP_problem
Choosing the fewest coins to make a given amount of money
the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem, a general
Change-making_problem
Mathematical and computational problem
maximizing the value of items that can fit in the bin is known as the knapsack problem. A variant of bin packing that occurs in practice is when items can
Bin_packing_problem
Security system
provable security. This system is based on a type of knapsack problem. Specifically, the underlying problem is this: given integers c,n,p and v0,...,vn, find
Naccache–Stern knapsack cryptosystem
Naccache–Stern_knapsack_cryptosystem
the knapsack problem is as follows: given a set of items of varying volume and a set of knapsacks of varying capacity, assign each item to a knapsack such
Distributed constraint optimization
Distributed_constraint_optimization
NP-hard knapsack problem can be solved by a dynamic programming algorithm requiring a number of steps polynomial in the size of the knapsack and the number
Weak_NP-completeness
NP-complete Some problems related to job-shop scheduling Knapsack problem, quadratic knapsack problem, and several variants Some problems related to multiprocessor
List_of_NP-complete_problems
Complexity class
(SAT) Knapsack problem Hamiltonian path problem Travelling salesman problem (decision version) Subgraph isomorphism problem Subset sum problem Clique
NP-completeness
ε. The knapsack problem, as well as some of its variants: 0-1 knapsack problem. Unbounded knapsack problem. Multi-dimensional knapsack problem with Delta-modular
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Topics referred to by the same term
Rhine-Westphalia the knapsack problem, a math problem the subset sum problem, a special case of the above Naccache-Stern knapsack cryptosystem, a cryptosystem
Knapsack_(disambiguation)
Optimization algorithm
partition problem (WCGTPP) Arc-weighted l-cardinality tree problem (AWlCTP) Multiple knapsack problem (MKP) Maximum independent set problem (MIS) Ant
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
Algorithmic paradigm for constraint satisfaction or enumeration problems
most convenient technique for parsing, for the knapsack problem and other combinatorial optimization problems. It is also the program execution strategy used
Backtracking
Computational property
NP-hardness of this unary version of the problem. For example, bin packing is strongly NP-complete while the 0-1 Knapsack problem is only weakly NP-complete. Thus
Strong_NP-completeness
solving an instance of the knapsack problem, which explains the term "knapsack auction". An example application of a knapsack auction is auctioning broadcast
Knapsack_auction
Problems which attempt to find the most efficient way to pack objects into containers
problem Close-packing of equal spheres Conway puzzle Covering problem Cutting stock problem Ellipsoid packing Kissing number problem Knapsack problem
Packing_problems
Sequence of operations for a task
value for many hard problems. For example, the Knapsack problem, where there is a set of items, and the goal is to pack the knapsack to get the maximum
Algorithm
Type of algorithm, produces approximately correct solutions
incorrect outcomes. To use a heuristic for solving a search problem or a knapsack problem, it is necessary to check that the heuristic is admissible.
Heuristic_(computer_science)
Type of cryptographic algorithm
Knapsack cryptosystems are cryptosystems whose security is based on the hardness of solving the knapsack problem. They remain quite unpopular because simple
Knapsack_cryptosystems
Inherent difficulty of computational problems
written that solve the problem in reasonable times in most cases. Similarly, algorithms can solve the NP-complete knapsack problem over a wide range of
Computational complexity theory
Computational_complexity_theory
Topics referred to by the same term
a banned political party in Kazakhstan Quadratic knapsack problem, an extension of knapsack problem that allows for quadratic terms in the objective function
QKP
Amount of resources to perform an algorithm
other NP problem. Many combinatorial problems, such as the Knapsack problem, the travelling salesman problem, and the Boolean satisfiability problem are NP-complete
Computational_complexity
Set of computational problems stated by Richard Karp (1973)
For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack. Satisfiability: the Boolean satisfiability problem for formulas
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Information systems supporting business or organizational decision-making activities
Judge–advisor system Knapsack problem Land allocation decision support system List of concept- and mind-mapping software Morphological analysis (problem-solving)
Decision_support_system
NP-complete problem in computer science
ISBN 9783540402862. Martello, Silvano; Toth, Paolo (1990). "4 Subset-sum problem". Knapsack problems: Algorithms and computer interpretations. Wiley-Interscience
Partition_problem
Mathematical riddle
polynomial time problem. If the capacity m is arbitrary, the problem is known to be NP-hard. Coin problem Knapsack problem Subset sum problem Jeffrey Shallit
Postage_stamp_problem
Complexity class
routing Scheduling Problems that are decidable but not NP-complete, often are optimization problems: Knapsack optimization problems Integer programming
NP-hardness
Concept in computational complexity theory
Paths, trees, and flowers]). D. Pisinger, 2003. "Where are the hard knapsack problems?" Technical Report 2003/08, Department of Computer Science, University
Cobham's_thesis
American mathematician (1914–2005)
Dantzig–Wolfe decomposition Knapsack problem Maximum flow problem Optimization (mathematics) Travelling salesman problem Shadow price Gass, Saul I. (2011)
George_Dantzig
Small breaded, deep-fried food
"Evaluation of Menu Planning Capability Based on Multi-dimensional 0/1 Knapsack Problem of Nutritional Management System". IAENG International Journal of Applied
Croquette
Computational problem used in cryptography
introduced cyclic lattices in his work in generalizing the compact knapsack problem to arbitrary rings. A cyclic lattice is a lattice that is closed under
Short integer solution problem
Short_integer_solution_problem
Topics referred to by the same term
Communist Party Pound sterling (non-standard code) Unbounded knapsack problem, a problem in combinatorial optimization Ubiquitous Knowledge Processing
UKP
Competitive algorithm for searching a problem space
always problem-dependent. For instance, in the knapsack problem one wants to maximize the total value of objects that can be put in a knapsack of some
Genetic_algorithm
Any algorithm which solves the search problem
include: Problems in combinatorial optimization, such as: The vehicle routing problem, a form of shortest path problem The knapsack problem: Given a set
Search_algorithm
Optimization by removing non-optimal solutions to subproblems
estimation 0/1 knapsack problem Set cover problem Feature selection in machine learning Structured prediction in computer vision Arc routing problem, including
Branch_and_bound
System that regulates the formation of blocks on a blockchain
such as Boolean satisfiability, capacitated vehicle routing, and the knapsack problem—and binds them together through an influence calculation that distributes
Proof_of_work
Open-source numerical computation software
1998). "An improved genetic algorithm for the multiconstrained 0-1 knapsack problem". 1998 IEEE International Conference on Evolutionary Computation Proceedings
Scilab
Computational method
MR 0682664. S2CID 5701340. M. van Hoeij: Factoring polynomials and the knapsack problem. Journal of Number Theory, 95, 167–189, (2002). Kinayman, Noyan; Aksun
Factorization_of_polynomials
Israeli computer scientist
computer scientist specializing in combinatorial optimization, including knapsack problems, interval scheduling, and the optimization of submodular set functions
Hadas_Shachnai
problem multidimensional knapsack problem 0-1 knapsack problem maximum cut problem quadratic assignment problem facility layout problem wind farm layout Hyper-heuristics
Hyper-heuristic
Linear programming for Combinatorial optimization
be implemented by solving the knapsack problem with sizes s and values y: if the optimal solution of the knapsack problem has a total value at most 1,
Configuration_linear_program
Optimization algorithms using quantum computing
utility-scale quantum optimization problems". Retrieved 2025-02-24. Implementation of the QAOA algorithm for the knapsack problem with Classiq[dead link]
Quantum optimization algorithms
Quantum_optimization_algorithms
Periodic auction which is cleared by mathematical optimization
Finding a market-clearing allocation corresponds to solution of a simple knapsack problem, and does not require much computation. By contrast, a smart market
Smart_market
Approach to public-key cryptography
a central hardness assumption is the elliptic curve discrete logarithm problem (ECDLP): given a public base point P {\displaystyle P} and another point
Elliptic-curve_cryptography
hierarchy Clique problem Hamiltonian cycle problem Hamiltonian path problem Integer factorization Knapsack problem Satisfiability problem 2-satisfiability
List of computability and complexity topics
List_of_computability_and_complexity_topics
Problem in social choice
voting, finding a utilitarian budget-allocation requires solving a knapsack problem, which is NP-hard in theory but can be solved easily in practice. There
Combinatorial participatory budgeting
Combinatorial_participatory_budgeting
Problem of sharing resources
apportionment Equity (economics) International trade Justice (economics) Knapsack problem Nash bargaining game Pizza theorem Price of fairness Blank, M. L.;
Fair_division
Computationally, Marxan provides solutions to a conservation version of the 0-1 knapsack problem, where the objects of interest are potential reserve sites with given
Marxan
Type of sealed-bid multiple-item auction
nothing). Note that the determination of winners is essentially a knapsack problem. Next, the formula for deciding payments gives: For bidder A: The payment
Vickrey–Clarke–Groves_auction
Algorithm for searching a problem space
classical NP problems. To cite some of them: graph partitioning, multidimensional knapsack, travelling salesman problem, quadratic assignment problem, set cover
Memetic_algorithm
Optimization replacing a function call with that function's source code
this sense, many inlining algorithms are usually modeled after the Knapsack problem. To decide which callsites are more valuable, an inlining algorithm
Inline_expansion
Complexity class of approximable problems
problem with a PTAS is the knapsack problem. A problem is said to be APX-hard if there is a PTAS reduction from every problem in APX to that problem,
APX
Pairing where no unchosen pair prefers each other over their choice
mathematics, economics, and computer science, the stable matching problem is the problem of finding a stable matching between two equally sized sets of elements
Stable_matching_problem
American operations manager
algorithm for solving large knapsack problems and which are used in almost every efficient algorithm for this type of problem. Other areas of Zemel's research
Eitan_Zemel
Italian electronic engineer
significant contributions in the areas of vehicle routing, knapsack and other cutting and packing problems, train scheduling, set covering, vertex coloring and
Paolo_Toth
Class of algorithms that find approximate solutions to optimization problems
even the best possible one. NP-hard problems vary greatly in their approximability; some, such as the knapsack problem, can be approximated within a multiplicative
Approximation_algorithm
Mathematical procedure
the knapsack problem. J. of Number Theory, 95, 167–189, (2002). Recognizing Numerical Constants by David H. Bailey and Simon Plouffe Ten Problems in Experimental
Integer_relation_algorithm
Operations research problem, paradigm of constrained scheduling problems
Kathryn A.; Thompson, Jonathan M. (2007). "A grasp-knapsack hybrid for a nurse-scheduling problem" (PDF). Journal of Heuristics. 15 (4). Springer: 351–379
Nurse_scheduling_problem
Black-box description of a convex set
that works in pseudopolynomial time can be implemented by solving a knapsack problem. This is used by the Karmarkar-Karp bin packing algorithms. Let f be
Separation_oracle
Topics referred to by the same term
phosphate (MgKPO4), a type of quick setting cement. Multidimensional knapsack problem, a problem in combinatorial optimization Kilopondmetre, a unit of torque
MKP
Season of television series
used: Merkle-Hellman, Wheat and Chessboard Problem, straddling checkerboard, substitution cipher, Bacon's cipher, knapsack problem and Lorentz force
Numbers_season_3
Mechanism for authenticating cryptographic keys
and has continued to function with little change. However, a related problem does occur: users, whether individuals or organizations, who lose track
Web_of_trust
Business management method
this way, DFSS is closely related to operations research (solving the knapsack problem), workflow balancing. DFSS is largely a design activity requiring tools
Design_for_Six_Sigma
Topics referred to by the same term
medication Boss Key Productions, a video game studio Bounded knapsack problem, a problem in combinatorial optimization This disambiguation page lists
BKP
Methods that imitate, replicate or use natural processes
performance of linkage-tree genetic algorithms for the multidimensional knapsack problem". Neurocomputing. 146: 17–29. doi:10.1016/j.neucom.2014.04.069. Engelbrecht
Natural_computing
Algebraic modeling language
post-processing results. The following is a simple OPL model (.mod) for a Knapsack problem: // Knapsack capacity, provided externally in the data file int capacity
Optimization Programming Language
Optimization_Programming_Language
Set of related approximation algorithms for the bin packing problem
we also have to return the configuration. This problem can be solved by solving a knapsack problem, where the item values are y 1 , … , y m {\displaystyle
Karmarkar–Karp bin packing algorithms
Karmarkar–Karp_bin_packing_algorithms
Type of stable matching problem
combinatorial game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching
Stable_roommates_problem
Italian scientist and engineer
he made significant contributions in the areas of knapsack and assignment problems, packing problems, and vehicle routing. As of 2023, he published 160
Silvano_Martello
Model of computational complexity
{\displaystyle n^{2}} lower bound for linear decision trees on the knapsack problem, generalized to algebraic decision trees by Steele and Yao. For Boolean
Decision_tree_model
cipher Exponentiating by squaring Knapsack problem Shor's algorithm Standard Model Symmetry in physics Burnside's problem Classification of finite simple
List_of_group_theory_topics
Unsolved problem in cryptography
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a
RSA_problem
improving or optimizing the performance of real-world systems. List of knapsack problems List of network theory topics List of graphical methods List of mathematics-based
Lists_of_mathematics_topics
Algorithm for public-key cryptography
responsible for finding their weaknesses. They tried many approaches, including "knapsack-based" and "permutation polynomials". For a time, they thought what they
RSA_cryptosystem
Japanese light novel series and its adaptations
premiered in June 2023. A third anime film, Rascal Does Not Dream of a Knapsack Kid, premiered in December 2023. A second anime television season, Rascal
Rascal_Does_Not_Dream
reduction from the knapsack problem. It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem. Sahni presents
Identical-machines_scheduling
Quantum-safe key encapsulation mechanism
problem, in conjunction with cyclotomic rings. Recently, there has also been a tight formal mathematical security reduction of the ring-LWE problem to
ML-KEM
Overuse of a shared resource
Secretary-General of the United Nations In addition, Hardin also pointed out the problem of individuals acting in rational self-interest by claiming that if all
Tragedy_of_the_commons
Device used to slow the motion of an object through an atmosphere
the backpack-style parachute by Charles Broadwick and Gleb Kotelnikov's knapsack parachute. After World War II, parachuting became a popular sport, and
Parachute
Combined science of forestry and informatics
problems, many of which can be formulated as optimization problems (e.g. generalized assignment problem, traveling salesman problem, knapsack problem
Forest_informatics
System that can issue, distribute and verify digital certificates
Certificate Validity reduced to 13 Months. An alternative approach to the problem of public authentication of public key information is the web-of-trust
Public_key_infrastructure
Cryptographic algorithm for digital signatures
AE CEILIDH EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern knapsack cryptosystem Three-pass protocol XTR SQIsign SPHINCS+ Theory Discrete logarithm
Elliptic Curve Digital Signature Algorithm
Elliptic_Curve_Digital_Signature_Algorithm
American computer scientist
Sahni, Sartaj (1969), "Computing partitions with applications to the knapsack problem", Journal of the Association for Computing Machinery, 21 (2): 277–292
Sartaj_Sahni
discrete logarithm problem in a finite group F2p+1. Knapsack-based hash functions—a family of hash functions based on the knapsack problem. The Zémor-Tillich
Security of cryptographic hash functions
Security_of_cryptographic_hash_functions
Standard example in game theory
paradox Centipede game Collective action problem Externality Folk theorem (game theory) Free-rider problem Gift-exchange game Hobbesian trap Innocent
Prisoner's_dilemma
Key agreement protocol
it), unless that party can solve the elliptic curve discrete logarithm problem. Bob's private key is similarly secure. No party other than Alice or Bob
Elliptic-curve_Diffie–Hellman
Optimization problem in computer science and operations research
reduction from the knapsack problem. It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem. Schulz and
Unrelated-machines_scheduling
U j {\displaystyle \sum w_{j}U_{j}} ) is equivalent to the Knapsack problem. The problem 1| r j {\displaystyle r_{j}} | ∑ U j {\displaystyle \sum U_{j}}
Single-machine_scheduling
KNAPSACK PROBLEM
KNAPSACK PROBLEM
Girl/Female
Indian, Telugu
Destroyer of Problems
Boy/Male
Arabic, Indian, Muslim
Problem Solver
Boy/Male
Indian, Tamil
People with this Name are Preferably Intelligent and Very Generous; Highly Knowledgeable in Problem Solving Skills
Boy/Male
Hindu, Indian
Problem
Surname or Lastname
English
English : metonymic occupational name for an itinerant merchant, from Old French talemasche ‘knapsack’ (a word of uncertain origin).Americanized form of one or more like-sounding Jewish surnames. Compare Talmage.
Boy/Male
Muslim
Problem solver
Girl/Female
Muslim/Islamic
Away from all Problems
Girl/Female
Bengali, Indian
Eternity; Problem Solver
KNAPSACK PROBLEM
KNAPSACK PROBLEM
Boy/Male
Hindu, Indian, Punjabi, Sikh, Traditional
Lord Rama; Peace; Attained at the Lord's Feet
Boy/Male
Muslim/Islamic
Haze Mist
Biblical
noise; tumult
Boy/Male
Native American
White eagle.
Girl/Female
American, Australian, British, Chinese, Christian, Danish, English, French, German, Greek, Italian, Latin, Portuguese, Spanish, Swedish
Pure; Maiden; Virgin
Boy/Male
Indian, Punjabi, Sikh
Of Auspicious Love
Boy/Male
Hindu, Indian
Winner of Every Heart
Boy/Male
Hindu, Indian
Brightness
Girl/Female
Hebrew American Italian Shakespearean Spanish
Devoted to God.
Male
Welsh
Welsh form of Latin Valentinus, FOLANT means "healthy, strong."
KNAPSACK PROBLEM
KNAPSACK PROBLEM
KNAPSACK PROBLEM
KNAPSACK PROBLEM
KNAPSACK PROBLEM
n.
To cause to stick; to bring to a stand; to pose; to puzzle; as, to stick one with a hard problem.
v. t.
To propose problems.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
a.
Susceptible of being solved; as, a soluble algebraic problem; susceptible of being disentangled, unraveled, or explained; as, the mystery is perhaps soluble.
a.
Having the nature of a problem; not shown in fact; questionable; uncertain; unsettled; doubtful.
n.
The act of solving, or the state of being solved; the disentanglement of any intricate problem or difficult question; explanation; clearing up; -- used especially in mathematics, either of the process of solving an equation or problem, or the result of the process.
n.
A problem of more than usual difficulty added to another on an examination paper.
n.
A problem to be solved, or an example to be wrought out.
n.
A bag or sack for carrying about the person, as a bag for carrying the necessaries for a journey; a knapsack; a beggar's receptacle for charity; a peddler's pack.
v. t.
A case of canvas or leather, for carrying on the back a soldier's necessaries, or the clothing, etc., of a traveler.
n.
A knapsack.
n.
To begin to deal with; as, to tackle the problem.
v. t.
To explain; to resolve; to unfold; to clear up (what is obscure or difficult to be understood); to work out to a result or conclusion; as, to solve a doubt; to solve difficulties; to solve a problem.
a.
Alt. of Problematical
n.
Same as Cossack.
a.
Questionable; equivocal; indefinite; problematical.
n.
One who proposes problems.
v. t.
To have just and adequate ideas of; to apprehended the meaning or intention of; to have knowledge of; to comprehend; to know; as, to understand a problem in Euclid; to understand a proposition or a declaration; the court understands the advocate or his argument; to understand the sacred oracles; to understand a nod or a wink.
n.
A bag or case, usually of stout cloth, in which a soldier carries his rations when on a march; -- distinguished from knapsack.
n.
The quality or state of being solvable; as, the solvability of a difficulty; the solvability of a problem.