Knapsack Problem: Introduction, Solution, and Applications

This document provides an overview of the knapsack problem, including its introduction, solution using dynamic programming, and real-world applications. The knapsack problem involves determining the number of items to include in a combination while ensuring that the overall weight does not exceed a specific limit. The document discusses the algorithm used for solving the problem and its effectiveness. It also explores the various applications of the knapsack problem in fields such as applied mathematics and computer science.

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KNAPSACK PROBLEM
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Introduction
Knapsack problem is applicable in the determining the number of items that is to be
included in a combination (Kapshikar 2018). The determination is carried out in such a way that
the overall weight of the combination is lower than or equal to a specific set limit. This ensures
that the resultant value Is large and is in line with the expected value (Wu and Lu 2018). This is a
severe financial problem and therefore arises in a situation where there exists a severe financial
constraint for that particular person (Gherboudj et al. 2012). An example of such case is studied
in areas such as combinatory, cryptography, computer science, daily fantasy sports and in
applied mathematics.
The computer science perspective of the knapsack problem is an exciting arena that can
be easily employed in providing a more precise analysis for the problem (Coello 2010). The
knapsack decision problem form is NP-complete (Liu et al. 2007). From the study of the
problem, there is no more explicit algorithm that is known to be fast and very precise for solving
the problem (Gherboudj et al. 2012). The analysis of the problem is known to be NP-complete,
whereas the problem optimization is known to be NP-hard (Moreno et al. 2010). The decision
problem, together with the resolution of the problem, are known to be at least difficult.
Irrespective of how optimal the algorithm can be, no known solution for the can precisely tell the
preciseness of the result of the answer given in the algorithmic process (Wu and Lu 2018). The
usability of the problem has, therefore, prompted many researchers to focus on the KPO1
problem. The problem research will, therefore, play a very critical role in drawing an analysis of
the mechanisms that can be used in the provision of the solution of the KPO1 problem (Coello
2010). The three methods that have been selectively employed in getting the resolution of the
problem include the heuristic methods, meta-heuristic methods and more exact solutions.

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The proposed solution for the KPO1 problem is the algorithm employed in dynamic
programming (Kapshikar 2018). Dominance relations can be complementarily used for the
programming algorithm improvements (Kapshikar 2018). This technique was initially proposed
for getting the solution to the KPO1 problem. Improvements have been improved n correlation to
mathematical formulations on linear programming. This research analysis has therefore focused
on the study of one of the proposed solutions to the Knapsack problem.
Problem definition
The knapsack problem is a problem in the area of the combinational algorithm. The
problem aims at the determination of the number of items to be included in the combination. The
problem is an illustration of an existing financial constraints that is to be solved by an individual
by a filling in items that are highly valuable (Kapshikar 2018). The problem is thus very
applicable at instances where an individual is supposed to perform a resource allocation
effectively. This Research evaluation will be primarily being focusing on the solution of the
Knapsack 0-1 problem. That involves the restriction of the xi of all the available items to either
one or zero. When a set of n items is used for the combination, a range of 1 to n is considered for
the analysis. Each item weight combination is taken as wi. The maximum weight capacity of the
combination is taken as W.

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Xi shows the number of actual values or the number of values which are included at the
Knapsack Representation (Wu. and Lu 2018). The main problem that is existing in the
manipulation to ensure the maximization of the summation of the values which are included in
the knapsack problem. This is done to ensure that the summation of the weight of the
combination is very minimal (Ialenti 2015). This value of the weight summation must be lower
in comparison to its capacity. The problem eliminates the restriction of the existence of only one
item. It, on the other hand, results into the restriction of the value xi for each of the item copies
towards the maximization of the value of the integer c.
Considering an unbounded knapsack problem, no bound is provided for each kind copies
of the items which are utilized and the formulation of the problem is done in correlation to the
only restriction of the value of xi.
The formulation for the unbounded knapsack problem can thus be stated as shown below.
Real world application of the problem.
The knapsack problem is widely applicable in very many fields in the modern world. The
knapsack problem is a very important problem that is widely applicable in the field of applied

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mathematics and computer science (Wu. and Lu 2018). Some of the main areas where the
problem is widely applicable include Home Energy management, Cognitive radio networks,
Resource management in software and also in power allocation management. A variant of the
knapsack problem is very applicable in most cognitive radio networks. This includes the areas of
centralized spectrum allocation usually conducted during the formulation of the
multidimensional of the multiple knapsack problem. The effectiveness of the application is
achieved through the incorporation of a heuristic algorithm that is very effective in the provision
of a precise solution for the problem (Kapshikar 2018). The algorithms that are therefore
proposed in these applications are consequently numerically compared for high levels of
accuracy on the results of the solution technique employed for that particular problem.
In the area of energy management, the knapsack problem plays an effective role in
correlation to the smart grid environment optimization (Ialenti 2015). Knapsack solutions are
therefore also very applicable in the achievement of smart solutions for different home
appliances. This, therefore, help in the creation of effective scenarios that are in total coherence
with the consumer's energy demands and preferences (Gherboudj et al. 2012). As analyzed
above, knapsack solutions, therefore, play an integral role in the design of energy controller
Technologies for different energy consumers (Wu. and Lu 2018). Through the analysis of the
knapsack problem, various software developers are also in an excellent position of ensuring
useful resource required for software development and also during the general testing process of
the software program.
Problem solution
Dynamic programming is the technique that has been utilized in the provision for the
solution to the problem. Dynamic programming is a method that is widely utilized in the

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provision of solutions for optimization problems (Moreno et al. 2010). The main idea that is
presented in dynamic programming is an idea that uses the sub problem, and the solution of the
problem are then tabulated on a table. This tabulation allows for the effective reuse of the
generated solutions of the problem. The first step in the dynamic program is the characterization
of the solution to the knapsack problem (Ialenti 2015). The second procedure for the dynamic
programming in deriving the solution for the knapsack problem is the recursive definition of the
optimal solution that is very effective in solving the problem.
When the problem of knapsack is decomposed into different components, an array V
[o…n, o…W] is constructed. The first consideration is made to the conditions (Kapshikar 2018).
1 <=i<=n
and for 0<=w<=W.
The storage in the entry V [I, W]. will be responsible for the storage of the combined
computing time for the file subsets, {1,2…. i}. The maximum size is set at a value of W.
If the array entries can be computed effectively, then for a given array entry V [n, W], the
highest computing time that is contained in the if the array files can therefore effectively fit in
the available storage (Coello 2010). This will consequently will be the main basis for the solution
to the stated problem.
The second step for arriving at the perfect solution of the knapsack problem involves the
recursive definition to the optimized solution that is initially obtained in the first step of the
general solution. Finding the problem solution is also carried out in a way that the problems are
disintegrated into much smaller programs.

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The preceding conditions effectively describe the initial settings to this second step of the
solution.
V[O, W]=0 for 0<=w<=W, no item is considered in this first manipulation.
V[I, W]=-∞, This is generally termed as an illegal manipulation to the problem solution (Coello
2010). Through the incorporation of these initial recursive solutions, the recursive formulation is
discussed conclusively in the preceding mathematical representation.
V[I, W]=(max V [i-1,W], vi+ V[i-1], w-wi])
The formulation is valid for the preceding conditions.
After the above formulations, an analysis is then conducted to tests for the level of
correctness of the method that has been employed for solving the problem. V[I, W].
The conditions for the evaluation is stated as, For the limits, 1<=i<=n, 0<=w<=W,
V[I, W], is, therefore, be noted by the preceding formulation and solution.
V[I,W]=(V[I-1],w],
vi+V[I-1, w-Wi]).
Through the provision of an appropriate computation for the value of V[I, W], it can be
noted that there are only two existing choices for the solution for the problem. The file of interest
in the derivation of this solution was initially denoted as I (Wu. and Lu 2018). Considering the
prevailing limitations in correlation to the finding of the solution to the problem, considerations
are, therefore made to other files that can significantly help in getting a precise answer to the
knapsack problem (Coello 2010). The best formulations that can be carried out with the data
1,2….i-1 can, therefore, be represented mathematically as V[I-1,w]. The consideration of the file

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I can only be made possible through the consideration of the condition wi<=W (Ialenti, A.,
2015). This manipulation, therefore, helps in getting the vi, which is part of the general
computing time of the problem (Kapshikar 2018). Through this formulation process, storage
bytes denoted by wi is spent in the process (Wu and Lu 2018). The perfect action that can,
therefore, be carried out with the remaining files includes V[I-1, WI-W].

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Through the above algorithm, the dynamic computational method is thus very effective
about finding the solution to the Knapsack problem (Wu. and Lu 2018). The algorithm has
effectively highlighted the necessary steps, very instrumental role in getting the knapsack
solution to the dynamic algorithm approach.
Conclusions
Knapsack problem can be effectively solved through the utilization of dynamic
programming as one of the most suitable techniques for getting the solution to the problem. A
solution that can be recursively defined can be described in correlation to a partial solution to the
problem. The obtained partial solutions can be stored and can be utilized in the future for getting
the solution to some problems. This is, therefore termed as the memorization of the sub problem
for the solution of the Knapsack problem. Through the provisions of the preceding analysis, it is

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worth noting that Knapsack problems are widely applicable in different areas such as the scoring
and the constructions of various heterogeneous tests and solutions. The problem solutions
technique is thus useful and can be utilized in making of different critical decisions such as
capital investments selection.

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