Questions tagged [knapsack-problems]

A problem in combinatorial optimization. Given a set of items with both weight and value, determine the number of each item to include in a collection so that the total weight is at most a given limit and the value of the collection is maximized.

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Proving that a problem is not FPT using reduction

In the Inclusive Vertex Cover problem, For a given graph $G=(V,E)$, each vertex $u\in V(G)$ has weight $u_{w} \in \mathbb{N}$ and value $u_{v}\in \mathbb{N}$. The value and weight of a set cover $S$ ...
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How do you prove that a solution to the 0-1 knapsack problem is optimal?

Given a boolean vector b representing a solution to the knapsack problem with n elements k capacity and where each element has integer weight and value. Proving that the solution is a solution is ...
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46 views

FPT algorithm for Knapsack

I tried to search whether Knapsack (the decision version) has a FPT algorithm, but didn't find anything on the topic. Can someone help with a reference?
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Path that stays within a convex polyhedron

Let $\mathcal{P},\mathcal{Q}$ denote two convex polyhedra in $\mathbb{R}^d$, which can be represented by a set of linear inequalities. Let $A \subset \mathbb{R}^d$ be a finite set of vectors. The ...
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79 views

Algorithm for Variant of 0-1 Knapsack Problem

Variant of 0-1 Knapsack Problem is when you can choose exactly $k$ items from $n$ items, and $k$ is positive integer parameter that came in the input. Is there an algorithm with running time ...
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50 views

How to create a subset with a given length and mean?

I have a set of numbers $P=\{p_1,\dotsc,p_{|P|}\}$, where $|P|$ is the length of the set. I want to select a subset, $S$, from $P$ such that its mean is approximately equal to a predefined value $\...
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44 views

0/1 knapsack problem: Greedy Algorithm Counterexample

While reading about 0/1 knapsack problem on the Internet, many tutorials considered value/weight ratio to solve the problem and I was wondering will it always contain the element with the greatest ...
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Filling Two Knapsacks with greedy algorithm

I've got the next problem, where I have two, instead of one, knapsacks. Formally, we have items $1, \ldots , n$ and each item $i$ has a positive integer weight $w_i \in \mathbb{N}$ and a positive ...
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108 views

How to derive max amount of items from DP table of Knapsack problem?

I have a little bit changed algorithm for 1-0 Knapsack problem. It calculates max count (which we can put to the knapsack) as well. I'm using it to find max subset sum which <= target sum. For ...
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99 views

Oven cooking queue problem: need help to determine its nature

CS community! I'm more of a practical programmer than a computer scientist, so I need your help to find a solution (or at least point me into the right direction) to an extremely practical problem ...
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558 views

Variant of the Knapsack Problem

I've got problem on integer programming, specifically with the following knapsack problem. I'd be happy to get some suggestions on how to solve the problem in a time efficient way. There are 120 ...
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150 views

“Knapsack problem” with repetition, “lesser or equal” constraint, and recording all valid combinations

In a game I am developing I came across an interesting problem, that seems like it could be solved using some modified variant of the knapsack problem, but it's a bit over my head. Let $x_i$, $ 1\leq ...
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154 views

Matching 2 sets of items by price

I'm trying to solve the following problem in the most efficient way I can find. I want to trade my items for someone elses items, every item have a price and a value. I want to maximize the value of ...
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66 views

Knapsack on two kinds of objects, where you cannot choose type 2 objects on their own

I had an online round at a company where I was asked this question. There are $N$ items, and you have to choose some items from them such that the total weight does not exceed $W$. Each item has three ...
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1answer
39 views

Optimal Selection of Non-Overlapping Jobs

I'm trying to find what the family of problem is - as well as an approach - for the following: I have a set of tasks T = [t1, ..., tn] to do, each of which has a corresponding reward ri. Each task ...
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190 views

what is approach of solving MINUS on SPOJ?

Link: http://www.spoj.com/problems/MINUS/ I have an Idea, after doing n-1 operations one can see that there is just + and - coefficients before each number that matters. For eg. given sequence: 12 ...
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0/1 Knapsack problem with minimal cost

so i have this problem where: I have to accomplish a challenge A with n quests. Each quest gives me: p points and needs t time to be done. The object is to complete the challenge A that needs M ...
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Brute force method to solve the 0-1 knapsack problem

I know that the brute force method is not the best way to solve the 0-1 knapsack problem. I'm not quite getting the dynamic programming idea, but would like to know the following: If the brute force ...
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50 views

How to check if an algorithm's running time is linear/polynomial in its input size? Multiple variables

I am reading a proof that the Subset Sum decision problem is NP-complete. I know that the time complexity is always calculated based on the number of bits of the input in binary, hence the $\log{W}$. ...
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Transforming a multidimensional 0-1 knapsack problem to a maximum weighted clique problem

Just out of theoretical curiosity, is there a way to transform a 0-1 multimensional knapsack problem to a maximum clique problem? Or maybe even easier, to a maximum weighted clique problem (the ...
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122 views

Analysis of the following integer program

So the knapsack problem has an integer programming formulation as follows, $$ \max_x v\cdot x\\s.t \\x_i \in \{0,1\}\\w\cdot x \leq C$$ Now consider the second integer program which might be a ...
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475 views

How to prove that the subset sum problem is polynomially reducible to the knapsack problem

I want to prove that the subset sum problem is polynomially reducible to the Knapsack problem. Overall I want to show that Knapsack is NP-complete. There are two parts to showing knapsack is NP-...
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33 views

Combinatorial optimization algorithm with constraints and objective function

I'm looking for an algorithm that will let me optimally select items from a set. These items have properties which are involved in defining constraints as well as the objective function. .e.g Say each ...
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51 views

Get $n^{\mathrm{th}}$ element of sorted subset sums

I have a sorted multiset (size < 100, real valued) and want to determine the $n^{\mathrm{th}}$ largest of all possible subset sums (including multiplicity in the sums). Attempt at solving : I have ...
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0/1 Knapsack problem. How does the total weight does not exceed the limit?

I am trying to wrap my head around the knapsack problem algorithm. I understood the most of it except one tiny thing. On the left is the [val,weight] and on the ...
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Creating neighbours of solutions to binary knapsack instances

I am currently trying to implement a Tabu Search algorithm for the binary knapsack problem. Part of my goal is to have a variety of different configurations of attributes used and stopping conditions. ...
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595 views

Solving the Knapsack problem in $O(v^*n^2)$, where $v^*$ is the maximum value of all $n$ items

For my study I am supposed to develop an dynamic programming algorithm which solves the following simplification of the Knapsack problem in $O(v^*n^2)$ time ($v^*$ is the maximum value of all elements)...
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20 views

Optimal way to pack items with multidimensional weight such that the number of items is minimized?

I am given a set of items S = {a1,a2,a3,...,an}. Each item has a corresponding M dimensional bit vector indicating the properties of that item. For example, if item x has corresponding vector: {0, 1, ...
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Looking for clarity on step in fractional knapsack problem proof

I'm reading through the proof for the fractional knapsack problem presented here: https://www.cs.rice.edu/~nakhleh/COMP182/Knapsack.pdf I follow the proof until the definition of the new set $Q = \...
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About the pseudo polynomial complexity of the KnapSack 0/1 problem

I have read Why is the dynamic programming algorithm of the knapsack problem not polynomial? and other related questions, so this is not a duplicate but just a related pair of questions to clear some ...
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110 views

0-1 Knapsack problem with item discounts

I recently encountered this kind of problem in a real world setting, and could not for the sake of me find any literature relating to the problem statement I came up with. An example will be included ...
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49 views

Partition a set of factors so that the difference between products is minimized

I'm sure this problem must be well-known... Given a collection $S$ of numbers, partition them into exactly two sub-collections, $A$ and $B$ (I mean, by definition $B$ is just $S-A$) such that the ...
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107 views

Knapsack Problem with Constraints on Item Values

Given $n$ items with weights $w_1,...,w_n$ and values $v_1,...,v_n$, and a weight limit $W$, the purpose is still maximizing the total value of items to be carried (while not exceeding the weight ...
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55 views

Distribution of resources from providers to maximum number of receivers

Consider there is a city with $n$ residents who are in need of internet and there are $m$ internet providers in the city. Here in the city every resident needs internet and every resident knows what ...
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68 views

Knapsack problem with a constraint

I am familiar with 0/1 knapsack problem. But if the following constraint is imposed... how do I solve the question? If you choose some item 'U' you won't be able to choose another item 'V'. For ...
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117 views

Dynamic Programming - Thief Variation Probem

I've encountered a Dynamic Programming problem which is a variation of the thief one. Say you are a thief and you are given a number of houses in a row you should rob : $$House_1,House_2 \dots ...
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46 views

Find the n best choices while maximizing value

Given: A list of slots (A, B, C, ...) Every slots supports a list of choices (C0, C1, C2, C3...) Every choice has a value All slots must be filled with at most n different choices. The sum of the ...
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373 views

How to realize applicable meet-in-the-middle algorithm for 0-1 Knapsack?

I am now studying Knapsack Problem (KP), and find the Meet-in-the-middle algorithm described in Wikipedia a little unclear that, how to realize it in the theoretical time complexity of $O^*(2^{n/2})$? ...
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How to the NP hard of a problem that search for a subset of points with maximum scores?

Suppose in a plane, there is a set of points, whose distance to $(0,0)$ is always 1: $[(0,1),(1,0),(0.707,0.707),(0.707,-0.707),...]$ Each point is assigned with a weight (possible negative): $[w(...
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TSP with knapsack combo

My problem statement is that my agent has to visit multiple pickup points once to collect orders, say Item A, Item B, Item C...... etc. We are using TSP for that. Now say Item A are stored in 3 ...
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Detecting conservation, loss, or gain in a crafting game with items and recipes

Suppose we're designing a game like Minecraft where we have lots of items $i_1,i_2,...,i_n\in I$ and a bunch of recipes $r_1,r_2,...,r_m\in R$. Recipes are functions $r:(I\times\mathbb{N})^n\...
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Why is a Knapsack problem not an LP problem?

We know that LP can solve optimization problems that have linear constraints and linear objective functions. A knapsack problem can be formulated into a linear objective function (because it is just ...
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Why is “encoding” important in time complexity?

I read many writing about the time complexity of 0-1 knapsack problem. (https://stackoverflow.com/questions/4538581/why-is-the-knapsack-problem-pseudo-polynomial#answer-4538668) In conclusion, the ...
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337 views

0-1 knapsack without repetition

My question is why O(nW) at the knapsack problem is pseudo-polynomial. I read lots of the explanation at stackoverflow, But I don't really understand it. (https://stackoverflow.com/questions/19647658/...
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Knapsack dynamic programming complexity issue for $W=1$, is it $O(n)$?

The 0-1 knapsack problem is given by $$\begin{align}&\text{maximize } \sum_{i=1}^n v_ix_i,\tag{P1}\\& \text{subject to } \sum_{i=1}^n w_i x_i \leq W,\\&\text{and } x_i \in \{0,1\}.\end{...
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Looking for an algorithm or similar mathematical problem for trading ownerships in shared property

Suppose there is $N$ property. Each property is owned by multiple person. (they have shared ownership) For example: $Person_1$ owns 22% of $Property_1$, $Person_2$ owns 35% of $Property_1$ and $...
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Maximum-density multiple-choice knapsack problem

I am looking for work done on solving a problem (specifically I'm looking for an approximation algorithm) which is very similar to a combination of two variations of the knapsack problem: maximum-...
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4k views

Find the lexicographically smallest order of N numbers such that the total of N numbers <= Threshold value

GIven a number N, Threshold T and an array A. Find the lexicographically smallest order of N numbers from A such that the total of these N numbers is <= T. This question is a simplification of ...
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993 views

Knapsack Problem with exact required item number constraint

How would we solve the knapsack problem if we now have to fix the number of items in the knapsack by a constant $L$? This is the same problem (max weight of $W$, every item have a value $v$ and weight ...
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120 views

Knapsack-like problem with profit formula

Given two sets of $N$ integers, weights and reps, that store info about some dumbbells, find out the maximum profit by taking at most $M$ dumbbells. Each dumbbell can be taken at most once. The ...

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