Questions tagged [subset-sum]
Questions about the NP-complete problem Subset Sum.
79
questions
1
vote
1answer
22 views
Reduction from SUBSET-SUM to 0-1-INT-PROG
The 0-1-INT-PROG problem is given an integer $m \times n$ matrix $A$ and an integer $m$-vector $b$, is there an integer $n$-vector $x$ with $A \cdot x \leq b$.
I am trying to prove that 0-1-INT-PROG ...
-2
votes
0answers
28 views
Is this decision problem NP-Complete? [duplicate]
Say we have a decision problem defined as:
LOADING: Given items of weights a1,...,am, vehicles of weight capacity B, can the items be loaded onto C or fewer vehicles? (All numbers are positive ...
0
votes
1answer
61 views
Minimum absolute value of subset sums of integer values
$f(x_1,...,x_m)=\min_{\emptyset\subset I\subseteq[m] }\left|\sum_{i\in I}x_i\right|, x_i\in \mathbb{Z}\setminus\{0\}$
How to prove $f\in \mathbf{POLY} \Leftrightarrow \mathbf{P}=\mathbf{NP}$?
When $\...
0
votes
2answers
74 views
Have I proven P equals NP if I find an amortized O(n) algorithm for Subset Sum
I have found an algorithm that runs quite fast on Subset Sum problem few years ago (sometime around 2016). It basically sorts the input set in descending order (instead of the regular ascending) and ...
0
votes
0answers
15 views
Given an abstract argumentation framework, is there a tool that can compute: conflict free, admissible and all extensions?
I am trying to find an online/offline tool that can compute the following:
Given an abstract argumentation framework <S, R>, where S = {a1, a2, a3, a4, a5} and the attack relation
R = {(a1, a2),(...
0
votes
1answer
227 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-...
1
vote
0answers
50 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 ...
0
votes
0answers
37 views
Subset selection with maximum sum and minimum variance?
So I am trying to tackle a combinatorial optimization problem and would like some insights on how to approach it. The problem statement is as follows: Consider a set of elements of size N, how do I ...
0
votes
0answers
16 views
Showing that Subset Sum Reduces to 3-Partition [duplicate]
Given a set of integers S (positive and negative, may contain duplicates) can S be divided into three disjoint subsets that all sum to the same value? Prove this problem is NP-complete.
This is a ...
0
votes
1answer
62 views
Reduce Subset-Sum to Sat
Is there a reduction from SUBSET-SUM to SAT?
Just general SAT, not 3-SAT.
Also the given multiset S only has positive integers.
SUBSET-SUM is defined as follows:
Input: a multiset S = { x1 , ... , xn }...
1
vote
1answer
39 views
Unconstrained subset sum vs constrained subset sum?
In class, we discussed two question types: constrained subset-sum and unconstrained subset-sum. Let me define the question specifically and then I will mention what I am confused by.
Question 1: ...
0
votes
0answers
32 views
Is is possible to create a SUBSET-SUM instance that each subset is “unique”?
Given a SUBSET-SUM instance $S$ with a weight $W$, is it possible to create, in polynomial time, a new non-empty instance $T$ (at most the same length as $S$) with weight $M$, that for each non-empty ...
0
votes
1answer
53 views
How can you modify a SUBSET-SUM instance so evaluating a set outputs either 0 or 1?
An SUBSET-SUM instance is a list of $n$ integers $\{ a_1, a_2,... a_n\}$. To evaluate a subset is to output the sum of a subset.
However, I want to know, is it possible to create a new instance $T$, ...
0
votes
0answers
26 views
Are inputs of NP problems too implicit?
I'm reading on wikipedia about P vs NP and I have a question.
Consider the zero sum problem: given a set of integer A, determine if there is a non empty subset such that the sum of all its elements is ...
0
votes
1answer
32 views
Multi-dimensional Knapsack with Minimum Value constraints for Dimensions
In MDK, we have a vector $W = \{W_1, W_2, ..., W_d\}$ where each element corresponds to the maximum weight for the respective dimension in the knapsack.
I want to add a conditional constraint: $V = {...
1
vote
1answer
87 views
Dataset of Hard Instances of SUBSET-SUM
I know for factoring we have the RSA Numbers, in which factoring one of them quickly (usually) indicates a breakthrough in the field. However, I want to know if there's something similar for SUBSET-...
1
vote
1answer
43 views
Converting a Mixed SUBSET-SUM Problem To All-Positive Case
Let's say we have a SUBSET-SUM problem with list {$x_1,x_2,x_3,...x_N$} and weight $W$, with some of $x_i<0$. Is there a known way, in polynomial time, to convert this problem into an equivalent ...
0
votes
2answers
54 views
Complexity of Subset Sum where the size of the subset is specified
I know it should be easy but I'm trying to determine the complexity of the following variant of Subset Sum.
Given a subset $S$ of positive integers and integers $k>0$ and $N>0$, is there a ...
0
votes
3answers
79 views
Find the smallest group of numbers with sum bigger then $X$
Given a list of numbers $S$ where $0 < s_i < 100$, find the minimum sum group of numbers with a sum bigger than $X$.
Each number can be used multiple times.
Ex: for $S = [3,4.1], X = 10$ the ...
0
votes
1answer
311 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 $O^*(2^{n/2})$? ...
0
votes
0answers
109 views
How to trace Subset from Boolean DP table in the Subset Sum Problem
I have seen that the Subset Sum Problem can be solved using Dynamic programming and we should look up the Last row's last column to return the result.
My questions are.
How did someone conclude that ...
2
votes
1answer
71 views
How to prove the NP-completeness of MOD-PARTITION
MOD-PARTITION: Given a set of integers $A={a_1,...,a_n}$, their weights $w = \{w_1, w_2, \dots, w_n\}$ and the number $k$, does there exist a subset $X$ of $A$ such that: $(\sum_{x \in X} w(x) * x) \...
2
votes
1answer
82 views
Enumerate all valid orders of subset sums
Given an positive integer $n$, we define an order of subset sums (or simply, an order, when there is no ambiguity) to be a sequence of all subsets of $\{1,\ldots,n\}$. For example, when $n=2$, the ...
4
votes
1answer
264 views
Two versions of Subset Sum Problem
I keep seeing two versions of the Subset Sum Problem. The first and seemingly least common is:
Given an integer bound $W$ and a collection of $n$ items, each with a positive integer weight $w_i$, ...
9
votes
2answers
435 views
Subset sum problem for permutations
Given permutations $g_1,\,\ldots, g_m \in S_n$ of size $n$ and target permutation $g \in S_n$, decide if there exists a subset of $\{g_1,\, \ldots, g_m\}$, which composition in some order (or, ...
2
votes
0answers
37 views
Finding a non negative combination of integers that adds up to a certain number [duplicate]
I have a set of positive numbers: ${n_1,n_2,...n_k}$ s.t. $n_1>n_2>\dots >n_k$.
I want to find an array of non-negative integers $c_1,c_2,\dots,c_k$ such that
$$n_1c_1 + n_2c_2 + \dots + ...
3
votes
1answer
26 views
Solve SUBSET SUM for Reciprocals of Primes
Let $p_1, ..., p_n$ distinct prime numbers with $P = \prod_{i=1}^{n}{p_i}$ and $A=(a_1, ..., a_n)$ with $a_i = P/p_i$.
Problem
Show the SUBSET SUM problem $(A, \alpha)$ can be solved in polynomial (...
1
vote
1answer
158 views
Return the subset with smallest cardinality of an array whose elements sum to at least a given value
Suppose we are given an array $A[1\ldots n]$ and a value $C$.
Is there an algorithm with linear expected runtime that can produce an array that is the subset with smallest cardinality of $A[1\ldots ...
2
votes
1answer
55 views
complexity of a variant of the subset sum problem
We have a set of positive integers $N=\{a_1,...,a_n\}$, we want to select a subset $N'$ of $N$ with maximum total sum of integers such that this sum should not exceed a given integer $B$.
What is the ...
2
votes
1answer
76 views
find maximum sum of xors
we are given an Array
Array size <= 10^4 .
0 <= A[i] <= 15
We need to partition the array into 4 subsets (each subset can have zero or more elements ). Take xor of each subset and sum ...
2
votes
2answers
136 views
Minimizing the iterative sum of pairs of numbers in a list
Given the tuple (list, value):
$$\left(\left[x_1, x_2, \cdots x_n\right], y\right)$$
You may choose two adjacent values in the list to modify the tuple as:
$$\left(\left[x_1, x_2, \cdots x_{i-1}, (...
1
vote
1answer
71 views
Divide a number in k powers of 2
Example
N = 9 and K=3
4 + 4 + 1 = 9 .
What I have tried.
We can not go on dividing with 2.
We can use unbounded knapsack with array elements from 2^0 to 2^32.
1
vote
0answers
129 views
multiset variant of subset sum problem known algorithms
I have been working in the time analysis for an exact solver I designed for the subset sum problem accepting multisets as input instances, and determined its time complexity to be dependent on the ...
2
votes
1answer
116 views
Find N best subset of quotations
I am faced with the following problem;
We are provided cost quotations for shipping cost per packet by various shipping companies, let's call these quotations $Q_1 ... Q_k$.
Each Quotation is a $M \...
1
vote
1answer
592 views
Generate all combinations of values that are less than array's elements and have a sum = target
I want to find a way to generate sets that contain elements that sum to a certain target. Initially, I have an array that contains elements representing the maximum value that can be stored in that ...
1
vote
1answer
59 views
Recover boolean vector from dot products
Question:
I want to determine a boolean vector $b \in \{0,1\}^n$ consisting of zeros and ones, but cannot access it directly. I can only call a black-box computer code which will take the dot product ...
0
votes
0answers
38 views
Finding fixed size submatrix with highest sum
I have a matrix, which has N rows and M columns. I need to find n rows and m columns, which has the highest sum. Matrix consists of positive numbers. Not optimal solutions are ok. For example N=M=4; n=...
0
votes
1answer
78 views
Prove Subset-Sum is NP-complete - Alternative reduction?
It is well known the Subset-Sum problem is NP-complete. This can be shown using a reduction from the 3SAT problem. I am wondering: is there any other NP-Complete problem that could be reduced to the ...
0
votes
1answer
294 views
Time complexity of subset sum problem with reals instead?
It is well known that the conventional subset sum problem with integers is NP-complete. What if the array elements can be any real numbers and also target sum can be any real number? Is it NP-complete ...
0
votes
1answer
514 views
Find all subsets with a given sum
How to choose from a set of positive numbers all the subsets that sum to some number x?
For example if the set $S=[1,1,2,3,4,5,6,7]$ and I'm searching for all the subsets that sum to $7$ I would have $...
1
vote
1answer
102 views
Finding the Range of Solutions for KSum Variation
Preface
I asked a question before where I was looking to understand the complexity class of the following problem:
Given a range of integers $\{a,a+1,...,b-1,b\}$, find a subset of size $k$ such ...
2
votes
1answer
101 views
Counting the number of subsets with positive sum
I have some constant vector $\mathbf{s}$ on $n$ dimensions, where every element of $\mathbf{s}$ is a real number, and I would like to multiply it by every possible $n$-dimensional binary vector $\...
1
vote
1answer
119 views
Job scheduling approximation
In the course notes for Stanford MS&E-319: https://web.stanford.edu/class/msande319/lec1.pdf
Lemma 5 is given as:
The approximation factor of the modified greedy [scheduling] algorithm is 4/3....
0
votes
1answer
1k views
Partition array into k subsets
We are given an array and a number K. Partition array into K subsets such that
let MaxSum be the maximum sum of among subsets.
We have to minimize summation =$$\sum_{i=1}^{k}MaxSum-sum(i) $$
Is ...
0
votes
1answer
47 views
invariant of bin packing
We are given an array of integers and a number K. We need to pack these integers into bins. The condition is that we have to use exactly K number of bins and each bin should have equal capacity. We ...
1
vote
0answers
28 views
Given a subset of numbers $1, \dots, n$, find the minimal subset of numbers $1, \dots, n$ sums of every subset of which cover all sums of first subset
Given a subset $A = \{a_1, a_2, \dots, a_k\}$ of numbers $\{1, 2, \dots, n\}$, find another subset $B = \{b_1, b_2, \dots, b_t\}$ of numbers $\{1, 2, \dots, n\}$ of minimal size (that is, minimise $t$)...
1
vote
1answer
59 views
Does a polynomial solution to weakly-NP Complete problem mean P = NP?
Suppose someone finds a polynomial solution to weakly-NP Complete problem does that mean P = NP.
4
votes
0answers
232 views
Subset Sum Search Problem for Input with At Most One Solution [closed]
Edit: This question has been reasked on TCS.
We first consider the search version of the subset sum problem: Given a set $S$ of $n$ naturals, find a subset of $S$ that sums to exactly $W$. My ...
2
votes
2answers
452 views
Check if K-Sum Variation is NP-Complete
Problem
Given a range of integers $\{a,a+1,...,b-1,b\}$, find a subset of size $k$ such that the sum is equal to $s$.
Question
This problem came from evaluating some scheduling algorithms that I am ...
0
votes
0answers
17 views
Intuitively, my problem is a mix of perfect hashing, tree spanning, combinatorial stuff - Ordered Decision Tree?
The problem I'm trying to solve is difficult to to give a single name, but I'll call it the ordered decision tree problem.
Imagine a row of commands:
...
