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Questions tagged [subset-sum]

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Is satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ NP Complete?

Question I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete. Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$. Where $\...
Mason's user avatar
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9 votes
0 answers
884 views

Faster pseudo-polynomial time algorithm for subset-sum?

Let $S(X) = \{\sum_{i\in Y} i | Y\subset X \}$, the set of subset sums of $X$. $S_n(X) = S(X)\cap \{1,\ldots,n\}$. Consider the following variant of subset sum. ALL-SUBSET-SUMS INPUT: positive ...
Chao Xu's user avatar
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3 votes
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Is the following equitable factoring problem $NP$-hard or in $P$?

Consider the following factoring problem: Given an integer $r$ and another integer $N$ along with all of its $n$ number of prime factors and their corresponding multiplicities $\{p_i,e_i\}_{i=1}^n$, ...
Turbo's user avatar
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3 votes
0 answers
317 views

Difficulty of graph coloring and independent set?

Given a graph on $n$ vertices it is strongly $NP$-complete to decide it is $3$-colorable while it is easy to decide it is $n$-colorable. Is there a parsimonious reduction from SUBSET-SUM to GRAPH-3-...
VS.'s user avatar
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3 votes
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260 views

new subset sum approach results

I have been working on a new approach for a subset sum exact solver, and the current state provides an algorithm operating on $O{n/2 \choose n/4}$, demonstrating as well the hardest target value is ...
John Seppard's user avatar
2 votes
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73 views

Partition of a set of integers into subsets where the max. of the subset-sums is minimum

Let $S$ be a set of $n$ positive integers, and $p$ be a partition of $S$ into $m$ mutually disjoint subsets, such that no subset contains more than $k$ elements. Let $\mathcal{P}$ denote the set of ...
Code-searcher's user avatar
2 votes
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30 views

Finding shortest calculation of the sum of a subset of a group, given sums for other previously summed subsets

Say $S=\{g\in G\}$ is a set of elements in an abelian group $G$ whose group operation $(+)$ is expensive to compute. Given a subset $T\subset S$, we want to compute the sum of $T$'s elements, $\...
Alex Coventry's user avatar
1 vote
0 answers
43 views

Understanding David Pisinger's balanced algorithm for the subset-sum problem with bounded weights

I'm trying to understand David Pisinger's balanced algorithm for the subset-sum problem with bounded weights, which can be found on page 5 of his paper Linear Time Algorithms for Knapsack Problems ...
Pablo Messina's user avatar
1 vote
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Unbounded Knapsack Instance with a Single Optimum that takes each Item Once?

Consider the Unbounded Knapsack Problem (UKP): We are given a set of $n$ items $I = \{1,\ldots,n\}$ of integral weights $w_1, \ldots, w_n \in \mathbb{N}$, integral profits $p_1, \ldots, p_n \in \...
John's user avatar
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Is this a variant of the set cover problem?

$\textbf{Decision Problem:}$ Given a finite set of elements $E$ and a collection $C$ of non empty sets, $C=\{E_1,...,E_n\}$, such that each $E_i$ covers at least one element of $E$. The goal is to ...
mahou_2019's user avatar
1 vote
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184 views

At what parameters is following $NP$-hard?

Problem Instances at given $\alpha>0$. $(1)$ Given $a_1,\dots,a_{n^\alpha}\in\Bbb Z$ with $|a_i|\in(2^{n-1},2^n-1)$ is there a subset of that sums to $0$? $(2)$ Given $a_1,\dots,a_{n}\in\Bbb Z$ ...
Turbo's user avatar
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1 vote
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Is this subset sum NP complete?

In $\mathsf{NP}$ complete subset sum problem we ask 'Given $n$ numbers in $\Bbb Z$ is there a subset that sums to $0$?'. Is problem 'Given $n$ of degree at most $d$ polynomials in $\Bbb Z[x]$ with ...
Turbo's user avatar
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0 votes
1 answer
120 views

Solution to Subset Sum Problem (in some sense) using Gaussian elimination modulo 2

Consider a set of natural numbers $S \in \mathbb{N}^n$ for some $n \in \mathbb{N}$. Assume that each number $s_i = S^T\cdot e_i$ meets $s_i \leq 2^m$ i.e. is written on at most $m$ number of digits, ...
C Marius's user avatar
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Unbounded Knapsack: Does Increasing capacity increase optimal value?

Our decision problem is as follows: given weights $\mathbf{w}$, values $\mathbf{v}$, and capacities $C_1$ and $C_2$, where $C_1 < C_2$, does the optimal value of unbounded knapsack with the above ...
happyfeet's user avatar
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Bin Covering problem with variable bin sizes

I have a decision problem that I cannot seem to map to a standard studied problem, although it seems similar to a few. I am wondering if anyone has come across this problem before, or if someone can ...
Rohan Bali's user avatar
0 votes
0 answers
147 views

What definition for $FPT$ algorithm for $KSUM$ gives $W[P]=FPT\implies KSUM$ is $FPT$?

In the definition on $KSUM$ problem we are given $n$ input integers and we have to decide if $K$ of them sum to $0$. $KSUM$ is $FPT$ if there is a $O(f(K)poly(n))$ algorithm for it. However Downey ...
Turbo's user avatar
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-3 votes
1 answer
193 views

Can we map this problem to subset-sum?

Let there be $n$ set of ordered pairs $s_1=\{(c_1,f_1),(c_1,f_2) ...(c_1,f_m)\}$, $s_2=\{(c_2,f_1),(c_2,f_2) ...(c_2,f_m)\}$, $s_3=\{(c_3,f_1),(c_3,f_2) ...(c_3,f_m)\}$, .... $s_n=\{(c_n,f_1)(c_n,f_2) ...
AVIK DUTTA's user avatar