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16 votes
3 answers
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Subset sum vs. Subset product (strong vs. weak NP hardness)

I was hoping that some one might be able to explain to me why exactly the subset product problem is strongly NP-hard while the subset sum problem is weakly NP-hard. Subset Sum: Given $X = \{x_1,...,...
RDN's user avatar
  • 325
15 votes
1 answer
952 views

Integer relation detection for Subset Sum or NPP?

Is there a way to encode an instance of Subset Sum or the Number Partition Problem so that a (small) solution to an integer relation yields an answer? If not definitely, then in some probabilistic ...
user834's user avatar
  • 2,806
13 votes
1 answer
387 views

Another variant of PARTITION

I've got a reduction of the following partition problem to a certain scheduling problem: Input: A list $a_1\leqslant\cdots\leqslant a_n$ of positive integers in non-decreasing order. Question: Does ...
Thomas Kalinowski's user avatar
13 votes
1 answer
652 views

Is DAG subset sum approximable?

We are given a directed acyclic graph $G=(V,E)$ with a number associated with each vertex ($g:V\to \mathbb{N}$), and a target number $T\in \mathbb{N}$. The DAG subset sum problem (might exist under a ...
R B's user avatar
  • 9,458
12 votes
0 answers
506 views

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
  • 229
10 votes
1 answer
443 views

An obstruction like ETH

We know under $ETH$ we cannot solve $K$-SUM in $f(K)poly(nK)$ time under any function $f(K)$ (usually $2^{O(K)}$). Is there any conjecture that prevents a $(\log n)^{O(K)}$ complexity (this is ...
Turbo's user avatar
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9 votes
2 answers
3k views

What are good approximation algorithms for the subset sum problem so far?

By "good", I mean either the algorithm provides a relatively tight bound or it has a relatively fast running time. Any reference is welcome.
sma's user avatar
  • 467
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
  • 4,479
8 votes
1 answer
891 views

Interesting SUBSET-SUM problems

I know the following variants of SUBSETSUM problems: $ \mathtt{UNARY\mbox{-}SUBSETSUM} \in \mathsf{L} $ (Elberfeld at. al., 2010), NP-complete $ \mathtt{SUBSETSUM} $, and NEXP-complete $ \mathtt{...
Abuzer Yakaryilmaz's user avatar
7 votes
2 answers
890 views

"Any" Subset Sum. Is it hard?

Here is a variant of the classic partition problem: Given a list of integers can it be partitioned into $S_1$, $S_2$, and $S_3$, with $S_1$ and $S_2$ nonempty, so the sum of elements in $S_1$ equals ...
Whosyourjay's user avatar
7 votes
1 answer
3k views

SUBSET-SUM with binary - unary - "positional" encoding

I don't know if my question is too simple (I'm not an expert, so feel free to delete it) SUBSET-SUM (given a set of integers $<x_{1},...,x_n>$, find if the sum of one or more non-empty subset ...
Marzio De Biasi's user avatar
6 votes
3 answers
1k views

Complexity of a subset sum variant

Given integers $a_1, \ldots, a_n, b \in \mathbb{N}$. What is the complexity of the following problem $$ \exists x_1, \ldots, x_n \in \mathbb{N} \text{ such that } a_1x_1 + \ldots a_nx_n = b? $$ I ...
Paul's user avatar
  • 61
6 votes
1 answer
271 views

Does this problem related to subset sum have a name?

I'm looking for research into this problem -- computational complexity, solution algorithms, approximation algorithms, etc. If it has a canonical name, that would help me look into prior research. ...
Kristov Atlas's user avatar
5 votes
1 answer
256 views

How many numbers are needed such that the possible subset sums cover $\{1, \frac{1}{2}, \frac{1}{3},\dots, \frac{1}{2^m}\}$?

For a multiset $N$ of positive numbers, the set of possible subset sums is $f(N)=\{s\in \mathbb{R}: \exists S\in 2^N, s=\sum_{a\in S} a\}$. We say $N$ generates $T$ if $T\subseteq f(N)$. For example, ...
Mengfan Ma's user avatar
5 votes
1 answer
436 views

On Zero sum perfect matching

Fix $c\geq1$. Input is a $m$ vertex complete graph with edges assigned $a_1,\dots,a_{\frac{m(m-1)}2}\in\Bbb Z$ in some order. Is it $\mathsf{NP}$-complete to decide if there is a perfect matching of ...
user avatar
5 votes
1 answer
783 views

Combining subset sum and subset product problems

The subset sum problem and subset product problem are NP-complete. Is the following problem polynomial-time solvable: given a set of positive integers, find a subset whose sum is $S$ and whose product ...
Dmitriy Finozhenok's user avatar
4 votes
1 answer
236 views

Is the problem NP-C or polynomially solvable?

I am considering a problem of the following: Given a set $X$ of integers and another integer B, are there two subsets of $X$, say $X_1$ and $X_2$, such that $X_1-X_2=B$ ? (Here, $X_i$ also denotes the ...
An Zhang's user avatar
4 votes
1 answer
652 views

Interesting real life problem similar to subsetsum /bin packing problem

I have a real life scenario, where I need to solve a construction related problem somewhat similar to bin packing problem.The situation is as follows : I have large number of cable reels/drums (let's ...
kaushik Ray's user avatar
3 votes
1 answer
441 views

Variant of Subset Sum Problem with Changing Bound

Given a sequence of decreasing integers, i.e., $a_1 \geq a_2 \geq \cdots \geq a_T $ and a positive real $k\geq 1$, find a subset $S$ such that $$\max_{S\subseteq \{1,\ldots,T\}} \sum_{i\in S} a_i$$ $$...
Tomlin's user avatar
  • 41
3 votes
1 answer
761 views

Subset Sum bounds

Are there any bounds for the subset problem with respect to the number of the terms involved in the sum and the range of the possible values?For example looking for some 2 terms whose sum equals 3 you ...
curious's user avatar
  • 173
3 votes
2 answers
1k views

Strongly NP-complete variants of subset sum or partition problem

Some problems have variants that appear to be harder. For instance, Graph Automorphism (GA) problem has quasi-polynomial time algorithm ( by Babai's GI result). However, the fixed-point free GA ...
Mohammad Al-Turkistany's user avatar
3 votes
0 answers
282 views

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
  • 13k
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
  • 539
3 votes
0 answers
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
1 answer
257 views

Is partitioning a multiset into two multisets with equal averages NP-complete?

The subset sum problem of partitioning a multiset of integers into two multisets with equal sums is NP-complete. Is the seemingly related problem of partitioning a multiset of integers into two ...
Romans Pancs's user avatar
2 votes
2 answers
747 views

Subset product problem

We have $m$ list of integers $L_1,\dots,L_m$ and each list $L_i$ contains $n$ distinct integers $L_i(1),\dots,L_i(n)$. You are given two integers $a$ and $b$ with $b<a$ and we know $n=O(2^{(\log b)...
Turbo's user avatar
  • 13k
2 votes
1 answer
1k views

Subset sum with many targets

Take a set $S = \{a_1, a_2, ..., a_n\}$ of integers $a_i$ and a set $G = \{b_1, b_2, ..., b_m\}$ of integers $b_j$. If we ask the question, is there a subset $C \subseteq S$ such that the sum $(\sum_{...
Jason Knight's user avatar
2 votes
1 answer
1k views

Multiple subset sum where subsets have complementary cardinality

$\underline{\mathsf{EQUAL\mbox{ }k-COMPLEMENTARY\mbox{ }SUBSET\mbox{ }SUM(EkCSS)} }$Problem: Input: $a_1,\dots,a_n,b\in \mathbb Z$, with distinct $a_i$ and $k\in\Bbb Z^+$. Output: $k$ $\mbox{ }\...
2 votes
0 answers
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
0 answers
116 views

Can this relaxed subset-sum problem be solved with a smaller dynamic program? [closed]

Cross-post from CS.SE In the subset sum problem, the input is a list of positive integers $x_1,\ldots,x_n$ and an integer $T$, and the goal is to decide whether there is a subset of sum exactly $T$. ...
Erel Segal-Halevi's user avatar
2 votes
0 answers
114 views

a direct polynomial reduction from 3EQU-SUM to EQU-SUM problem [closed]

Given a multiset of integers $S$, in the Equ-Sum problem we want to check whether or not $S$ can be divided into two disjoint subsets, say $X_1$, $X_2$ such that $\sum_{x_i \in X_1}x_i = \sum_{x_j \...
Fatemeh Ghasemi's user avatar
2 votes
0 answers
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
2 votes
1 answer
536 views

Complexity class of a subset problem similar to subset sum

What could be the complexity class of the following problem: Given a positive integer $K$, a positive integer $d$ (say $2$) and a set $S$ of all non-negative integers less than $K$ find a $S' \...
gautam's user avatar
  • 41
1 vote
1 answer
474 views

NP-hardness of minimizing sum of weighted product

Consider a total of $d$ items, $\{I_1,I_2,\cdots,I_d\}$, each having a weight $w_i$ (a positive integer), and a total of $m$ bins, $\{B_1,B_2,\cdots,B_m\}$. We would like to distribute the items into ...
NeedHelp'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
0 answers
66 views

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
  • 412
1 vote
0 answers
57 views

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
2 answers
322 views

Two valued variant of subset sum problem

I'm interested in the complexity of the following problem: Given a multiset $S$ containing only two positive integers $a$ and $b$, find a $k$-partition of $S$ that maximizes the sum of part with ...
ashtavakra's user avatar
1 vote
0 answers
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
  • 13k
1 vote
0 answers
18 views

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
  • 13k
0 votes
2 answers
679 views

NP completeness of linear $0-1$ assignment problem

Supposing we have a linear equation in $n^2$ variables with integer (negatives allowed) coefficients of at most $m$ bits each. Partition $\Pi_1$ the variables into $n$ disjoint sets of $n$ variables ...
Turbo's user avatar
  • 13k
0 votes
1 answer
186 views

Subset Sum Problem and hard looking instances that are not really hard

I have been working in a subset sum solver (some new approach) and while working on the time complexity analysis I found what I describe below. Maybe this could explain why some "hard looking" ...
Jesus Salas's user avatar
0 votes
1 answer
321 views

A dominate vector subset sum problem

Let $k$ be some constants (e.g. one can take $k=2$ for simplexity), for any $u,v\in \mathbb{R}$, we say $u$ dominate $v$ if $\forall 1\le i\le k,~ u[i]\ge v[i]$, write it as $u\succ v$. Consider the ...
Paul's user avatar
  • 271
0 votes
1 answer
223 views

Complexity of $r$-sum as a function of integer sizes

Given $n$ distinct integers whose absolute value is of size $c_{n,k}=\lceil n^{1/k}\rceil$ bits ($c_{n,k}$th bit position is always $1$ for absolute value) we know that using dynamic programming we ...
user avatar
0 votes
2 answers
93 views

strong NP-completeness of multi-dimensional Equal-Subset-Sum

I want to show that the multi-dimensional Equal-Subset-Sum is NP-complete in the strong sense: Given a set of $d$-dimensional vectors of non-negative integers, does there exist two distinct nonempty ...
caduk's user avatar
  • 101
0 votes
1 answer
38 views

Feature selection problem under promise

Are there well used examples of feature selection problem where the problem is defined under certain promise? Let's say the task is to select the minimum number of features such that the mutual ...
Omar Shehab's user avatar
0 votes
1 answer
91 views

reducing this problem to a decision problem

Before I can define my problem, let's make a simple definition. An expression $e$ is a conjunction of inequalities of the form $x~ op~ v$ where: $x$ is a variable, $op\in[<,>,\leq,\geq,=]$, and $...
mahou_2019's user avatar
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
  • 141
0 votes
0 answers
46 views

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
0 votes
0 answers
64 views

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