# Questions tagged [subset-sum]

Questions about the NP-complete problem Subset Sum.

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### Find the smallest subarray with sum larger than a threshold

Given a set of $n$ positive numbers $\{a_1,\ldots,a_n\}$ and a positive target $T$, find a subset $S$ from $\{a_1,\ldots,a_n\}$ of contiguous elements, that is $S=\{a_i,a_{i+1},a_{i+2},\ldots\}$ for ...
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### HAMILTONIAN PATH AND SUBSETSUM

If we were to discover a deterministic algorithm capable of deciding, in polynomial time, whether a given graph contains a Hamiltonian path, would that imply that the problem SUBSETSUM belongs to P? ...
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### Subset sum problem with big items

Consider the variant of the Subset Sum problem, where the input is a list of $2 m + 1$ positive integers of sum $2 S$, and the goal is to find a subset with the largest sum that is at most $S$. The ...
1 vote
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### Is this variant of multiset covering problem NP-hard?

Consider this variant of multiset covering problem. Input: a collection of sets $S = \{s_1, s_2, \ldots, s_n\}$ and a universal set $U$, in which $s_k \subseteq U$ and $s_k \neq \emptyset$ for all $k$...
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### 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 ...
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### Efficient algorithm for finding the target sum

Task. Find such natural numbers a1,. . . , am , that none of them would be included in the list of excluded numbers, a1 + · · · + am = N and max{a1 , . . . , am} would be as small as possible. Numbers ...
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### List of weakly NP-HARD problems

I need a list of at least 10 weakly NP-HARD problems. I already know the Knapsack problem, partition problem and subset sum problem. Please introduce other weakly NP-hard problems to me.
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1 vote
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### Finding equal-sum subsets from two arrays

Consider the following problem: You are given two integer arrays $A$ and $B$ of size $N$ and $M$, respectively. You are guaranteed that $1 <= A[i] <= M$ and $1 <= B[i] <= N$ for all $i$ (...
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### I'm looking for an algorithm to find all subsets of a set S with subset sums between a given min and max

Given a set S of numbers, min and max. Find all sebsets of numbers from S with a subset sum larger or equal to min and smaller or equal to max. The following article says that there's an algorithm ...
1 vote
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1 vote
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### Is there a pseudopolynomial time algorithm for this subset sum variant?

The subset sum problem is: given a list of $n$ positive integers, and a positive number $T$, find a sub-list with largest sum that is at most $T$. The problem can be found in time polynomial in $n$ ...
1 vote
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### Given a set, generate all permutations whose sums are less or equal to a given number

I am looking for a way to generate every permutation (so order does matter) of a set of positive numbers whose sum is less than (or equal to) a given limit. I need to find the permutations themselves, ...
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### Python closest subset sum function

I'm writing a closestSubset(s,A) function that takes an integer s and an array of positive integers A and returns an array consisting of elements of A which add up to s. If there is no subset that ...
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### Is this set covering problem NP-Hard?

Consider this variant of set covering problem. Input: a collection of sets $S = \{s_1, s_2, \ldots, s_n\}$ and a universal set $U$, in which $s_k \subseteq U$ for all $k$. The problem is, divide $S$ ...
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### Is there a name for this modification to the subset sum problem?

Let $S = \{\{x_{1},y_{1},z_{1}\},\{x_{2},y_{2},z_{2}\}, \ldots, \{x_{n},y_{n},z_{n}\}\}$ and a target $t$. Let $S_i$ be the subset list $\{x_{i},y_{i},z_{i}\}$. Find a subset sum that sums to $t$ such ...
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### Subset sum with only two item types

Suppose we have $r$ copies of the integer $a$ and $t$ copies of the integer $b$, and a capacity $C$. We would like to find the maximum sum of the given integers, that is at most $C$. This is a special ...
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### Subset Sum With Interval Integer Target

Define the subset sum with interval integer target problem (SSIITP) as follows: SSIITP Input: A multiset $S = \{a_1, …, a_p\}$ of positive integers $a_i$. An integer $T$. SSIITP Output: True, if ...
1 vote
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### Convert float array to lower or higher integer, find sum(integers) == round(sum(floats)), reducible to subset sum?

You have an array of floats, for example: ...
1 vote
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### In terms of P=?NP, would a P time solution to Subset-Sum have to work in P time when there is no subset that sums to T in the input?

This question is asking for clarification on what P=?NP is asking specifically. I've read the official problem description: here and it seems like P=?NP is primarily concerned with inputs that result ...
325 views

### Complexity of a variant of Subset Sum problem

This is the variant of SSP: Given $n$ positive integer points $a_1, \ldots, a_n$ which are all at most $n$, does there exist a subset $\{a_i\}_{i \in P}$, such that its summation is exactly $n+1$? My ...
1 vote
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### Using FFT as a black box to solve subset sum. How is this done? Given a set of numbers, S, and a target value T

Given a set of numbers, S {s1, s2, ... sn} and a value T, I am looking to determine if any three elements in the set add up to value T. It is valid to have repeats like 2+2+2 would be fine for ...
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### Subset Sum With Interval Target

Define the subset sum with interval target problem (SSITP) as follows: SSITP Input: A multiset $S = \{a_1, …, a_p\}$ of positive integers $a_i$ such that $\sum_{a_i \in S} a_i = T$. SSITP Output: ...
1 vote
148 views

### Does an FPTAS exist for the multiple subset sum problem when m is fixed and c is not a variable?

From Wikipedia Multiple subset sum: The multiple subset sum problem (MSSP) is a generalization of the subset sum problem (SSP): given a multiset $S$ of $n$ integers, and an integer $m$, the goal is to ...
1 vote
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### SUBSET SUM reduction to PARTITION

This is the PARTITION problem: Given a multiset S of positive integers, decide if it can be partitioned into two equal-sum subsets. This is the SUBSET SUM problem: Given a multiset S of integers ...
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### Is there an FPTAS for 3-way number partitioning?

The maximization problem of the 3-way number partitioning reads as follows: given $n$ positive integers, partition them into 3 subsets such that the smallest sum is as large as possible. It is known ...
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### Prove SubsetSum is polyequivalent to SubsetSum with surplus

I'm solving problem 13.17 of What can be computed?, which is asking to prove $\text{SubsetSum} \equiv_{P} \text{SubsetSumWithFives}$. Here is the definition of SubsetSumWithFives. SUBSETSUMWITHFIVES: ...
1 vote
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### Algorithm for computing the sum of symmetric sums (better than $\mathcal{O}(2^N)$ )

Let denote $\mathbf{x} = \{x_1,x_2,...,x_N \}$ with $x_i \in \Bbb R$ for $i=1,...,N$ and $f(\mathbf{x},n)$ be the $n$-th symmetric sum of the set $\mathbf{x}$  f(\mathbf{x},n) = \sum_{\sigma_1,...,\...
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### The subset sum problem is not in P because the question is about lossy compressed data? Why not?

Where is there a gap or error in my reasoning? The subset sum problem deals with a set of n numbers, which is the result of lossy compression of an array r of numbers (r = (2^n)-1). The compression ...
1 vote
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 ...