Questions tagged [subset-sum]
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62
questions
0
votes
1answer
50 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
25 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
53 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
59 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
101 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
1answer
258 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
$$n_1c_1 + n_2c_2 + \dots + ...
0
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0answers
24 views
Subset sum reduction to 3SAT [duplicate]
I have gone over numerous proofs that reduce 3SAT to Subset sum reduction and then claim equivalence, however the other direction is never coherently explained in these proofs. In particular the proof ...
3
votes
1answer
24 views
Solve SUBSET SUM for Reciprocals of Primes
Let $p_1, ..., p_n$ distinct prime numbers with $P = \prod_{i=1}^{n}{p_i}$ and $A=(a_1, ..., a_n)$ with $a_i = P/p_i$.
Problem
Show the SUBSET SUM problem $(A, \alpha)$ can be solved in polynomial (...
1
vote
1answer
104 views
Return the subset with smallest cardinality of an array whose elements sum to at least a given value
Suppose we are given an array $A[1\ldots n]$ and a value $C$.
Is there an algorithm with linear expected runtime that can produce an array that is the subset with smallest cardinality of $A[1\ldots ...
2
votes
1answer
41 views
complexity of a variant of the subset sum problem
We have a set of positive integers $N=\{a_1,...,a_n\}$, we want to select a subset $N'$ of $N$ with maximum total sum of integers such that this sum should not exceed a given integer $B$.
What is the ...
2
votes
1answer
70 views
find maximum sum of xors
we are given an Array
Array size <= 10^4 .
0 <= A[i] <= 15
We need to partition the array into 4 subsets (each subset can have zero or more elements ). Take xor of each subset and sum ...
2
votes
2answers
115 views
Minimizing the iterative sum of pairs of numbers in a list
Given the tuple (list, value):
$$\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
69 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
100 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
114 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
404 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
33 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
34 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
61 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
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1answer
171 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
136 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
93 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
54 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
96 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
45 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
24 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
49 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
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0answers
190 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
265 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
14 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:
...
2
votes
3answers
484 views
N subsets with a given sum?
How to efficiently Ā¹ā¾ choose from a set of numbers $S$, a given number $n$ of disjoint subsets, each with a given sum $K$ of chosen elements?
Ā¹ā¾ Not as in $P$, I just want something smarter than $O(n^...
1
vote
0answers
61 views
Find a partition of multiset of binomial coefficients with constriants
Given the multiset $S$ where the elements are defined by the binomial coefficient ${n \choose k}$ where $n \in \mathbb{N}$ and $
0\leq k \leq n$ find the partition $P$ of $S$ such that the sum of ...
1
vote
1answer
150 views
Reducing subset sum to even subset sum
I'm trying to learn reduction. I have this problem called "even subset sum" that's very similar to subset sum. It's the same problem as as subset sum except that the only numbers allowed are even ...
1
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0answers
104 views
Subset sum problem with an extra condition [closed]
A sub-set sum problem is described that giving a set of integers $S$ and a constant $C$, find a subset $s$ of $S$ so that the sum of integers in $s$ is maximum but not greater than $C$.
$$maximize\...
2
votes
1answer
984 views
Proof of NP Completeness of set-partition problem
I have reduced subset sum problem to set partition problem but do not know whether it is correct and so I need your help.
MY METHOD
In subset sum problem we have to find a subset $S_1$ of set $S$ so ...
1
vote
1answer
58 views
Subset sum problem, given that a valid subset exists
I have a problem at work. I need to find a subset of a set of positive integers that sums to a certain value. I know there is a subset but I need to find it. Is this new problem the same as the ...
1
vote
1answer
150 views
Subset sum exponential solution - how does the sorting work?
The wiki for the subset sum problem found here it states that you take the list of N elements and split it into two lists of N/2 elements. You then generate all the subsets for each list (each having ...
1
vote
1answer
47 views
How to find sets of polynomially bounded numbers whose subset sums are different?
Let $n$ be any positibe integer and set $N=\{1,\dots,n\}$. Now select two arbitrary but different subsets of $N$, say $S$ and $S'$. We are interested in finding a function $\pi(A)=\sum_{i\in A}a_i$ ...
1
vote
1answer
147 views
Is this an $NP$-complete problem: Product-2-Partition
I want to prove the NP-hardness of some problem P in scheduling theory. I was trying with Partition, 3-Partition and Subset product, But neither was successful.
Now, I can reduce a problem, say ...
2
votes
1answer
488 views
Maximum Equal Sum K Subsequences
Given an array we need to find maximum equal sum $K$ subsequences, i.e. we want the sum to be maximized such that there are exactly $K$ non-overlapping subsequences each having the same sum.
Example:
...
2
votes
3answers
3k 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 ...
1
vote
2answers
1k views
Understanding algorithm for maximum sum of non-consecutive elements
There is a well-known problem in CS of finding the maximum sum of non-consecutive integers in a list. There is even an SO post about how to solve it:
https://stackoverflow.com/questions/4487438/...
1
vote
1answer
29 views
Finding the maximum possible size of S, where S is a set of pairwise-disjoint subsets of the list, and every element of S sums to k
Say I had a list of numbers in the range of 1-20 for example: [5, 16, 17, 3, 2, 14, 4, 9, 11, 19], and an integer k (let's say k = 40)
How would I find the maximum possible size of S, where S is a ...
2
votes
1answer
53 views
Can the subset sum problem work with an undefined set?
To give a brief outline of my actual problem, I'm attempting to identify possible impurities in proteins. I have a theoretical mass and an experimental mass, and the difference between the two would ...
2
votes
1answer
241 views
Subset sum into a consecutive range vs. standard subset sum
The following problems live in integer domain.
I want to find a subset of $\{x_0, x_1, \ldots, x_{n-1}\}$ such that the subset's elements sum to any number in a prescribed interval $[X,X+k]$, $k\geq ...
2
votes
0answers
1k views
k-way set partition problem--dynamic programming solution
I was reading up on the set partition problem on this site of Wikipedia: https://en.m.wikipedia.org/wiki/Partition_problem
Among other things, they present a DP approach to solving the equal subset ...
0
votes
2answers
117 views
Determine if an array is divisible in pairs of equal sum
I'm currently studying the time complexity of the solution provided by my teacher to this problem, and I can't understand the logic:
Let A be an unordered array of positive natural numbers. Write an ...
2
votes
0answers
218 views
Modified Subset Sum Problem
Given an array of $n$ integers $A$, and some value $m$, determine if it is possible, by using certain amounts of each element, to get a total sum equal to $m$. Consider that you can use any amount of ...
