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Decision problem in computer science
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers
Subset_sum_problem
Mathematical problem
by David J. Grynkiewicz in 2005). Barycentric-sum problem Davenport constant Subset sum problem Zero-sum Ramsey theory Erdős, Paul; Ginzburg, A.; Ziv,
Zero-sum_problem
Problem in combinatorial optimization
knapsack problem is often used to refer specifically to the subset sum problem. The subset sum problem is one of Karp's 21 NP-complete problems. Knapsack
Knapsack_problem
Mathematical optimization problem
multiple subset sum problem is an optimization problem in computer science and operations research. It is a generalization of the subset sum problem. The
Multiple_subset_sum
NP-complete problem in computer science
The partition problem is a special case of two related problems: In the subset sum problem, the goal is to find a subset of S whose sum is a certain target
Partition_problem
Complexity class
polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if H is NP-hard, then
NP-hardness
Problem in computer science
maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within
Maximum_subarray_problem
Set whose elements all belong to another set
spacePages displaying short descriptions of redirect targets Subset sum problem – Decision problem in computer science Subsumptive containment – System of
Subset
Form of public key cryptography
key for decryption. It is based on the subset sum problem (a special case of the knapsack problem). The problem is as follows: given a set of integers
Merkle–Hellman knapsack cryptosystem
Merkle–Hellman_knapsack_cryptosystem
Mathematical result on systems of linear equations
subset sum problem can be reduced to the problem of computing the required partition C1, C2, ..., Cn of columns: Given an input set S for the subset sum
Rado's theorem (Ramsey theory)
Rado's_theorem_(Ramsey_theory)
Complexity class used to classify decision problems
given subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore
NP_(complexity)
Mathematical problem in number theory
theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural
Waring's_problem
Topics referred to by the same term
fibered sum in category theory QCD sum rules, in quantum field theory Riemann sum, in calculus Rule of sum, in combinatorics Subset sum problem, in cryptography
Sum
scheduling Partition problem Quadratic assignment problem Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem Variations on the
List_of_NP-complete_problems
Classical problem in combinatorics
collection, referred to as S, of a given m subsets whose union equals the universe, the set cover problem is to identify a smallest sub-collection of
Set_cover_problem
Set disjoint from its sumset with itself
number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation
Sum-free_set
weakly NP-complete problem is the subset sum problem. The related term strongly NP-complete (or unary NP-complete) refers to those problems that remain NP-complete
Weak_NP-completeness
Computer that uses photons or light waves
attacked in this way was the Hamiltonian path problem. The simplest problem is the subset sum problem. An optical device solving an instance with four
Optical_computing
One-way cryptographic tool
For example, an early suggestion was to use schemes based on the subset sum problem. This turned out rather quickly to be unsuitable. As of 2004[update]
Trapdoor_function
Mathematical problem involving optimal stopping theory
known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. Its solution is also
Secretary_problem
Prime numbers that occupy prime-numbered positions
on calculations involving the subset sum problem) to show that every integer greater than 96 may be represented as a sum of distinct super-prime numbers
Super-prime
Mathematical riddle
polynomial time problem. If the capacity m is arbitrary, the problem is known to be NP-hard. Coin problem Knapsack problem Subset sum problem Jeffrey Shallit
Postage_stamp_problem
NP-hard problem in combinatorial optimization
tours, each visiting only a subset of the vertices; arguably, it is this global requirement that makes TSP a hard problem. The MTZ and DFJ formulations
Travelling_salesman_problem
Type of vector space in math
{\displaystyle \sum _{b\in B}\left|x(b)\right|^{2}=\sup \sum _{n=1}^{N}\left|x(b_{n})\right|^{2}} the supremum being taken over all finite subsets of B. It follows
Hilbert_space
partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are as similar as possible
Multiway_number_partitioning
Cryptographic hash function
designed such that the problem of finding collisions should be reducible to a known and hard mathematical problem (the subset sum problem). It means that if
Elliptic_curve_only_hash
Unsolved problem in computer science
solve SUBSET-SUM in polynomial time is b bits long, the above algorithm will try at least 2b − 1 other programs first. A decision problem is a problem that
P_versus_NP_problem
Numerical optimization process
A sum-of-squares optimization program is an optimization problem with a linear cost function and constraints that certain polynomials constructed from
Sum-of-squares_optimization
knapsack problem: If for each item the profit and weight are equal, we get the subset sum problem (often the corresponding decision problem is given instead):
List_of_knapsack_problems
Optimization problem
VRPP are: Orienteering Problem (OP), where a price constraint (or time constraint) is given and the goal is to maximize the sum of collected profits while
Vehicle_routing_problem
Complexity class
Knapsack problem Hamiltonian path problem Travelling salesman problem (decision version) Subgraph isomorphism problem Subset sum problem Clique problem Vertex
NP-completeness
Least-weight tree connecting graph vertices
theory, a minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects
Minimum_spanning_tree
In number theory, a limitation of sieve theory
problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was
Parity_problem
Process in machine learning and statistics
particular type of model or typical problem. Filter methods use a proxy measure instead of the error rate to score a feature subset. This measure is chosen to
Feature_selection
Gluing graphs at complete subgraphs
or more disconnected components, and no subset of the cycle has the same property) are exactly the clique-sums of cliques and maximal planar graphs, again
Clique-sum
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set. The covering problem of Rado:
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topics referred to by the same term
Port Supply-side platform, in online advertising Subset sum problem, an NP-complete decision problem Six-state protocol, a quantum key distribution protocol
SSP
Form of encryption that allows computation on ciphertexts
assumed hardness of two problems: certain worst-case problems over ideal lattices, and the sparse (or low-weight) subset sum problem. Gentry's Ph.D. thesis
Homomorphic_encryption
Sequence of integers
represented as a sum of at most three prime numbers. However, finding such a representation could involve solving instances of the subset sum problem, which is
Pillai_sequence
numbers. This algorithm can be generalized to a solution for the subset sum problem. Korf, Richard E. (2009). Multi-Way Number Partitioning (PDF). IJCAI
Pseudopolynomial time number partitioning
Pseudopolynomial_time_number_partitioning
is the sum of any subset of the previous ones. The sum of the reciprocals of the numbers in any sum-free sequence is less than 2.8570. The sum of the
List_of_sums_of_reciprocals
Problem in graph theory
Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset S of the vertex set such that the number
Maximum_cut
Collection of possible string theory vacua
vacua, the problem of finding one with a sufficiently small cosmological constant is NP complete. This is a version of the subset sum problem. A possible
String_theory_landscape
Rawlsian decision rule for social choice
several contexts: Division of a single homogeneous resource; Fair subset sum problem; Egalitarian cake-cutting; Egalitarian item allocation; Egalitarian
Egalitarian_rule
Addition of several numbers or other values
f d μ {\displaystyle \sum _{k\mathop {=} a}^{b}f(k)=\int _{[a,b]}f\,d\mu } where [ a , b ] {\displaystyle [a,b]} is the subset of the integers from a
Summation
Set of computational problems stated by Richard Karp (1973)
Coloring Problem) Clique cover Exact cover Hitting set Steiner tree 3-dimensional matching Knapsack (Karp's definition of Knapsack is closer to Subset sum) Job
Karp's 21 NP-complete problems
Karp's_21_NP-complete_problems
Optimization problem in computer science
"Lattice basis reduction: Improved practical algorithms and solving subset sum problems" (PDF). Mathematical Programming. 66 (1–3): 181–199. doi:10.1007/bf01581144
Lattice_problem
which the sums in the two subsets are equal; see problem [SP12]. There are many algorithms that aim to find a balanced partition in which the sum is as nearly-equal
Balanced_number_partitioning
no such subset). This optimization problem is similar to the decision problem SUBSET-SUM. Given a set of integers, SUBSET-SUM is the problem of finding
NP-equivalent
Sums vector sets A and B by adding each vector in A to each vector in B
{\displaystyle A+B=\{a+b\mid a\in A,\ b\in B\}.} In geometry, the Minkowski sum of two subsets A and B of a Euclidean space is the set of the points whose position
Minkowski_addition
Computational problem used in cryptography
a_{n-1})\mid \sum _{i=0}^{n-1}a_{i}x^{i}\in I\right\}\subset \mathbb {Z} ^{n}.} Theorem: L ⊂ Z n {\displaystyle {\mathfrak {L}}\subset \mathbb {Z} ^{n}}
Short integer solution problem
Short_integer_solution_problem
Function used in computer cryptography
of random linear codes, the hardness of certain lattice problems, and the subset sum problem (Naccache–Stern knapsack cryptosystem). There is an explicit
One-way_function
Topics referred to by the same term
Rhine-Erft district, North Rhine-Westphalia the knapsack problem, a math problem the subset sum problem, a special case of the above Naccache-Stern knapsack
Knapsack_(disambiguation)
Complexity class
constraints?" For example: Are there any subsets of a list of integers that add up to zero? (subset sum problem) Are there any Hamiltonian cycles in a given
♯P
knapsack problem. Parametric knapsack problem. Symmetric quadratic knapsack problem. Count-subset-sum (#SubsetSum) - finding the number of distinct subsets with
Fully polynomial-time approximation scheme
Fully_polynomial-time_approximation_scheme
Problem in mathematical sociology
{\displaystyle \sum _{i=1}^{n}c_{i}\geq E} , that is, the total claims are (weakly) larger than the estate. A division rule is a function that maps a problem instance
Bankruptcy_problem
Theorem in arithmetic combinatorics
finite field sum-product problem in F {\displaystyle \mathbb {F} } typically fall into two categories: when A ⊂ F {\displaystyle A\subset \mathbb {F} }
Erdős–Szemerédi_theorem
Process of calculating the causal factors that produced a set of observations
classical linear inverse problem in exploration seismology: the amplitude recorded at one time for a given source-receiver pair is the sum of contributions arising
Inverse_problem
Payments from a bank account exceeding the balance
smallest will maximize the number (but not necessarily value—see subset sum problem) of overdrawn debits on a customer's account. This situation can arise
Overdraft
Arithmetic operation
exponent) for bn is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient
Exponentiation
efficiently. The brute-force algorithm to solve this problem is to identify all possible subsets of the items without exceeding the capacity and select
Quadratic_knapsack_problem
Infinite sum
finite subset A 0 {\displaystyle A_{0}} of I {\displaystyle I} such that S − ∑ i ∈ A a i ∈ V for every finite superset A ⊇ A 0 . {\displaystyle S-\sum _{i\in
Series_(mathematics)
Type of computational problem
is conflict-free covering. In this problem: There is a set O of m objects, and a conflict-graph GO on O. A subset Q of O is called conflict-free if it
Covering_problems
In geometry, set whose intersection with every line is a single line segment
because the empty set annihilates every other subset: For every subset S of a vector space, its sum with the empty set is empty: S + ∅ = ∅ {\displaystyle
Convex_set
Problem in computational complexity theory
of n elements each, are there n² distinct x + y for x ∈ X, y ∈ Y? Subset sum problem Grønlund & Pettie 2018. Freund 2017. Gold & Sharir 2017. Chan 2020
3SUM
Set of objects whose state must satisfy limits
inference Eight queens puzzle Map coloring problem Maximum cut problem Sudoku, crosswords, futoshiki, Kakuro (Cross Sums), Numbrix/Hidato, Zebra Puzzle, and
Constraint satisfaction problem
Constraint_satisfaction_problem
Unsolved problem in mathematics Which finite groups are BI-groups? More unsolved problems in mathematics Babai's problem is a problem in algebraic graph
Babai's_problem
Concept in mathematical optimization
{x} \in \mathbf {X} } is the optimization variable chosen from a convex subset of R n {\displaystyle \mathbb {R} ^{n}} , f {\displaystyle f} is the objective
Karush–Kuhn–Tucker_conditions
Subset of a graph's vertices, including at least one endpoint of every edge
solution by selecting the subset of vertices whose variables are nonzero. The decision variant of the vertex cover problem is NP-complete, which means
Vertex_cover
satisfiability problem Subset sum problem 3SUM Traveling salesman problem Vertex cover problem One-way function Set cover problem Independent set problem Probabilistic
List of computability and complexity topics
List_of_computability_and_complexity_topics
18 mathematical problems stated in 1998
Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list
Smale's_problems
On solvability of Diophantine equations
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge
Hilbert's_tenth_problem
Strongly NP-complete problem in computer science
unary. The 3-partition problem is similar to the partition problem, in which the goal is to partition S into two subsets with equal sum, and the multiway number
3-partition_problem
Subset of a graph's edges
cover – the edge cover problem is a special case of the set cover problem: the elements of the universe are vertices, and each subset covers exactly two elements
Edge_cover
finite simple groups Herzog–Schönheim conjecture Subset sum problem Whitehead problem Word problem for groups Amenable group Capable group Commensurability
List_of_group_theory_topics
Task of computing complete subgraphs
In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete
Clique_problem
Study of structures where a subset must sum to zero
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial
Zero-sum_Ramsey_theory
Divergent sum of positive unit fractions
infinite series formed by summing all positive unit fractions: ∑ i = 1 ∞ 1 i = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ . {\displaystyle \sum _{i=1}^{\infty }{\frac
Harmonic_series_(mathematics)
Term for issues in the First World
has been called a subset of the fallacy of relative privation and is also used to acknowledge gratefulness for not having worse problems, such as those in
First_World_problem
Every graph has evenly many odd vertices
Seven Bridges of Königsberg Problem, which subsequently formalized Eulerian Tours, other applications of the degree sum formula include proofs of certain
Handshaking_lemma
space of dimension d, the problem is to determine, given a finite subset of vectors S and a convex subset A, the number of subsets of S whose summation is
Littlewood–Offord_problem
Progression-free set of numbers
based hardness of the subset sum problem. These sets can also be applied in recreational mathematics to a mathematical chess problem of placing as few queens
Salem–Spencer_set
Counting technique in combinatorics
sum _{k=1}^{n}\left((-1)^{k-1}\sum _{I\subseteq \{1,\ldots ,n\} \atop |I|=k}\mathbb {P} (A_{I})\right),} where the last sum runs over all subsets I
Inclusion–exclusion_principle
Matroid theory
matroid-constrainted partitioning problems are: The (max,sum) objective is the maximum over all subsets, of the total weight in the subset. When the items represent
Matroid-constrained number partitioning
Matroid-constrained_number_partitioning
Category of routing problem minimizing total distance and time
the rural postman problem (RPP) requires a subset of the edges to be traversed with the minimum length cycle. Arc routing problems impact strategic, tactical
Arc_routing
Mathematical models of strategic interactions
science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the
Game_theory
German theoretical computer scientist
complexity and a near-linear pseudopolynomial time algorithm for the subset sum problem. Furthermore, he was appointed as a professor at Saarland University
Karl_Bringmann
Field of artificial intelligence around Go computer programs
much worse than computers at solving, for example, instances of the subset sum problem. AlphaGo, a machine learning program by Google DeepMind, and the first
Computer_Go
Infinite series that is not convergent
results. A major problem was Euler's idea that any divergent series should have a natural sum, without first defining what is meant by the sum of a divergent
Divergent_series
assistance of ChatGPT 5.2 and the Aristotle Lean API. Erdős problem 347 on the subset sums of sequences with ratio limit 2 was proved in January 2026 by
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
American computer scientist
theory, and on improved exponential time exact algorithms for the subset sum problem, among his many other research results. Concepts in Discrete Mathematics
Sartaj_Sahni
Theorem on the existence of finite sets of integers >1 whose reciprocals sum to 1
there is a finite monochromatic subset S {\displaystyle S} of these integers such that. ∑ n ∈ S 1 n = 1. {\displaystyle \sum _{n\in S}{\frac {1}{n}}=1.} In
Erdős–Graham_problem
Problem in computer science
m } {\displaystyle S=\{S_{1},S_{2},\ldots ,S_{m}\}} . Objective: Find a subset S ′ ⊆ S {\displaystyle S'\subseteq S} of sets, such that | S ′ | ≤ k {\displaystyle
Maximum_coverage_problem
Mathematical set containing no elements
∅ . {\displaystyle V=\varnothing .} By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ {\displaystyle
Empty_set
Finite sum of distinct unit fractions
are partitioned into finitely many subsets, then one of the subsets has a finite subset of itself whose reciprocals sum to one. That is, for every r > 0
Egyptian_fraction
Set of edges without common vertices
This reduces the problem of finding a large matching to finding augmented paths. In any graph without isolated vertices, the sum of the matching number
Matching_(graph_theory)
Number that is abundant but not semiperfect
the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to
Weird_number
x , y , z ) {\displaystyle (x,y,z)} in the subset x + y + z = b {\displaystyle x+y+z=b} holds. This problem is labeled as [SP16] in. Take X = { 3 , 4
Numerical 3-dimensional matching
Numerical_3-dimensional_matching
Computational problem in graph theory
u t . {\displaystyle |f|=\sum _{v:\ (s,v)\in E}f_{sv}=\sum _{u:\ (u,t)\in E}f_{ut}.} Definition. The maximum flow problem is to route as much flow as
Maximum_flow_problem
Graphical aid for deriving some concepts in combinatorics
number of k-tuples of positive integers whose sum is n is equal to the number of (k − 1)-element subsets of a set with n − 1 elements. For example, if
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Female
English
Short form of English Susan, SUE means "lily."
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Boy/Male
Hindu, Indian, Marathi
Fragrance; Flower; Sum; Total
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Female
Thai/Siamese
Thai name SOM means "orange (the fruit)."
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Girl/Female
Indian, Kannada, Korean, Telugu
The Sun; Obedient
Boy/Male
Sikh
Sun, Godly, Warrior, Brave, A musical note
Boy/Male
Hebrew American
Sun child; bright sun.
Boy/Male
Hindu, Indian, Tamil
Sun; Sunset
Boy/Male
Australian, Biblical, Danish, German, Swedish
Mame; Renown; Sun Child; Little Sun
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Surname or Lastname
English (Sussex)
English (Sussex) : unexplained.
Surname or Lastname
English
English : regional name for someone from the county of Sussex, named ‘(territory of) the South Saxons’, from Old English sūth + Seaxe.
Girl/Female
Australian, Danish, Swedish
Sun
Boy/Male
American, Arabic, British, Czechoslovakian, Danish, Dutch, English, Finnish, French, German, Hawaiian, Hebrew, Hindu, Indian, Iranian, Jamaican, Malayalam, Parsi, Sanskrit, Swedish, Tamil, Telugu, Urdu
Told by God; God has Listen; To Hear; Sun; His Name is God; Sun Child; Little Sun; Strong Person; Heard of God; God; Good Person
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
Girl/Female
Tamil
Good work, The Goddess who appreciates good deeds
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Lord of Peace
Girl/Female
Gaelic
Thin.
Girl/Female
Tamil
Best, The Goddess who is above the five elements
Boy/Male
Tamil
Shashanka | ஷஷாஂகா
Other name of Lord Shiva, The Moon
Girl/Female
Arabic, Muslim
Magnificence; Splendour
Girl/Female
Hindu, Indian, Marathi
Born in Water; Goddess Lakshmi
Girl/Female
Hindi
Ford.
Boy/Male
Indian
Beautiful, Magnificent, Shining
Boy/Male
Muslim
Glorious, Magnificent, Splendid, Brilliant, Shining
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
SUBSET SUM-PROBLEM
n.
The aggregate of two or more numbers, magnitudes, quantities, or particulars; the amount or whole of any number of individuals or particulars added together; as, the sum of 5 and 7 is 12.
n.
See Gum tree, below.
imp. & p. p.
of Sublet
v. i.
To exude or from gum; to become gummy.
n.
A vegetable secretion of many trees or plants that hardens when it exudes, but is soluble in water; as, gum arabic; gum tragacanth; the gum of the cherry tree. Also, with less propriety, exudations that are not soluble in water; as, gum copal and gum sandarac, which are really resins.
v. t.
To subjoin; to subnect.
v. t.
To smear with gum; to close with gum; to unite or stiffen by gum or a gumlike substance; to make sticky with a gumlike substance.
v. t.
To expose to the sun's rays; to warm or dry in the sun; as, to sun cloth; to sun grain.
n.
A russet color; a pigment of a russet color.
n.
The setting of the sun; sunset.
n.
An apple, or a pear, of a russet color; as, the English russet, and the Roxbury russet.
v. t.
To submit; to make accountable.
n.
The principal points or thoughts when viewed together; the amount; the substance; compendium; as, this is the sum of all the evidence in the case; this is the sum and substance of his objections.
n.
The direct light or warmth of the sun; sunshine.
a.
Old-fashioned; queer; odd; as, a rum idea; a rum fellow.
n.
A quantity of money or currency; any amount, indefinitely; as, a sum of money; a small sum, or a large sum.
v. i.
To form a scum; to become covered with scum. Also used figuratively.