Welcome to the Complexity Garden, the botanical companion to the Complexity Zoo. This field guide lists rigorously defined computational problems, together with their relations to complexity classes, their best algorithmic results, and their relations to other problems. There are now 39 problems and (one can only hope) counting. The initial members either have or might have some exotic property related to speedup (such as O-speedup, monotone-nonmonotone gap, or P-nonuniformity), or provide a canonical example of a complexity class (i.e. NP-complete). However, all additions are welcome, and it is hoped that this will grow into a comprehensive collection.
Gardener: Hunter Monroe (volunteers welcome)
To create a new problem, click on the edit link of the problem before or after the one that you want to add and copy the format, and save. The preferred format of each problem is as follows: description of the problem, relations to complexity classes, their best algorithmic results (upper-bounds and lower-bounds e.g. see the Set Cover problem), memberships and properties, and relations to other problems. Then, add the problem to the table of contents and increment the total number of problems. After this, you can use the side edit links to edit the individual sections. For more on using the wiki language, see our simple wiki help page.
If your problem is not on the list and you suspect that it might be an open problem you can check the Open Problem Garden.
Table of Contents
Algebraic problems: Ring Automorphism - Ring Isomorphism - Approximate Shortest Lattice Vector - Group Nonmembership - Integer Factorization - Integer Factor ≤ k - Integer Multiplication - Matrix Multiplication - Permanent - Square Root mod n
Sets and partitions: Set Cover
Uncategorized problems: Boolean Matrix Multiplication - 3SUM - Boolean Convolution - Boolean Sorting - Discrete Logarithm - Equality - k-Local Hamiltonians - k-Round Sorting - Linear Programming - Majority - Parity - Short Implicant - Stochastic Games - Tautology - Threshold(k)
: Count perfect matchings
The #Perfect Matching problem is to count the number of perfect matchings in a bipartite graph. It is the counting version of Perfect Matching.
Memberships: ∈ #P-complete
Related Problems: #Perfect Matching is equivalent to Permanent.
: Count satisfying truth assignments
This is the counting version of SAT: given a Boolean formula, compute how many satisfying truth assignments it has.
Memberships: ∈ #P-complete
: Weighted SAT
Memberships: ∈ #WT
3SUM: Do there exist members of a list satisfying ?
Given a list of integers, do there exist elements such that ? This problem is important enough to computational geometry that there is defined a class of problems to which it is reducible: 3SUM-hard.
Approximate Shortest Lattice Vector: Does the shortest vector exceed kn?
Given a lattice L in Zn and an integer n, does the shortest vector of L have (Euclidean) length at most kn?
Boolean Convolution: Convolution of two n-bit integers.
Compute the convolution of two n-bit integers in binary notation , where the multiplicands are respectively the i-1 and j-1 bits of the two integer inputs, and k is the k-1 bit of the output. It is a Boolean function with n input bits and 2n-1 output bits.
It has monotone circuit complexity (Weiss) and nonmonotone circuit complexity (Furer [Fur07]). means iteratively taking of n until the result is less than 2. Nonmonotone circuits use the discrete Fourier transform (Wegener [Weg87]).
Boolean Matrix Multiplication: Monotone product of Boolean matrices
Boolean Matrix Multiplication is a Boolean function with 2n2 input bits (two nxn matrices) and n2 output bits (an nxn matrix). The function is defined using the usual row-column rule, but using OR instead of binary addition.
Boolean Sorting: Sort n input bits.
A Boolean function with n inputs and n outputs that puts the 0s before the 1s. For example, 101 → 011 and 11000 → 00011.
It has monotone circuit complexity (Lamagna and Savage [LS74]) and nonmonotone circuit complexity (Muller and Preparata [MP75]). The nonmonotone circuits use binary rather than unary addition to count the 0 inputs.
Clique-Like: Tardos' Clique-like approximation
Clique-Like is a Boolean function with inputs (an adjacency matrix) and one output.
Constraint Satisfaction Problem: Generalized version of satisfiability problems
Given two sets of relations , where is the instance (not to be confused with an instance of CSP) and is the template, over the same vocabulary, where the vocabulary defines the names and arities of allowed relations, is there a mapping such that for all relations , ?
See Also: [FV93].
Discrete Logarithm: Reverse exponentiation
The discrete logarithm problem is to solve for x in the equation ax = b in some number-theoretic abelian group, typically either the group of units of a finite field or an elliptic curve over a finite field. Like the related Integer Factorization, the fastest classical algorithm is the number field sieve, with heuristic time complexity . Also like Integer Factorization, Shor's algorithm solves Discrete Logarithm in quantum polynomial time. In fact, Shor's algorithm solves Discrete Logarithm even in a black-box abelian group, provided that group elements have unique names.
Equality: Are two strings equal?
If Alice has a string and Bob has a string , define EQUALITY(x, y) = 1 if and only if x = y.
Graph Automorphism: Is there a nontrivial automorphism?
Graph Automorphism is equivalent to Graph Isomorphism in the case where the two graphs are identical.
Graph Isomorphism: Are two graphs isomorphic?
Given two graphs, are they isomorphic, i.e., is there a bijection between their vertices which preserves edges?
Luks showed that Graph Isomorphism for bounded-valence graphs is in P non-uniformly (the exponent of algorithm's running time depends on the bound). Combining Luks' algorithm with a trick due to Zemlyachenko yields a time complexity upper bound of for graphs with v vertices. However, some practical Graph Isomorphism algorithms, such as NAUTY, seem to run much faster than this rigorous upper bound.
Like Perfect Matching, it is not known to be complete for a natural complexity class.
Group Non-Membership: Is an element of G a member of a subgroup H?
Defined by [Wat00]. Let be a group, whose elements are represented by polynomial-size strings. We're given a "black box" that correctly multiplies and inverts elements of . Then given elements and , we can asked to find if (the subgroup generated by ).
Hamiltonian circuit: Exists one Hamiltonian circuit?
Given a Graph , exists one Hamiltonian circuit in ?
Integer Factorization: Find an integer's prime factors
Algorithms: The fastest known algorithm for integer factorization is the number field sieve. It has heuristic randomized time complexity for inputs with n digits.
On the other hand, Shor's algorithm famously solves Integer Factorization in quantum polynomial time.
Integer Factor ≤ k : Is there a factor ≤ k?
With an upper bound on the desired prime factor, this problem is subtly different from Integer Factorization.
Algorithms: The fastest algorithm is either the number field sieve or Lenstra's elliptic curve method, depending on the relative size of n and k. Lenstra's algorithm has heuristic randomized time complexity if n and k have n and k digits, respectively.
Integer Multiplication: Product of two n-bit integers
k-Local Hamiltonians: Find the smallest eigenvalue of m k-local Hamiltonians acting on n qubits.
Given an n-qubit Hilbert space, as well as a collection H1,...,Hm of Hamiltonians (i.e. Hermitian positive semidefinite matrices), each of which acts on at most k qubits of the space. Also given real numbers a,b such that . Decide whether the smallest eigenvalue of is less than a or greater than b, promised that one of these is the case.
- QMA-complete for k = 5 [KSV02].
- QMA-complete for k = 3 [KR03].
- QMA-complete for k = 2, under the assumption that P≠QMA [KKR04].
k-Round Sorting: Sort inputs in ≤ k rounds
There are k-round sorting networks not known to be constructible in deterministic polynomial time which outperform the best-known networks constructible in deterministic polynomial time [GGK03].
Linear Programming: Maximize a linear function with linear constraints
The Linear Programming problem is to maximize a linear function in a convex polytope. It was famously shown to be in FP by Leonid Khachiyan by means of the ellipsoid method. The main algorithms used in practice are simplex and interior-point methods.
Memberships: ∈ FP
Majority: Are most inputs 1?
Majority is a Boolean function with n input bits and 1 output bit. The output is 1 if the majority of input bits are 1. Examples: 001 → 0, 1100 → 1.
Matrix Multiplication: Multiply two matrices
Multiply two dense matrices over a field . Matrix Multiplication has O-speedup among Strassen-type bilinear algorithms [CW82]. Determining the minimal number of multiplications needed to compute a bilinear form (of which Matrix Multiplication is one) is NP-complete ([Has90]). This suggests that Matrix Multiplication is P-nonuniform over bilinear algorithms if NPcoNP.
Parity: Is the number of 1 inputs odd?
Parity is a Boolean function with n inputs and 1 output. The output is 1 if the number of 1 inputs is odd. Examples: 001 → 1, 11011 → 0.
Related Problems: Equals XOR. See the Complexity Zoology entry.
Perfect Matching: Is there a perfect matching?
Perfect Matching is a Boolean function with n2 inputs (an adjacency matrix) describing a bipartite graph and one output which is 1 if the graph has a perfect matching.
Perfect Matching reduces to Linear Programming, and is therefore in P, although specialized algorithms are also known.
It has monotone circuit complexity of (Razborov [Raz85b]) but polynomial nonmonotone circuit complexity.
Permanent: What is a 0-1 matrix's permanent
The permanent of an n-by-n 0-1 matrix (ai,j) is defined as where σ ranges over all permutations of the numbers 1, 2, ..., n. The value of the permanent is equivalent to #Perfect Matching
Memberships: ∈ #P
Ring Automorphism: Does a ring have a non-trivial automorphism?
A special case of Ring Isomorphism where the two rings investigated are the same, and where we are looking for non-trivial automorphisms.
Ring Isomorphism: Are two rings isomorphic?
Given two rings, are they isomorphic to each other? The counting version of the problem, #RI, asks how many different isomorphisms exist between the two rings.
QBF: Quantified Boolean Formula
Given a Boolean formula with universal and existential quantifiers, is it true? Quantified Boolean Formula (QBF) is the canonical PSPACE-complete problem.
Note that [Pap94] calls this problem QSAT to emphasize its relationship to SAT. In particular, any instance of QBF can be written as , where is a Boolean formula as in SAT, and where is either a universal or existential qualifier, depending on whether is even or odd. This characterization lends itself well as a candidate for reductions from two-player games.
Memberships: ∈ PSPACE
Quantum k-SAT: Quantum Analog for SAT
Given a set of measurements, each i of which involves no more than k qubits and accepts with probability Pi, and given the promise that either there exists a state such that the number of accepting measurements is "very large," or that for all states, the number of accepting measurements is very small. We are asked which of the two promise conditions holds.
Properties: QMA-complete for k = 3.
See also: Democritus Lecture 13.
SAT: Is there a satisfying truth assignment?
The SAT problem is to decide whether a given Boolean formula has any satisfying truth assignments. SAT is in NP, since a "yes" answer can be proved by just exhibiting a satisfying assignment.
Memberships: ∈ NP-complete
Set Cover: Cover a set with some of its subsets
Given a universe and a family of subsets of , a cover is a subfamily of sets whose union is . In the set covering decision problem, the input is a pair and an integer ; the question is whether there is a set covering of size or less. In the set covering optimization problem, the input is a pair , and the task is to find a set covering that uses the fewest sets.
Algorithms: A simple greedy algorithm yields an -approximation (and hence, -approximation)where is the size of largest subset in and is th Harmonic number.
Set covering cannot be approximated in polynomial time to within a factor of , unless NP has quasi-polynomial time algorithms [Fie98]. In addition, Set covering cannot be approximated in polynomial time to within a factor of , where is a constant, unless PNP. The largest value of is proved in [AMS06].
= -SAT: Satisfiability with clauses of length
For some natural number , an instance of -SAT is an instance of SAT in conjunctive normal form (CNF) where all clauses are the logical OR of Boolean variables.
Short Implicant: Is there a short implicant?
Given a DNF expression Φ, is there a conjunction of at most k negated or non-negated literals that implies Φ?
Stochastic Games: Is there a first player advantage?
White, Black, and Nature alternate moving a token on the edges of a directed graph. Nature's moves are random. Given a graph, start and target nodes for the token, does White have a strategy which will make the token reach the target with probability ≥ 1/2?
Tautology: Are all truth assignments satisfying?
Tautology is coNP-complete.
Memberships: ∈ coNP
Square Root mod n: Find square roots mod n
The Square Root mod n problem is to solve for x in the equation x2 = a mod N. Rabin noted that Integer Factorization BPP-reduces to Square Root mod n and vice-versa; the problems are equivalent.
Memberships: ∈ FP
Properties: Is a one-way function?
Threshold(k): Are ≥ k inputs 1?
Threshold(k) is a Boolean function with n input bits and one output bit.
Memberships: ∈ P
Unique k-SAT: k-SAT with uniqueness promise.
Intuitively, adding this promise restricts us to only the hardest instances, as the more satisfying assignments there are, the easier it should be to find one. This intuition is justified by [CIK+03], where it is also shown that if Unique -SAT can be solved in deterministic time for all , then so can -SAT for all .