computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...

, problems that are in the

complexity class In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms o ...

NP but are neither in the class P nor

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...

are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by

Richard E. Ladner Richard Emil Ladner is an American computer scientist known for his contributions to both theoretical computer science and assistive technology. Ladner is a professor emeritus at the University of Washington. Biography Richard Ladner was born as ...

, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since it is also true that if NPI problems exist, then P ≠ NP, it follows that P = NP if and only if NPI is empty. Under the assumption that P ≠ NP, Ladner explicitly constructs a problem in NPI, although this problem is artificial and otherwise uninteresting. It is an open question whether any "natural" problem has the same property:

Schaefer's dichotomy theorem In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem states necessary and sufficient conditions under which a finite set ''S'' of relations over the Boolean domain yields polynomial-time or NP-complete prob ...

provides conditions under which classes of constrained Boolean satisfiability problems cannot be in NPI. Some problems that are considered good candidates for being NP-intermediate are the

graph isomorphism problem The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational compl ...

, factoring, and computing the

discrete logarithm In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b ...

List of problems that might be NP-intermediate

Algebra and number theory

* Factoring integers *

Discrete Log Problem In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ...

and others related to cryptographic assumptions * Isomorphism problems: Group isomorphism problem,

Group automorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two gro ...

Ring isomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preservi ...

Ring automorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...

* Linear divisibility: given integers

x

and

y

, does

y

have a divisor congruent to 1 modulo

x

Boolean logic

* IMSAT, the

Boolean satisfiability problem In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisf ...

for "intersecting monotone CNF":

conjunctive normal form In Boolean logic, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. As a cano ...

, with each clause containing only positive or only negative terms, and each positive clause having a variable in common with each negative clause * Minimum Circuit Size Problem * Monotone self-duality: given a CNF formula for a Boolean function, is the function invariant under a transformation that negates all of its variables and then negates the output value?

Computational geometry and computational topology

* Computing the

rotation distance In discrete mathematics and theoretical computer science, the rotation distance between two binary trees with the same number of nodes is the minimum number of tree rotations needed to reconfigure one tree into another. Because of a combinatorial e ...

between two

binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...

s or the flip distance between two triangulations of the same convex polygon * The turnpike problem of reconstructing points on line from their distance multiset * The

cutting stock problem In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing material wasted. It is an optimization problem ...

with a constant number of object lengths * Knot triviality * Finding a simple closed quasigeodesic on a convex polyhedron

Game theory

* Determining winner in

parity game A parity game is played on a colored directed graph, where each node has been colored by a priority – one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph. The own ...

s, in which graph vertices are labeled by which player chooses the next step, and the winner is determined by the parity of the highest-priority vertex reached * Determining the winner for stochastic graph games, in which graph vertices are labeled by which player chooses the next step, or whether it is chosen randomly, and the winner is determined by reaching a designated sink vertex.

Graph algorithms

Graph isomorphism problem The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational compl ...

* Planar minimum bisection * Deciding whether a graph admits a

graceful labeling In graph theory, a graceful labeling of a graph with edges is a labeling of its vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute diff ...

* Recognizing

leaf power In the mathematical area of graph theory, a -leaf power of a tree is a graph whose vertices are the leaves of and whose edges connect pairs of leaves whose distance in is at most . That is, is an induced subgraph of the graph power , induce ...

s and -leaf powers * Recognizing graphs of bounded

clique-width In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be bounded even for dense graphs. It is defined as the minimum n ...

* Finding a

simultaneous embedding Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between ...

with fixed edges

Miscellaneous

* Problems in

TFNP In computational complexity theory, the complexity class TFNP is the class of total function problems which can be solved in nondeterministic polynomial time. That is, it is the class of function problems that are guaranteed to have an answer, and t ...

* Pigeonhole subset sum: given

n

positive integers whose sum is less than

2^n-1

, find two distinct subsets with the same sum * Finding the

Vapnik–Chervonenkis dimension Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the capacity (complexity, expressive power, richness, or flexibility) of a set of functions that can be learned by a statistical binary classification algorithm ...

of a given family of sets

References

External links

*
Basic structure, Turing reducibility and NP-hardness
* {{DEFAULTSORT:Np-Intermediate Complexity classes