Quasi-polynomial time

In computational complexity theory and the analysis of algorithms, an algorithm is said to take quasi-polynomial time if its time complexity is quasi-polynomially bounded. That is, there should exist a constant $$c$$ such that the worst-case running time of the algorithm, on inputs of size $n$, has an upper bound of the form $$2^{O\bigl((\log n)^c\bigr)}.$$

The decision problems with quasi-polynomial time algorithms are natural candidates for being NP-intermediate, neither having polynomial time nor likely to be NP-hard.

Complexity class
The complexity class QP consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of DTIME as follows.
 * $$\mathsf{QP} = \bigcup_{c \in \mathbb{N}} \mathsf{DTIME} \left(2^{(\log n)^c}\right)$$

Examples
An early example of a quasi-polynomial time algorithm was the Adleman–Pomerance–Rumely primality test; however, the problem of testing whether a number is a prime number has subsequently been shown to have a polynomial time algorithm, the AKS primality test.

In some cases, quasi-polynomial time bounds can be proven to be optimal under the exponential time hypothesis or a related computational hardness assumption. For instance, this is true for finding the largest disjoint subset of a collection of unit disks in the hyperbolic plane, and for finding a graph with the fewest vertices that does not appear as an induced subgraph of a given graph.

Other problems for which the best known algorithm takes quasi-polynomial time include:
 * The planted clique problem, of determining whether a random graph has been modified by adding edges between all pairs of a subset of its vertices.
 * Monotone dualization, several equivalent problems of converting logical formulas between conjunctive and disjunctive normal form, listing all minimal hitting sets of a family of sets, or listing all minimal set covers of a family of sets, with time complexity measured in the combined input and output size.
 * Parity games, involving token-passing along the edges of a colored directed graph. The paper giving a quasi-polynomial algorithm for these games won the 2021 Nerode Prize.
 * Computing the Vapnik–Chervonenkis dimension of a family of sets.
 * Finding the smallest dominating set in a tournament, a subset of the vertices of the tournament that has at least one directed edge to all other vertices.

Problems for which a quasi-polynomial time algorithm has been announced but not fully published include:
 * The graph isomorphism problem, determining whether two graphs can be made equal to each other by relabeling their vertices, announced in 2015 and updated in 2017 by László Babai.
 * The unknotting problem, recognizing whether a knot diagram describes the unknot, announced by Marc Lackenby in 2021.

In approximation algorithms
Quasi-polynomial time has also been used to study approximation algorithms. In particular, a quasi-polynomial-time approximation scheme (QPTAS) is a variant of a polynomial-time approximation scheme whose running time is quasi-polynomial rather than polynomial. Problems with a QPTAS include minimum-weight triangulation, and finding the maximum clique on the intersection graph of disks.

More strongly, the problem of finding an approximate Nash equilibrium has a QPTAS, but cannot have a PTAS under the exponential time hypothesis.