superpolynomially

Superpolynomially refers to a class of computational problems whose solution time grows faster than any polynomial function of the input size, but not as fast as an exponential function. More formally, a problem is superpolynomially solvable if its time complexity is bounded by $O(n^{c \log n})$ for some constant $c$, where $n$ is the size of the input. This complexity class lies between polynomial time (P) and exponential time (EXPTIME).

Problems that are superpolynomially solvable are considered intractable for practical purposes, as their running time increases very rapidly with the input size. However, they are not as definitively difficult as problems requiring exponential time. Superpolynomial complexity is often encountered in theoretical computer science when analyzing the limitations of certain algorithms or the inherent difficulty of specific problems. For instance, some decision problems related to satisfiability in certain logical theories can exhibit superpolynomial complexity.

The concept is important for understanding the landscape of computational complexity. While P represents problems efficiently solvable, and EXPTIME represents problems generally considered unsolvable in practice, superpolynomial time occupies an intermediate, challenging space. It suggests that while algorithms might not be strictly exponential, they still present significant computational hurdles for larger inputs.