Examples of higher order algorithms ($\mathcal{O}(n^4)$ or larger)

In most computer science cirriculums, students only get to see algorithms that run in very lower time complexities. For example these generally are

• Constant time $\mathcal{O}(1)$: Ex sum of first $n$ numbers

• Logarithmic time $\mathcal{O}(\log n)$: Ex binary searching a sorted list

• Linear time $\mathcal{O}(n)$: Ex Searching an unsorted list

• LogLinear time $\mathcal{O}(n\log n)$: Ex Merge Sort

• Quadratic time $\mathcal{O}(n^2)$: Ex Bubble/Insertion/Selection Sort

• (Rarely) Cubic time $\mathcal{O}(n^3)$: Ex Gaussian Elimination of a Matrix

However it can be shown that
$$
\mathcal{O}(1)\subset \mathcal{O}(\log n)\subset \ldots \subset \mathcal{O}(n^3)\subset \mathcal{O}(n^4)\subset\mathcal{O}(n^5)\subset\ldots\subset \mathcal{O}(n^k)\subset\ldots
$$

so it would be expected that there would be more well known problems that are in higher order time complexity classes, such as $\mathcal{O}(n^8)$.

What are some examples of algorithms that fall into these classes $\mathcal{O}(n^k)$ where $k\geq 4$?

Brute-force algorithms can be considered as a good example to achieve the mentioned running times (i.e. $\Omega(n^4)$).

Suppose given the sequence $\sigma=\langle a_1,a_2,\dots , a_n\rangle$
of real numbers, you want to find, if exists $k$ elements ($k\geq
4$, and $k$ is
constant ) from $\sigma$ such that $$\sum_{i=1}^{k}a_i=0.$$

Obviously, a simple brute-force algorithm for this problem check all $\binom{n}{k}$ subsets of the input and detect whether the elements in at least one of them sum to $0$. If $k$ is a constant then $\binom{n}{k}=\Theta(n^k)$. For example if $k=10$ then the running time of your algorithm is $\Theta(n^{10}).$ Finally, you find algorithm for your desired running time.

StackExchange Computer Science Q#131467, answer score: 3