A personal favorite of mine is fixed-parameter tractability. Instead of analysing the worst case running time solely on the input length we introduce a *parameter* and study the time complexity in both. Intuitively, the parameter „should“ be small and is often chosen to be the solution size or some specific measure of the input.

Consider the vertex cover problem where you want to cover all edges of a graph by at most k vertices (that are incident to all edges). It is a classical NP-complete problem and the naive bruteforce solver needs something like n^k time. Which is already huge for small k, say, 10.

A very simple fixed-parameter tractable (fpt) algorithm for this problem achieves a worst case time of 2^k * n. For huge graphs and small k (again, let’s say 10) this is a massive improvement.

This is a very active field, where we have a good understanding which problems allow have such a worst case time and which not (under some complexity theoretic assumptions of course). It incorporates also the idea of restricting the input to only specific „simple“ instances gradually. This happens if you add graph measures as a parameter.

Many NP-hard graph problems are in P if restricted to planar graphs. But what if the instances are „almost“ planar? If you choose a parameter that measures the structure of a graph such that the measure is low if the graph is planar and high if it isn’t, any fpt algorithm for this parameterization works on any graph; fast if it is planar, and fast-ish if it is close to being planar.

Of course, this is theory with the similar metaphysical caveats classical complexity theorem has. However, it results in interesting algorithmcsl tools and interacts nicely with specific fields of structural graph theory.