Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the $\mathcal{O}(n^3)$ time algorithm due to Floyd and Warshall (1962). Somewhat faster algorithms exist for the vertex-weighted version if fast matrix multiplication may be used. Yuster (SODA 2009) gave an algorithm running in time $\mathcal{O}(n^{2.842})$, but no combinatorial, truly subcubic algorithm is known. Motivated by the recent framework of efficient parameterized algorithms (or "FPT in P"), we investigate the influence of the graph parameters clique-width ($cw$) and modular-width ($mw$) on the running times of algorithms for solving All-Pairs Shortest Paths. We obtain efficient (and combinatorial) parameterized algorithms on non-negative vertex-weighted graphs of times $\mathcal{O}(cw^2n^2)$, resp. $\mathcal{O}(mw^2n + n^2)$. If fast matrix multiplication is allowed then the latter can be improved to $\mathcal{O}(mw^{1.842}n + n^2)$ using the algorithm of Yuster as a black box. The algorithm relative to modular-width is adaptive, meaning that the running time matches the best unparameterized algorithm for parameter value $mw$ equal to $n$, and they outperform them already for $mw \in \mathcal{O}(n^{1 - \varepsilon})$ for any $\varepsilon > 0$.

We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum $s$-$t$ Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time $O(n+m)$ for graphs with $n$ vertices and $m$ edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes. Throughout, we obtain efficient parameterized algorithms of running times $O(f(mw)n+m)$, $O(n+f(mw)m)$ , or $O(f(mw)+n+m)$ for graphs of modular-width $mw$. Our algorithm for Maximum Matching, running in time $O(mw^2\log mw \cdot n+m)$, is both faster and simpler than the recent $O(mw^4n+m)$ time algorithm of Coudert et al. (SODA 2018). For several other problems, e.g., Triangle Counting and Maximum $b$-Matching, we give adaptive algorithms, meaning that their running times match the best unparameterized algorithms for worst-case modular-width of $mw=Θ(n)$ and they outperform them already for $mw=o(n)$, until reaching linear time for $mw=O(1)$.