Sensitivity, Proximity and FPT Algorithms for Exact Matroid Problems

We consider the problem of finding a basis of a matroid with weight exactly equal to a given target. Here weights can be discrete values from$\{-\Delta,\ \ldots,\ \Delta\}$or more generally m-dimensional vectors of such discrete values. We resolve the parameterized complexity completely, by presenting an FPT algorithm parameterized by$\Delta$and$m$for arbitrary matroids. Prior to our work, no such algorithms were known even when weights are in$\{0,1\}$, or arbitrary$\Delta$and$m=1$. Our main technical contributions are new proximity and sensitivity bounds for matroid problems, independent of the number of elements. These bounds imply FPT algorithms via matroid intersection.