SEARCH-SPACE REDUCTION VIA ESSENTIAL VERTICES

We investigate preprocessing for vertex-subset problems on graphs. While the notioof kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a nonempty vertex set hat belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of r ex Seta c-essential vertex as one that is contained in all c-approximate solutions. For several classic comshow that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of nonessential vertices in the solution.