Summary: The question if a given partial solution to a problem can be extended reasonably occurs in many algorithmic approaches for optimization problems. For instance, when enumerating minimal vertex covers of a graph \(G = (V, E)\), one usually arrives at the problem to decide for a vertex set \(U \subseteq V\) (pre-solution), if there exists a minimal vertex cover \(S\) (i.e., a vertex cover \(S \subseteq V\) such that no proper subset of \(S\) is a vertex cover) with \(U \subseteq S\) (minimal extension of \(U\)). We propose a general, partial-order based formulation of such extension problems which allows to model parameterization and approximation aspects of extension, and also highlights relationships between extension tasks for different specific problems. As examples, we study a number of specific problems which can be expressed and related in this framework.
In particular, we discuss extension variants of the problems dominating set and feedback vertex/edge set. All these problems are shown to be \(\mathsf{NP} \)-complete even when restricted to bipartite graphs of bounded degree, with the exception of our extension version of feedback edge set on undirected graphs which is shown to be solvable in polynomial time. For the extension variants of dominating and feedback vertex set, we also show \(\mathsf{NP} \)-completeness for the restriction to planar graphs of bounded degree. As non-graph problem, we also study an extension version of the bin packing problem. We further consider the parameterized complexity of all these extension variants, where the parameter is a measure of the pre-solution as defined by our framework.