1 research outputs found
A Unifying Model for Locally Constrained Spanning Tree Problems
Given a graph and a digraph whose vertices are the edges of , we
investigate the problem of finding a spanning tree of that satisfies the
constraints imposed by . The restrictions to add an edge in the tree depend
on its neighborhood in . Here, we generalize previously investigated
problems by also considering as input functions and on that
give a lower and an upper bound, respectively, on the number of constraints
that must be satisfied by each edge. The produced feasibility problem is
denoted by \texttt{G-DCST}, while the optimization problem is denoted by
\texttt{G-DCMST}. We show that \texttt{G-DCST} is NP-complete even under strong
assumptions on the structures of and , as well as on functions
and . On the positive side, we prove two polynomial results, one for
\texttt{G-DCST} and another for \texttt{G-DCMST}, and also give a simple
exponential-time algorithm along with a proof that it is asymptotically optimal
under the \ETH. Finally, we prove that other previously studied constrained
spanning tree (\textsc{CST}) problems can be modeled within our framework,
namely, the \textsc{Conflict CST}, the \textsc{Forcing CS, the \textsc{At Least
One/All Dependency CST}, the \textsc{Maximum Degree CST}, the \textsc{Minimum
Degree CST}, and the \textsc{Fixed-Leaves Minimum Degree CST}.Comment: 28 pages, 6 figure