4 research outputs found
Vertex Cover and Feedback Vertex Set Above and Below Structural Guarantees
Vertex Cover parameterized by the solution size k is the quintessential fixed-parameter tractable problem. FPT algorithms are most interesting when the parameter is small. Several lower bounds on k are well-known, such as the maximum size of a matching. This has led to a line of research on parameterizations of Vertex Cover by the difference of the solution size k and a lower bound. The most prominent cases for such lower bounds for which the problem is FPT are the matching number or the optimal fractional LP solution. We investigate parameterizations by the difference between k and other graph parameters including the feedback vertex number, the degeneracy, cluster deletion number, and treewidth with the goal of finding the border of fixed-parameter tractability for said difference parameterizations. We also consider similar parameterizations of the Feedback Vertex Set problem
Induced Matching below Guarantees: Average Paves the Way for Fixed-Parameter Tractability
In this work, we study the Induced Matching problem: Given an undirected
graph and an integer , is there an induced matching of size at
least ? An edge subset is an induced matching in if is a
matching such that there is no edge between two distinct edges of . Our work
looks into the parameterized complexity of Induced Matching with respect to
"below guarantee" parameterizations. We consider the parameterization for an upper bound on the size of any induced matching. For instance,
any induced matching is of size at most where is the number of
vertices, which gives us a parameter . In fact, there is a
straightforward -time algorithm for Induced
Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we
ask: Is Induced Matching FPT for a parameter smaller than ? In
search for such parameters, we consider and ,
where is the maximum matching size and is the maximum
independent set size of . We find that Induced Matching is presumably not
FPT when parameterized by or . In contrast to
these intractability results, we find that taking the average of the two helps
-- our main result is a branching algorithm that solves Induced Matching in
time. Our algorithm makes use
of the Gallai-Edmonds decomposition to find a structure to branch on