2 research outputs found
Incremental Optimization of Independent Sets under Reachability Constraints
We introduce a new framework for reconfiguration problems, and apply it to
independent sets as the first example. Suppose that we are given an independent
set of a graph , and an integer which represents a lower
bound on the size of any independent set of . Then, we are asked to find an
independent set of having the maximum size among independent sets that are
reachable from by either adding or removing a single vertex at a time
such that all intermediate independent sets are of size at least . We show
that this problem is PSPACE-hard even for bounded pathwidth graphs, and remains
NP-hard for planar graphs. On the other hand, we give a linear-time algorithm
to solve the problem for chordal graphs. We also study the fixed-parameter
(in)tractability of the problem with respect to the following three parameters:
the degeneracy of an input graph, a lower bound on the size of the
independent sets, and a lower bound on the solution size. We show that the
problem is fixed-parameter intractable when only one of , , and is
taken as a parameter. On the other hand, we give a fixed-parameter algorithm
when parameterized by ; this result implies that the problem parameterized
only by is fixed-parameter tractable for planar graphs, and for bounded
treewidth graphs
Decremental Optimization of Dominating Sets Under the Reconfiguration Framework
Given a dominating set, how much smaller a dominating set can we find through
elementary operations? Here, we proceed by iterative vertex addition and
removal while maintaining the property that the set forms a dominating set of
bounded size. This can be seen as the optimization variant of the dominating
set reconfiguration problem, where two dominating sets are given and the
question is merely whether they can be reached one from another through
elementary operations. We show that this problem is PSPACE-complete, even if
the input graph is a bipartite graph, a split graph, or has bounded pathwidth.
On the positive side, we give linear-time algorithms for cographs, trees and
interval graphs. We also study the parameterized complexity of this problem.
More precisely, we show that the problem is W[2]-hard when parameterized by the
upper bound on the size of an intermediary dominating set. On the other hand,
we give fixed-parameter algorithms with respect to the minimum size of a vertex
cover, or where is the degeneracy and is the upper bound of
output solution.Comment: 15 pages, 8 figure