Incremental optimization of independent sets under the reconfiguration framework

Abstract

Suppose that we are given an independent set $$I_0$$ of a graph G, and an integer $$l\ge 0$$ . Then, we are asked to find an independent set of G having the maximum size among independent sets that are reachable from $$I_0$$ by either adding or removing a single vertex at a time such that all intermediate independent sets are of size at least $$l$$ . 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 parameterized complexity of the problem with respect to the following three parameters: the degeneracy $$d$$ of an input graph, a lower bound $$l$$ on the size of independent sets, and a lower bound $$s$$ on the size of a solution reachable from $$I_0$$ . We show that the problem is fixed-parameter intractable when only one of $$d$$ , $$l$$ , or $$s$$ is taken as a parameter. On the other hand, we give a fixed-parameter algorithm when parameterized by $$s+d$$ ; this result implies that the problem parameterized only by $$s$$ is fixed-parameter tractable for planar graphs, and for bounded-treewidth graphs.