Incremental optimization of independent sets under the reconfiguration framework

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 \(d\), \(l\), and \(s\) 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.