Incremental optimization of independent sets under the reconfiguration framework

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