Incremental Optimization of Independent Sets under Reachability Constraints
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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 I_0 of a graph G, and an integer l > 0 which represents a lower bound on the size of any independent set of G. 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 fixed-parameter (in)tractability 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 the independent sets, and a lower bound s on the solution size. 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.
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