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 ≥ 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 indep...

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Bibliographic Details
Published in:Journal of combinatorial optimization Vol. 43; no. 5; pp. 1264 - 1279
Main Authors: Ito, Takehiro, Mizuta, Haruka, Nishimura, Naomi, Suzuki, Akira
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2022
Springer Nature B.V
Subjects:
Algorithms
Combinatorics
Convex and Discrete Geometry
Graphs
Lower bounds
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Parameterization
Parameters
Reconfiguration
Theory of Computation
Combinatorial reconfiguration
Independent set
Graph algorithms
ISSN:1382-6905, 1573-2886
Online Access:Get full text
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Summary:Suppose that we are given an independent set I 0 of a graph G , and an integer 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 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.
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ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-020-00630-z