TY - JOUR

T1 - Incremental optimization of independent sets under the reconfiguration framework

AU - Ito, Takehiro

AU - Mizuta, Haruka

AU - Nishimura, Naomi

AU - Suzuki, Akira

N1 - Funding Information:
Research by Takehiro Ito is partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan. Research by Haruka Mizuta is partially supported by JSPS KAKENHI Grant Number JP19J10042, Japan. Research by Akira Suzuki is partially supported by JST CREST Grant Number JPMJCR1402, and JSPS KAKENHI Grant Numbers JP17K12636, JP18H04091 and JP20K11666, Japan. Research by Naomi Nishimura is partially supported by the Natural Sciences and Engineering Research Council of Canada Grant Number RGPIN-2016-03621.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2022/7

Y1 - 2022/7

N2 - Suppose that we are given an independent set I 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 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. 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.

AB - Suppose that we are given an independent set I 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 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. 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.

KW - Combinatorial reconfiguration

KW - Graph algorithms

KW - Independent set

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U2 - 10.1007/s10878-020-00630-z

DO - 10.1007/s10878-020-00630-z

M3 - Article

AN - SCOPUS:85089993553

VL - 43

SP - 1264

EP - 1279

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

IS - 5

ER -