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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| Published in: | Journal of combinatorial optimization Vol. 43; no. 5; pp. 1264 - 1279 |
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| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
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01.07.2022
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| ISSN: | 1382-6905, 1573-2886 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. Suppose that we are given an independent set I0 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 I0 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 I0. 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. |
| Author | Mizuta, Haruka Ito, Takehiro Suzuki, Akira Nishimura, Naomi |
| Author_xml | – sequence: 1 givenname: Takehiro surname: Ito fullname: Ito, Takehiro organization: Tohoku University – sequence: 2 givenname: Haruka surname: Mizuta fullname: Mizuta, Haruka email: haruka.mizuta.s4@dc.tohoku.ac.jp organization: Tohoku University – sequence: 3 givenname: Naomi surname: Nishimura fullname: Nishimura, Naomi organization: University of Waterloo – sequence: 4 givenname: Akira surname: Suzuki fullname: Suzuki, Akira organization: Tohoku University |
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| Keywords | Combinatorial reconfiguration Independent set Graph algorithms |
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| References_xml | – volume: 78 start-page: 274 issue: 1 year: 2017 end-page: 297 ident: CR17 article-title: On the parameterized complexity of reconfiguration problems publication-title: Algorithmica doi: 10.1007/s00453-016-0159-2 – volume: 25 start-page: 159 issue: 1 year: 2006 end-page: 185 ident: CR20 article-title: Dynamic local search for the maximum clique problem publication-title: J Artif Intell Res doi: 10.1613/jair.1815 – volume: 22 start-page: 1082 year: 1970 end-page: 1096 ident: CR14 article-title: -degenerate graphs publication-title: Can J Math doi: 10.4153/CJM-1970-125-1 – volume: 93 start-page: 1 year: 2018 end-page: 10 ident: CR25 article-title: Reconfiguration in bounded bandwidth and tree-depth publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2017.11.003 – ident: CR2 – ident: CR12 – ident: CR10 – year: 1979 ident: CR7 publication-title: Computers and intractability: a guide to the theory of NP-completeness – ident: CR6 – volume: 35 start-page: 85 year: 1960 end-page: 90 ident: CR5 article-title: Intersection theorems for systems of sets publication-title: J Lond Math Soc doi: 10.1112/jlms/s1-35.1.85 – ident: CR8 – volume: 35 start-page: 39 year: 1983 end-page: 61 ident: CR21 article-title: Graph minors. I. Excluding a forest publication-title: J Combin Theory Ser B doi: 10.1016/0095-8956(83)90079-5 – year: 1999 ident: CR4 publication-title: Graph classes: a survey doi: 10.1137/1.9780898719796 – ident: CR23 – volume: 80 start-page: 576 year: 2018 end-page: 607 ident: CR22 article-title: A theory and algorithms for combinatorial reoptimization publication-title: Algorithmica doi: 10.1007/s00453-017-0274-8 – ident: CR19 – volume: 412 start-page: 1054 issue: 12–14 year: 2011 end-page: 1065 ident: CR9 article-title: On the complexity of reconfiguration problems publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2010.12.005 – volume: 439 start-page: 9 year: 2012 end-page: 15 ident: CR13 article-title: Complexity of independent set reconfigurability problems publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2012.03.004 – ident: CR3 – ident: CR15 – year: 2006 ident: CR18 publication-title: Invitation to fixed-parameter algorithms doi: 10.1093/acprof:oso/9780198566076.001.0001 – volume: 95 start-page: 122 year: 2018 end-page: 131 ident: CR16 article-title: Reconfiguration on sparse graphs publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2018.02.004 – ident: CR11 – ident: CR24 – volume: 18 start-page: 525 issue: 4 year: 2012 end-page: 547 ident: CR1 article-title: Fast local search for the maximum independent set problem publication-title: J Heurist doi: 10.1007/s10732-012-9196-4 – ident: 630_CR23 – ident: 630_CR6 – volume: 35 start-page: 85 year: 1960 ident: 630_CR5 publication-title: J Lond Math Soc doi: 10.1112/jlms/s1-35.1.85 – ident: 630_CR15 doi: 10.1137/1.9781611975031.13 – volume: 439 start-page: 9 year: 2012 ident: 630_CR13 publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2012.03.004 – volume: 95 start-page: 122 year: 2018 ident: 630_CR16 publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2018.02.004 – ident: 630_CR2 doi: 10.1007/978-3-319-68705-6_10 – ident: 630_CR12 doi: 10.1007/978-3-030-26176-4_26 – volume: 80 start-page: 576 year: 2018 ident: 630_CR22 publication-title: Algorithmica doi: 10.1007/s00453-017-0274-8 – volume-title: Invitation to fixed-parameter algorithms year: 2006 ident: 630_CR18 doi: 10.1093/acprof:oso/9780198566076.001.0001 – volume: 78 start-page: 274 issue: 1 year: 2017 ident: 630_CR17 publication-title: Algorithmica doi: 10.1007/s00453-016-0159-2 – volume-title: Computers and intractability: a guide to the theory of NP-completeness year: 1979 ident: 630_CR7 – ident: 630_CR19 doi: 10.3390/a11040052 – volume: 22 start-page: 1082 year: 1970 ident: 630_CR14 publication-title: Can J Math doi: 10.4153/CJM-1970-125-1 – ident: 630_CR24 doi: 10.1017/CBO9781139506748.005 – volume: 93 start-page: 1 year: 2018 ident: 630_CR25 publication-title: J Comput Syst Sci doi: 10.1016/j.jcss.2017.11.003 – volume: 25 start-page: 159 issue: 1 year: 2006 ident: 630_CR20 publication-title: J Artif Intell Res doi: 10.1613/jair.1815 – volume: 35 start-page: 39 year: 1983 ident: 630_CR21 publication-title: J Combin Theory Ser B doi: 10.1016/0095-8956(83)90079-5 – ident: 630_CR3 doi: 10.1007/978-3-662-55751-8_12 – ident: 630_CR11 doi: 10.1007/978-3-319-06089-7_24 – volume-title: Graph classes: a survey year: 1999 ident: 630_CR4 doi: 10.1137/1.9780898719796 – ident: 630_CR10 doi: 10.1007/978-3-319-13075-0_17 – ident: 630_CR8 – volume: 412 start-page: 1054 issue: 12–14 year: 2011 ident: 630_CR9 publication-title: Theor Comput Sci doi: 10.1016/j.tcs.2010.12.005 – volume: 18 start-page: 525 issue: 4 year: 2012 ident: 630_CR1 publication-title: J Heurist doi: 10.1007/s10732-012-9196-4 |
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| Snippet | 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... Suppose that we are given an independent set I0 of a graph G, and an integer l≥0. Then, we are asked to find an independent set of G having the maximum size... |
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| SubjectTerms | 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 |
| Title | Incremental optimization of independent sets under the reconfiguration framework |
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