TY - JOUR
T1 - Parameterized complexity of independent set reconfiguration problems
AU - Ito, Takehiro
AU - Kamiński, Marcin
AU - Ono, Hirotaka
AU - Suzuki, Akira
AU - Uehara, Ryuhei
AU - Yamanaka, Katsuhisa
N1 - Funding Information:
We thank anonymous referees of the early version [14] and of this journal version for their helpful suggestions. In particular, one of their suggestions led us to an improvement of the running time of our FPT algorithms. This work is partially supported by the (Polish) National Science Center under Grant Number 2013/09/B/ST6/03136 , and by JSPS/MEXT KAKENHI, Japan Grant Numbers JP24106004 , JP25104521 , JP25330003 , JP26330009 , JP26730001 , JP15H00849 , JP16K00002 , JP16K00004 , JP17H01698 , JP17K12636 , JP17K19960 , and JP18H04091 , Japan.
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/9/15
Y1 - 2020/9/15
N2 - Suppose that we are given two independent sets I0 and Ir of a graph such that |I0|=|Ir|, and imagine that a token is placed on each vertex in I0. Then, the TOKEN JUMPING problem is to determine whether there exists a sequence of independent sets which transforms I0 into Ir so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W[1]-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I0 and Ir with the minimum number of token movements. We finally show that one of the results for TOKEN JUMPING can be extended to a more generalized reconfiguration problem for independent sets, called TOKEN ADDITION AND REMOVAL.
AB - Suppose that we are given two independent sets I0 and Ir of a graph such that |I0|=|Ir|, and imagine that a token is placed on each vertex in I0. Then, the TOKEN JUMPING problem is to determine whether there exists a sequence of independent sets which transforms I0 into Ir so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W[1]-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I0 and Ir with the minimum number of token movements. We finally show that one of the results for TOKEN JUMPING can be extended to a more generalized reconfiguration problem for independent sets, called TOKEN ADDITION AND REMOVAL.
KW - Combinatorial reconfiguration
KW - Fixed parameter tractability
KW - Graph algorithm
KW - Independent set
UR - http://www.scopus.com/inward/record.url?scp=85080096619&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85080096619&partnerID=8YFLogxK
U2 - 10.1016/j.dam.2020.01.022
DO - 10.1016/j.dam.2020.01.022
M3 - Article
AN - SCOPUS:85080096619
SN - 0166-218X
VL - 283
SP - 336
EP - 345
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
ER -