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

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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 -