TY - GEN
T1 - Shortest Reconfiguration of Matchings
AU - Bousquet, Nicolas
AU - Hatanaka, Tatsuhiko
AU - Ito, Takehiro
AU - Mühlenthaler, Moritz
N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2019.
PY - 2019
Y1 - 2019
N2 - Imagine that unlabelled tokens are placed on edges forming a matching of a graph. A token can be moved to another edge provided that the edges containing tokens remain a matching. The distance between two configurations of tokens is the minimum number of moves required to transform one into the other. We study the problem of computing the distance between two given configurations. We prove that if source and target configurations are maximal matchings, then the problem admits no polynomial-time sublogarithmic-factor approximation algorithm unless $$\mathsf{P}= \mathsf{NP}$$. On the positive side, we show that for matchings of bipartite graphs the problem is fixed-parameter tractable parameterized by the size d of the symmetric difference of the two given configurations. Furthermore, we obtain a $$d^\varepsilon $$ -factor approximation algorithm for the distance of two maximum matchings of bipartite graphs for every $$\varepsilon > 0$$. The proofs of our positive results are constructive and can hence be turned into algorithms that output shortest transformations. Both algorithmic results rely on a close connection to the Directed Steiner Tree problem. Finally, we show that determining the exact distance between two configurations is complete for the class $$\mathsf{D}^\mathsf{P}$$, and determining the maximum distance between any two configurations of a given graph is $$\mathsf{D}^\mathsf{P}$$ -hard.
AB - Imagine that unlabelled tokens are placed on edges forming a matching of a graph. A token can be moved to another edge provided that the edges containing tokens remain a matching. The distance between two configurations of tokens is the minimum number of moves required to transform one into the other. We study the problem of computing the distance between two given configurations. We prove that if source and target configurations are maximal matchings, then the problem admits no polynomial-time sublogarithmic-factor approximation algorithm unless $$\mathsf{P}= \mathsf{NP}$$. On the positive side, we show that for matchings of bipartite graphs the problem is fixed-parameter tractable parameterized by the size d of the symmetric difference of the two given configurations. Furthermore, we obtain a $$d^\varepsilon $$ -factor approximation algorithm for the distance of two maximum matchings of bipartite graphs for every $$\varepsilon > 0$$. The proofs of our positive results are constructive and can hence be turned into algorithms that output shortest transformations. Both algorithmic results rely on a close connection to the Directed Steiner Tree problem. Finally, we show that determining the exact distance between two configurations is complete for the class $$\mathsf{D}^\mathsf{P}$$, and determining the maximum distance between any two configurations of a given graph is $$\mathsf{D}^\mathsf{P}$$ -hard.
KW - Approximation hardness
KW - Fixed-parameter tractability
KW - Matchings
KW - Reconfiguration
UR - http://www.scopus.com/inward/record.url?scp=85072858169&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85072858169&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-30786-8_13
DO - 10.1007/978-3-030-30786-8_13
M3 - Conference contribution
AN - SCOPUS:85072858169
SN - 9783030307851
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 162
EP - 174
BT - Graph-Theoretic Concepts in Computer Science - 45th International Workshop, WG 2019, Revised Papers
A2 - Sau, Ignasi
A2 - Thilikos, Dimitrios M.
PB - Springer Verlag
T2 - 45th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2019
Y2 - 19 June 2019 through 21 June 2019
ER -