TY - GEN
T1 - Parameterized Complexity of Perfectly Matched Sets
AU - Agrawal, Akanksha
AU - Bhattacharjee, Sutanay
AU - Jana, Satyabrata
AU - Sahu, Abhishek
N1 - Publisher Copyright:
© Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu.
PY - 2022/12/1
Y1 - 2022/12/1
N2 - For an undirected graph G, a pair of vertex disjoint subsets pA, Bq is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (“|B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2Op?kq ¨ nOp1q, and ii) Kb,b-free graphs. We obtain a linear kernel for planar graphs and kOpdq-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.
AB - For an undirected graph G, a pair of vertex disjoint subsets pA, Bq is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (“|B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2Op?kq ¨ nOp1q, and ii) Kb,b-free graphs. We obtain a linear kernel for planar graphs and kOpdq-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.
KW - Apex-minor-free graphs
KW - Interval Graphs
KW - Parameterized Complexity
KW - Perfectly Matched Sets
KW - Planar graphs
KW - d-degenerate graphs
UR - http://www.scopus.com/inward/record.url?scp=85144205995&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.IPEC.2022.2
DO - 10.4230/LIPIcs.IPEC.2022.2
M3 - Conference contribution
AN - SCOPUS:85144205995
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022
A2 - Dell, Holger
A2 - Nederlof, Jesper
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022
Y2 - 7 September 2022 through 9 September 2022
ER -