TY - JOUR
T1 - Split contraction
T2 - The untold story
AU - Agrawal, Akanksha
AU - Lokshtanov, Daniel
AU - Saurabh, Saket
AU - Zehavi, Meirav
N1 - Funding Information:
A preliminary version of this paper appeared in the proceedings of the 34th International Symposium on Theoretical Aspects of Computer Science (STACS 2017). The research leading to these results has received funding from the European Research Council (ERC) via Grant PARAP-PROX, Reference No. 306992. Authors’ addresses: A. Agrawal and M. Zehavi, Department of Computer Science, Ben-Gurion University of the Negev, Be’er Sheva, Israel; emails: agrawal@post.bgu.ac.il, meiravze@bgu.ac.il; D. Lokshtanov, Department of Computer Science, University of California Santa Barbara, Santa Barbara, USA; email: daniello@ucsb.edu; S. Saurabh, The Institute of Mathematical Science, HBNI, Chennai, India, Department of Informatics, University of Bergen, Bergen, Norway, and UMI ReLax; email: saket@imsc.res.in. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2019 Association for Computing Machinery. 1942-3454/2019/05-ART18 $15.00 https://doi.org/10.1145/3319909
Publisher Copyright:
© 2019 Association for Computing Machinery.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this article, we examine an important family of graphs, namely, the family of split graphs, which in the context of edge contractions is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, Split Contraction asks whether there exists X ⊆ E(G) such that G/X is a split graph and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. Guo and Cai [Theoretical Computer Science, 2015] claimed that Split Contraction is fixed-parameter tractable. However, our findings are different. We show that Split Contraction, despite its deceptive simplicity, is W[1]-hard. Our main result establishes the following conditional lower bound: Under the Exponential Time Hypothesis, Split Contraction cannot be solved in time 2o(ℓ2) · nO(1), where ℓ is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2o(ℓ2) · nO(1) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.
AB - The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this article, we examine an important family of graphs, namely, the family of split graphs, which in the context of edge contractions is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k, Split Contraction asks whether there exists X ⊆ E(G) such that G/X is a split graph and |X| ≤ k. Here, G/X is the graph obtained from G by contracting edges in X. Guo and Cai [Theoretical Computer Science, 2015] claimed that Split Contraction is fixed-parameter tractable. However, our findings are different. We show that Split Contraction, despite its deceptive simplicity, is W[1]-hard. Our main result establishes the following conditional lower bound: Under the Exponential Time Hypothesis, Split Contraction cannot be solved in time 2o(ℓ2) · nO(1), where ℓ is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2o(ℓ2) · nO(1) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.
KW - Edge contraction
KW - Parameterized complexity
KW - Split contraction
KW - Split graphs
UR - http://www.scopus.com/inward/record.url?scp=85074832879&partnerID=8YFLogxK
U2 - 10.1145/3319909
DO - 10.1145/3319909
M3 - Article
AN - SCOPUS:85074832879
SN - 1942-3454
VL - 11
JO - ACM Transactions on Computation Theory
JF - ACM Transactions on Computation Theory
IS - 3
M1 - 18
ER -