TY - JOUR
T1 - On the Parameterized Complexity of Contraction to Generalization of Trees
AU - Agarwal, Akanksha
AU - Saurabh, Saket
AU - Tale, Prafullkumar
N1 - Funding Information:
The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreements no. 306992 (PARAPPROX).
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/4/15
Y1 - 2019/4/15
N2 - For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a “parameterized way”. Let T ℓ be the family of graphs such that each graph in T ℓ can be made into a tree by deleting at most ℓ edges. Thus, the problem we study is T ℓ -Contraction. We design an FPT algorithm for T ℓ -Contraction running in time O((2ℓ+2)O(k+ℓ)⋅nO(1)). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T ℓ -Contraction of size O([k(k+2ℓ)](⌈αα−1⌉+1)).
AB - For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S ⊆ E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a “parameterized way”. Let T ℓ be the family of graphs such that each graph in T ℓ can be made into a tree by deleting at most ℓ edges. Thus, the problem we study is T ℓ -Contraction. We design an FPT algorithm for T ℓ -Contraction running in time O((2ℓ+2)O(k+ℓ)⋅nO(1)). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T ℓ -Contraction of size O([k(k+2ℓ)](⌈αα−1⌉+1)).
KW - Fixed parameter tractability
KW - Generalization of trees
KW - Graph algorithms
KW - Graph contraction
UR - http://www.scopus.com/inward/record.url?scp=85056343280&partnerID=8YFLogxK
U2 - 10.1007/s00224-018-9892-z
DO - 10.1007/s00224-018-9892-z
M3 - Article
AN - SCOPUS:85056343280
VL - 63
SP - 587
EP - 614
JO - Theory of Computing Systems
JF - Theory of Computing Systems
SN - 1432-4350
IS - 3
ER -