TY - GEN

T1 - On the Parameterized Complexity of Contraction to generalization of trees

AU - Agrawal, Akanksha

AU - Saurabh, Saket

AU - Tale, Prafullkumar

N1 - Publisher Copyright:
© 2018 Akanksha Agrawal, Saket Saurabh, and Prafullkumar Tale.

PY - 2018/2/1

Y1 - 2018/2/1

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 Tℓ be the family of graphs such that each graph in Tℓ can be made into a tree by deleting at most ℓ edges. Thus, the problem we study is Tℓ-Contraction. We design an FPT algorithm for Tℓ-Contraction running in time O((2√ℓ + 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 Tℓ-Contraction of size O([k(k + 2ℓ)]([α/α-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 Tℓ be the family of graphs such that each graph in Tℓ can be made into a tree by deleting at most ℓ edges. Thus, the problem we study is Tℓ-Contraction. We design an FPT algorithm for Tℓ-Contraction running in time O((2√ℓ + 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 Tℓ-Contraction of size O([k(k + 2ℓ)]([α/α-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=85044784464&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2017.1

DO - 10.4230/LIPIcs.IPEC.2017.1

M3 - Conference contribution

AN - SCOPUS:85044784464

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 12th International Symposium on Parameterized and Exact Computation, IPEC 2017

A2 - Lokshtanov, Daniel

A2 - Nishimura, Naomi

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 12th International Symposium on Parameterized and Exact Computation, IPEC 2017

Y2 - 6 September 2017 through 8 September 2017

ER -