TY - GEN
T1 - Conflict free feedback vertex set
T2 - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
AU - Agrawal, Akanksha
AU - Jain, Pallavi
AU - Kanesh, Lawqueen
AU - Lokshtanov, Daniel
AU - Saurabh, Saket
N1 - Funding Information:
Funding This research has received funding from the European Research Council under ERC grant no. 306992 PARAPPROX, ERC grant no. 715744 PaPaALG, ERC grant no. 725978 SYSTEMATICGRAPH, and DST, India for SERB-NPDF fellowship [PDF/2016/003508].
Publisher Copyright:
© Akanksha Agrawal, Pallavi Jain, Lawqueen Kanesh, Daniel Lokshtanov, and Saket Saurabh.
PY - 2018/8/1
Y1 - 2018/8/1
N2 - In this paper we study recently introduced conflict version of the classical Feedback Vertex Set (FVS) problem. For a family of graphs F, we consider the problem F-CF-Feedback Vertex Set (F-CF-FVS, for short). The F-CF-FVS problem takes as an input a graph G, a graph H ∈ F (where V (G) = V (H)), and an integer k, and the objective is to decide if there is a set S ⊆ V (G) of size at most k such that G − S is a forest and S is an independent set in H. Observe that if we instantiate F to be the family of edgeless graphs then we get the classical FVS problem. Jain, Kanesh, and Misra [CSR 2018] showed that in contrast to FVS, F-CF-FVS is W[1]-hard on general graphs and admits an FPT algorithm if F is the family of d-degenerate graphs. In this paper, we relate F-CF-FVS to the Independent Set problem on special classes of graphs, and obtain a complete dichotomy result on the Parameterized Complexity of the problem F-CF-FVS, when F is a hereditary graph family. In particular, we show that F-CF-FVS is FPT parameterized by the solution size if and only if F+Cluster IS is FPT parameterized by the solution size. Here, F+Cluster IS is the Independent Set problem in the (edge) union of a graph G ∈ F and a cluster graph H (G and H are explicitly given). Next, we exploit this characterization to obtain new FPT results as well as intractability results for F-CF-FVS. In particular, we give an FPT algorithm for F+Cluster IS when F is the family of Ki,j-free graphs. We show that for the family of bipartite graph B, B-CF-FVS is W[1]-hard, when parameterized by the solution size. Finally, we consider, for each 0 < < 1, the family of graphs F, which comprise of graphs G such that |E(G)| ≤ |V (G)|2−, and show that F-CF-FVS is W[1]-hard, when parameterized by the solution size, for every 0 < < 1.
AB - In this paper we study recently introduced conflict version of the classical Feedback Vertex Set (FVS) problem. For a family of graphs F, we consider the problem F-CF-Feedback Vertex Set (F-CF-FVS, for short). The F-CF-FVS problem takes as an input a graph G, a graph H ∈ F (where V (G) = V (H)), and an integer k, and the objective is to decide if there is a set S ⊆ V (G) of size at most k such that G − S is a forest and S is an independent set in H. Observe that if we instantiate F to be the family of edgeless graphs then we get the classical FVS problem. Jain, Kanesh, and Misra [CSR 2018] showed that in contrast to FVS, F-CF-FVS is W[1]-hard on general graphs and admits an FPT algorithm if F is the family of d-degenerate graphs. In this paper, we relate F-CF-FVS to the Independent Set problem on special classes of graphs, and obtain a complete dichotomy result on the Parameterized Complexity of the problem F-CF-FVS, when F is a hereditary graph family. In particular, we show that F-CF-FVS is FPT parameterized by the solution size if and only if F+Cluster IS is FPT parameterized by the solution size. Here, F+Cluster IS is the Independent Set problem in the (edge) union of a graph G ∈ F and a cluster graph H (G and H are explicitly given). Next, we exploit this characterization to obtain new FPT results as well as intractability results for F-CF-FVS. In particular, we give an FPT algorithm for F+Cluster IS when F is the family of Ki,j-free graphs. We show that for the family of bipartite graph B, B-CF-FVS is W[1]-hard, when parameterized by the solution size. Finally, we consider, for each 0 < < 1, the family of graphs F, which comprise of graphs G such that |E(G)| ≤ |V (G)|2−, and show that F-CF-FVS is W[1]-hard, when parameterized by the solution size, for every 0 < < 1.
KW - Conflict-free
KW - FPT algorithm
KW - Feedback vertex set
KW - W[1]-hardness
UR - http://www.scopus.com/inward/record.url?scp=85053188261&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.MFCS.2018.53
DO - 10.4230/LIPIcs.MFCS.2018.53
M3 - Conference contribution
AN - SCOPUS:85053188261
SN - 9783959770866
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018
A2 - Potapov, Igor
A2 - Worrell, James
A2 - Spirakis, Paul
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Y2 - 27 August 2018 through 31 August 2018
ER -