TY - GEN

T1 - SIMULTANEOUS Feedback Vertex set

T2 - 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016

AU - Agrawal, Akanksha

AU - Lokshtanov, Daniel

AU - Mouawad, Amer E.

AU - Saurabh, Saket

N1 - Publisher Copyright:
© Akanksha Agrawal, Daniel Lokshtanov, Amer E. Mouawad, and Saket Saurabh; licensed under Creative Commons License CC-BY.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - For a family of graphs F, a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex SET, and ODD CYCLE TRANSVERSAL. A graph G = (V,∪αi=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colored graph. A natural extension of the F-DELETION problem to edge-colored graphs is the α-SIMULTANEOUS F-DELETION problem. In the latter problem, we are given an α-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph Gi \ S, where Gi = (V, Ei) and 1 ≤ i ≤ α, is in F. In this work, we study α-SIMULTANEOUS F-DELETION for F being the family of forests. In other words, we focus on the α-SIMULTANEOUS FEEDBACK VERTEX SET (α-SIMFVS) problem. Algorithmically, we show that, like its classical counterpart, α-SIMFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α. In particular, we give an algorithm running in 2O(αk)nO(1) time and a kernel with O(αk3(α+1)) vertices. The running time of our algorithm implies that α-SimFVS is FPT even when α ∈ o(log n). We complement this positive result by showing that for α ∈ O(log n), where n is the number of vertices in the input graph, α-SIMFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

AB - For a family of graphs F, a graph G, and a positive integer k, the F-Deletion problem asks whether we can delete at most k vertices from G to obtain a graph in F. F-Deletion generalizes many classical graph problems such as Vertex Cover, Feedback Vertex SET, and ODD CYCLE TRANSVERSAL. A graph G = (V,∪αi=1Ei), where the edge set of G is partitioned into α color classes, is called an α-edge-colored graph. A natural extension of the F-DELETION problem to edge-colored graphs is the α-SIMULTANEOUS F-DELETION problem. In the latter problem, we are given an α-edge-colored graph G and the goal is to find a set S of at most k vertices such that each graph Gi \ S, where Gi = (V, Ei) and 1 ≤ i ≤ α, is in F. In this work, we study α-SIMULTANEOUS F-DELETION for F being the family of forests. In other words, we focus on the α-SIMULTANEOUS FEEDBACK VERTEX SET (α-SIMFVS) problem. Algorithmically, we show that, like its classical counterpart, α-SIMFVS parameterized by k is fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed constant α. In particular, we give an algorithm running in 2O(αk)nO(1) time and a kernel with O(αk3(α+1)) vertices. The running time of our algorithm implies that α-SimFVS is FPT even when α ∈ o(log n). We complement this positive result by showing that for α ∈ O(log n), where n is the number of vertices in the input graph, α-SIMFVS becomes W[1]-hard. Our positive results answer one of the open problems posed by Cai and Ye (MFCS 2014).

KW - Edge-colored graphs

KW - Feedback Vertex SET

KW - Kernel

KW - Parameterized complexity

UR - http://www.scopus.com/inward/record.url?scp=84961639103&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.STACS.2016.7

DO - 10.4230/LIPIcs.STACS.2016.7

M3 - Conference contribution

AN - SCOPUS:84961639103

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 33rd Symposium on Theoretical Aspects of Computer Science, STACS 2016

A2 - Vollmer, Heribert

A2 - Ollinger, Nicolas

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

Y2 - 17 February 2016 through 20 February 2016

ER -