TY - JOUR

T1 - Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

AU - Lokshtanov, Daniel

AU - Panolan, Fahad

AU - Saurabh, Saket

AU - Sharma, Roohani

AU - Zehavi, Meirav

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a d-degenerate graph G and an integer k, outputs an independent set Y, such that for every independent set X in G of size at most k, the probability that X is a subset of Y is at least (((d+1)kk) . k(d+1))-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = s1, t1 , s2, t2, .... , s , t of terminal pairs, and an integer k, returns an induced subgraph G∗ of G that maintains all the inclusion minimal multicuts of G of size at most k and does not contain any (k+2)-vertex connected set of size 2O(k). In particular, G∗ excludes a clique of size 2O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for STABLE s-t SEPARATOR, STABLE ODD CYCLE TRANSVERSAL, and STABLE MULTICUT on general graphs, and for STABLE DIRECTED FEEDBACK VERTEX SET on d-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013{. All of our algorithms can be derandomized at the cost of a small overhead in the running time.

AB - We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a d-degenerate graph G and an integer k, outputs an independent set Y, such that for every independent set X in G of size at most k, the probability that X is a subset of Y is at least (((d+1)kk) . k(d+1))-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = s1, t1 , s2, t2, .... , s , t of terminal pairs, and an integer k, returns an induced subgraph G∗ of G that maintains all the inclusion minimal multicuts of G of size at most k and does not contain any (k+2)-vertex connected set of size 2O(k). In particular, G∗ excludes a clique of size 2O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for STABLE s-t SEPARATOR, STABLE ODD CYCLE TRANSVERSAL, and STABLE MULTICUT on general graphs, and for STABLE DIRECTED FEEDBACK VERTEX SET on d-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013{. All of our algorithms can be derandomized at the cost of a small overhead in the running time.

KW - Independece covering family

KW - parameterized algorithms

KW - stable OCT

KW - stable multicut

KW - stable s-t separator

UR - http://www.scopus.com/inward/record.url?scp=85088702843&partnerID=8YFLogxK

U2 - 10.1145/3379698

DO - 10.1145/3379698

M3 - Article

AN - SCOPUS:85088702843

SN - 1549-6325

VL - 16

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 3

M1 - 31

ER -