TY - GEN

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:
© Copyright 2018 by SIAM.

PY - 2018/1/1

Y1 - 2018/1/1

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)k kí k(d + 1)-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = ffs1; t1g; fs2; t2g; : : : ; fs'; t'gg 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)k kí k(d + 1)-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = ffs1; t1g; fs2; t2g; : : : ; fs'; t'gg 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.

UR - http://www.scopus.com/inward/record.url?scp=85045557016&partnerID=8YFLogxK

U2 - 10.1137/1.9781611975031.177

DO - 10.1137/1.9781611975031.177

M3 - Conference contribution

AN - SCOPUS:85045557016

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2785

EP - 2800

BT - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

A2 - Czumaj, Artur

PB - Association for Computing Machinery

T2 - 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018

Y2 - 7 January 2018 through 10 January 2018

ER -