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Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms
ACM Transactions on Algorithms ( IF 1.3 ) Pub Date : 2020-06-07 , DOI: 10.1145/3379698 Daniel Lokshtanov 1 , Fahad Panolan 2 , Saket Saurabh 3 , Roohani Sharma 4 , Meirav Zehavi 5
ACM Transactions on Algorithms ( IF 1.3 ) Pub Date : 2020-06-07 , DOI: 10.1145/3379698 Daniel Lokshtanov 1 , Fahad Panolan 2 , Saket Saurabh 3 , Roohani Sharma 4 , Meirav Zehavi 5
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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 k ) . k (d+1)) -1 . The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G , a set T = {s_1, t_1} , {s_2, t_2}, …. , {s_ℓ , t_ℓ} 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 2 O(k) . In particular, G* excludes a clique of size 2 O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for S TABLE s-t S EPARATOR , S TABLE O DD C YCLE T RANSVERSAL , and S TABLE M ULTICUT on general graphs, and for S TABLE D IRECTED F EEDBACK V ERTEX S ET 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.
更新日期：2020-06-07
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