Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

Abstract

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.

Parameterized Complexity of Conflict-Free Matchings and Paths

“…These algorithms are based on the computation of an independence covering family, a notion which was recently introduced by Lokshtanov et al [25]. We note that even for nowhere dense graphs, such an independence covering family can be computed efficiently [25]. Since our algorithms are based on computation of independence covering families, hence, our results hold even when the conflict graph is a nowhere dense graph.…”

“…In this section, we show that CF-MM and CF-SP both are in FPT, when the conflict graph H is a d-degenerate graphs. These algorithms are based on the notion of independence covering family, which was introduced in [25], defined below.…”

“…We use Alg-CF-MM (Alg-CF-SP) together with Independence Covering Lemma of [25] to obtain algorithms for CF-MM (CF-SP) when the conflict graph is d-degenerate or nowhere dense graph. Towards this we state some lemmata from [25] that we use in our algorithms.…”

“…For conflict graphs being d-degenerate, we obtain FPT algorithms for both CF-MM and CF-SP. These algorithms are based on the computation of an independence covering family, a notion which was recently introduced by Lokshtanov et al [25]. We note that even for nowhere dense graphs, such an independence covering family can be computed efficiently [25].…”

A Framework for Approximation Schemes on Disk Graphs

“…We will now combine the arguments given so far in this subsection, in order to prove the following lemma. Having small treewidth is particularly useful due to the following proposition, which is a consequence of, e.g., Courcelle's theorem [8] (or, more efficiently, it can be proved using dynamic programming directly as in [30,15]). Proposition 5.29 ([8]).…”