The SCA problem can be solved in time and admits a polynomial kernel with vertices and bits and the algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.
Abstract
Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.
We consider Directed Feedback Vertex Set on planar digraphs, parameterized by the solution size $k$. We give a randomized algorithm with one-sided error running in time $(2+\sqrt5)^k n^{O(1)}= 4.24^k n^{O(1)}$, and a deterministic algorithm running in time $8.04^k n^{O(1)}$. Both algorithms use polynomial space. To the...
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