Exact Algorithms via Multivariate Subroutines

We consider the family of $\Phi$-Subset problems, where the input consists of an instance $I$ of size $N$ over a universe $UI$ of size $n$ and the task is to check whether the universe contains a subset with property $\Phi$ (e.g., $\Phi$ could be the property of being a feedback vertex set for the input graph of size at most $k$). Our main tool is a simple randomized algorithm which solves $\Phi$-Subset in time $(1+b-\frac{1}{c})ⁿ N^{O(1)}$, provided that there is an algorithm for the $\Phi$-Extension problem with running time $b^{n-|X|} c^k N^{O(1)}$. Here, the input for $\Phi$-Extension is an instance $I$ of size $N$ over a universe $UI$ of size $n$, a subset $X\subseteq UI$, and an integer $k$, and the task is to check whether there is a set $Y$ with $X\subseteq Y \subseteq UI$ and $|Y\setminus X|\le k$ with property $\Phi$. We derandomize this algorithm at the cost of increasing the running time by a subexponential factor in $n$, and we adapt it to the enumeration setting where we need to enumerate all subsets of the universe with property $\Phi$. This generalizes the results of Fomin et al. [STOC 2016] who proved the case where $b=1$. As case studies, we use these results to design faster deterministic algorithms for: - checking whether a graph has a feedback vertex set of size at most $k$ - enumerating all minimal feedback vertex sets - enumerating all minimal vertex covers of size at most $k$, and - enumerating all minimal 3-hitting sets. We obtain these results by deriving new $b^{n-|X|} c^k N^{O(1)}$-time algorithms for the corresponding $\Phi$-Extension problems (or enumeration variant). In some cases, this is done by adapting the analysis of an existing algorithm, or in other cases by designing a new algorithm. Our analyses are based on Measure and Conquer, but the value to minimize, $1+b-\frac{1}{c}$, is unconventional and requires non-convex optimization.