Exact Algorithms via Multivariate Subroutines *
We consider the family of Φ-Subset problems, where the input consists of an instance I of size N over a universe U I of size n and the task is to check whether the universe contains a subset with property Φ (e.g., Φ 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 Φ-Subset in time (1 + b − 1 c) n N O(1) , provided that there is an algorithm for the Φ-Extension problem with running time b
... running time b n−|X| c k N O(1). Here, the input for Φ-Extension is an instance I of size N over a universe U I of size n, a subset X ⊆ U I , and an integer k, and the task is to check whether there is a set Y with X ⊆ Y ⊆ U I and |Y \ X| ≤ k with property Φ. We also 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 Φ. This generalizes the results of Fomin et al. [STOC 2016] who proved them for the case 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 Φ-Extension problems (or the enumeration variant). In some cases, this is done by simply adapting the analysis of an existing algorithm, in other cases it is done by designing a new algorithm. Our analyses are based on Measure and Conquer, but the value to minimize, 1 + b − 1 c , is unconventional and leads to non-convex optimization problems in the analysis.