eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-01
50:1
50:19
10.4230/LIPIcs.ESA.2022.50
article
Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search
Esmer, Barış Can
1
2
https://orcid.org/0000-0001-5694-1465
Kulik, Ariel
1
https://orcid.org/0000-0002-0533-3926
Marx, Dániel
1
https://orcid.org/0000-0002-5686-8314
Neuen, Daniel
3
https://orcid.org/0000-0002-4940-0318
Sharma, Roohani
4
https://orcid.org/0000-0003-2212-1359
CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Saarbrücken Graduate School of Computer Science, Saarland Informatics Campus, Germany
School of Computing Science, Simon Fraser University, Burnaby, Canada
Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time.
We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1.
We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol244-esa2022/LIPIcs.ESA.2022.50/LIPIcs.ESA.2022.50.pdf
parameterized approximations
exponential approximations
monotone local search