Abstract
The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a selfreduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time.
In this paper, we study a setting that is more typical for NPcomplete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rankbased approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use
 𝒪(tlog n + log²{n}) random bits on graphs of treewidth at most t;
 𝒪(√n) random bits on planar or Hminor free graphs; and
 𝒪(n)random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed H we obtain an algorithm for detecting a Hamiltonian cycle in an Hminorfree graph that runs in deterministic time 2^{𝒪(√n)} and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use 𝒪(d) random bits and assign polynomiallybounded weights.
We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.
BibTeX  Entry
@InProceedings{nederlof_et_al:LIPIcs.STACS.2022.50,
author = {Nederlof, Jesper and Pilipczuk, Micha{\l} and Swennenhuis, C\'{e}line M. F. and W\k{e}grzycki, Karol},
title = {{Isolation Schemes for Problems on Decomposable Graphs}},
booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
pages = {50:150:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772228},
ISSN = {18688969},
year = {2022},
volume = {219},
editor = {Berenbrink, Petra and Monmege, Benjamin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15860},
URN = {urn:nbn:de:0030drops158601},
doi = {10.4230/LIPIcs.STACS.2022.50},
annote = {Keywords: Isolation Lemma, Derandomization, Hamiltonian Cycle, Exact Algorithms}
}
Keywords: 

Isolation Lemma, Derandomization, Hamiltonian Cycle, Exact Algorithms 
Collection: 

39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) 
Issue Date: 

2022 
Date of publication: 

09.03.2022 