Abstract
Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2 + k clauses? Or, if an nvertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2 + k for some k ≥ 1 and how difficult is it to find such a vertex cover?
The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on wellstudied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2connected nvertex graph G with minimum degree δ(G) ≤ n/2, the length of the longest cycle L is at least 2δ(G). Thus the "essential" part of finding the longest cycle is in approximating the "offset" k = L  2δ(G). The main result of this paper is the aboveguarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably wellbehaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k)).
BibTeX  Entry
@InProceedings{fomin_et_al:LIPIcs.ICALP.2023.60,
author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill},
title = {{Approximating Long Cycle Above Dirac’s Guarantee}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {60:160:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772785},
ISSN = {18688969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18112},
URN = {urn:nbn:de:0030drops181128},
doi = {10.4230/LIPIcs.ICALP.2023.60},
annote = {Keywords: Longest path, longest cycle, approximation algorithms, above guarantee parameterization, minimum degree, Dirac theorem}
}
Keywords: 

Longest path, longest cycle, approximation algorithms, above guarantee parameterization, minimum degree, Dirac theorem 
Collection: 

50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) 
Issue Date: 

2023 
Date of publication: 

05.07.2023 