Ultimate Greedy Approximation of Independent Sets in Subcubic Graphs

We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. It is known for its inherent hardness of approximation. We focus on the well known minimum-degree greedy algorith...

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Veröffentlicht in:Algorithmica Jg. 86; H. 11; S. 3518 - 3578
Hauptverfasser: Krysta, Piotr, Mari, Mathieu, Zhi, Nan
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.11.2024
Springer Nature B.V
Springer Verlag
Schlagworte:
Algorithm Analysis and Problem Complexity
Algorithms
Approximation
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Design analysis
Graphs
Greedy algorithms
Hardness
Lower bounds
Mathematics of Computing
Staples
Theory of Computation
Upper bounds
Greedy algorithms
68Q25
Bounded degree graphs
Approximation algorithms
Independent set problem
68W25
90C27
ISSN:0178-4617, 1432-0541
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Zusammenfassung:We study the approximability of the maximum size independent set (MIS) problem in bounded degree graphs. This is one of the most classic and widely studied NP-hard optimization problems. It is known for its inherent hardness of approximation. We focus on the well known minimum-degree greedy algorithm for this problem. This algorithm iteratively chooses a minimum degree vertex in the graph, adds it to the solution and removes its neighbors, until the remaining graph is empty. The approximation ratios of this algorithm have been widely studied, where it is augmented with an advice that tells the greedy algorithm which minimum degree vertex to choose if it is not unique. Our main contribution is a new mathematical theory for the design of such greedy algorithms for MIS with efficiently computable advice and for the analysis of their approximation ratios. Using this theory we obtain the ultimate approximation ratio of 5/4 for greedy algorithms on graphs with maximum degree 3, which completely solves an open problem from the paper by Halldórsson and Yoshihara (in: Staples, Eades, Katoh, Moffat (eds) Algorithms and computations—ISAAC ’95, in 2026 LNCS, Springer, Berlin, Heidelberg, 1995) . Our algorithm is the fastest currently known algorithm for MIS with this approximation ratio on such graphs. We also obtain a simple and short proof of the ( Δ + 2 ) / 3 -approximation ratio of any greedy algorithms on graphs with maximum degree Δ , the result proved previously by Halldórsson and Radhakrishnan (Nord J Comput 1:475–492, 1994) . We almost match this ratio by showing a lower bound of ( Δ + 1 ) / 3 on the ratio of a greedy algorithm that can use an advice. We apply our new algorithm to the minimum vertex cover problem on graphs with maximum degree 3 to obtain a substantially faster 6/5-approximation algorithm than the one currently known. We complement our algorithmic, upper bound results with lower bound results, which show that the problem of designing good advice for greedy algorithms for MIS is computationally hard and even hard to approximate on various classes of graphs. These results significantly improve on the previously known hardness results. Moreover, these results suggest that obtaining the upper bound results on the design and analysis of the greedy advice is non-trivial.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-024-01268-7