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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Abstract	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.
AbstractList	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 $(\Delta+2)/3$-approximation ratio of any greedy algorithms on graphs with maximum degree $\Delta$, 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 $(\Delta+1)/3$ 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. 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. 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.
Author	Krysta, Piotr Zhi, Nan Mari, Mathieu
Author_xml	– sequence: 1 givenname: Piotr surname: Krysta fullname: Krysta, Piotr organization: Augusta University, University of Liverpool – sequence: 2 givenname: Mathieu orcidid: 0000-0001-8074-0241 surname: Mari fullname: Mari, Mathieu email: mathieu.mari@lirmm.fr organization: LIRMM, Université de Montpellier, CNRS – sequence: 3 givenname: Nan surname: Zhi fullname: Zhi, Nan organization: University of Liverpool
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Cites_doi	10.1007/BF01994876 10.1002/rsa.3240050504 10.4086/toc.2011.v007a003 10.1142/S0129054118500168 10.1016/S0020-0190(96)00208-6 10.1007/BF01580444 10.1137/S089548010240415X 10.1007/BF02523693 10.1016/S0893-9659(97)00044-X 10.1016/0166-218X(83)90080-X 10.1016/0012-365X(83)90273-X 10.1016/S0304-3975(98)00158-3 10.1007/978-3-540-27796-5_5 10.1137/0403025 10.1007/BFb0015418 10.1016/0095-8956(75)90089-1 10.1006/jcss.2000.1727 10.1016/0304-3975(76)90059-1 10.1137/S0097539700381097 10.1137/S0097539795286612 10.1145/2811255 10.1007/BF01874388 10.1007/s002240000113 10.1145/195058.195221 10.1007/978-3-540-45077-1_4 10.1007/978-1-4684-2001-2_9 10.1145/2746539.2746607 10.1007/BF02392825 10.1007/3-540-58218-5_18 10.1145/1132516.1132612
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Snippet	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...
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SubjectTerms	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
Title	Ultimate Greedy Approximation of Independent Sets in Subcubic Graphs
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