Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Approximation algorithms
Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
Many hard examples in exact phase transitions
Theoretical Computer Science
Theoretical Computer Science
Approximating the minimum weight weak vertex cover
Theoretical Computer Science - Computing and combinatorics
Random constraint satisfaction: Easy generation of hard (satisfiable) instances
Artificial Intelligence
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
A better list heuristic for vertex cover
Information Processing Letters
Mean analysis of an online algorithm for the vertex cover problem
Information Processing Letters
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Introduction to Algorithms, Third Edition
Introduction to Algorithms, Third Edition
A tight analysis of the maximal matching heuristic
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
A list heuristic for vertex cover
Operations Research Letters
Implementation and comparison of heuristics for the vertex cover problem on huge graphs
SEA'12 Proceedings of the 11th international conference on Experimental Algorithms
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The vertex cover is a well-known NP-complete minimization problem in graphs that has received a lot of attention these last decades. Many algorithms have been proposed to construct vertex cover in different contexts (offline, online, list algorithms, etc.) leading to solutions of different level of quality. This quality is traditionally measured in terms of approximation ratio, that is, the worst possible ratio between the quality of the solution constructed and the optimal one. For the vertex cover problem the range of such known ratios are between 2 (conjectured as being the smallest constant ratio) and Δ, the maximum degree of the graph. Based on this measure of quality, the hierarchy is almost clear (the smaller the ratio is, the better the algorithm is). In this article, we show that this measure, although of great importance, is too macroscopic and does not reflect the practical behavior of the methods. We prove this by analyzing (known and recent) algorithms running on a particular class of graphs: the paths. We obtain closed and exact formulas for the mean of the sizes of vertex cover constructed by these different algorithms. Then, we assess their quality experimentally in several well-chosen class of graphs (random, regular, trees, BHOSLIB benchmarks, trap graphs, etc.). The synthesis of all these results lead us to formulate a “practical hierarchy” of the algorithms. We remark that it is, more or less, the opposite to the one only based on approximation ratios, showing that worst-case analysis only gives partial information on the quality of an algorithm.