Approximation algorithms for NP-hard problems
Approximation algorithms for NP-hard problems
Using homogeneous weights for approximating the partial cover problem
Journal of Algorithms
Approximation algorithms
Some optimal inapproximability results
Journal of the ACM (JACM)
STACS '98 Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
The t-Vertex Cover Problem: Extending the Half Integrality Framework with Budget Constraints
APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
Approximation algorithms for partial covering problems
Journal of Algorithms
A general approach for incremental approximation and hierarchical clustering
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
An incremental model for combinatorial maximization problems
WEA'06 Proceedings of the 5th international conference on Experimental Algorithms
A primal-dual approximation algorithm for partial vertex cover: making educated guesses
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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In the classical k-vertex cover problem, we wish to find a minimum weight set of vertices that covers at least k edges. In the incremental version of the k-vertex cover problem, we wish to find a sequence of vertices, such that if we choose the smallest prefix of vertices in the sequence that covers at least k edges, this solution is close in value to that of the optimal k-vertex cover solution. The maximum ratio is called competitive ratio. Previously the known upper bound of competitive ratio was 4α, where α is the approximation ratio of the k-vertex cover problem. And the known lower bound was 1.36 unless P = NP, or 2-ε for any constant ε assuming the Unique Game Conjecture. In this paper we present some new results for this problem. Firstly we prove that, without any computational complexity assumption, the lower bound of competitive ratio of incremental vertex cover problem is φ, where φ = √5+1/2 ≈ 1.618 is the golden ratio. We then consider the restricted versions where k is restricted to one of two given values(Named 2-IVC problem) and one of three given values(Named 3-IVC problem). For 2-IVC problem, we give an algorithm to prove that the competitive ratio is at most φα. This incremental algorithm is also optimal for 2-IVC problem if we are permitted to use non-polynomial time. For the 3-IVC problem, we give an incremental algorithm with ratio factor (1 + √2)α.