Generalized submodular cover problems and applications
Theoretical Computer Science
Tight Approximation Results for General Covering Integer Programs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Approximating covering integer programs with multiplicity constraints
Discrete Applied Mathematics
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
Approximation algorithms for partial covering problems
Journal of Algorithms
Approximating k-median with non-uniform capacities
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Covering Problems with Hard Capacities
SIAM Journal on Computing
Approximation algorithms for combinatorial problems
Journal of Computer and System Sciences
An improved approximation algorithm for vertex cover with hard capacities
Journal of Computer and System Sciences
Approximation of partial capacitated vertex cover
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Scheduling to minimize power consumption using submodular functions
Proceedings of the twenty-second annual ACM symposium on Parallelism in algorithms and architectures
Energy efficient scheduling via partial shutdown
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete 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 this paper, we consider generalizations of classical covering problems to handle hard capacities. In the hard capacitated set cover problem, additionally each set has a covering capacity which we are not allowed to exceed. In other words, after picking a set, we may cover at most a specified number of elements. Based on the classical results by Wolsey, an O(logn) approximation follows for this problem. Chuzhoy and Naor [FOCS 2002], first studied the special case of unweighted vertex cover with hard capacities and developed an elegant 3 approximation for it based on rounding a natural LP relaxation. This was subsequently improved to a 2 approximation by Gandhi et al. [ICALP 2003]. These results are surprising in light of the fact that for weighted vertex cover with hard capacities, the problem is at least as hard as set cover to approximate. Hence this separates the unweighted problem from the weighted version. The set cover hardness precludes the possibility of a constant factor approximation for the hard-capacitated vertex cover problem on weighted graphs. However, it was not known whether a better than logarithmic approximation is possible on unweighted multigraphs, i.e., graphs that may contain parallel edges. Neither the approach of Chuzhoy and Naor, nor the follow-up work of Gandhi et al. can handle the case of multigraphs. In fact, achieving a constant factor approximation for hard-capacitated vertex cover problem on unweighted multigraphs was posed as an open question in Chuzhoy and Naor's work. In this paper, we resolve this question by providing the first constant factor approximation algorithm for the vertex cover problem with hard capacities on unweighted multigraphs. Previous works cannot handle hypergraphs which is analogous to consider set systems where elements belong to at most f sets. In this paper, we give an O(f) approximation algorithm for this problem. Further, we extend these works to consider partial covers.