Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
SIAM Journal on Discrete Mathematics
SIAM Journal on Computing
Robust subgraphs for trees and paths
ACM Transactions on Algorithms (TALG)
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Robust matchings and matroid intersections
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
A General Approach for Incremental Approximation and Hierarchical Clustering
SIAM Journal on Computing
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
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In this paper, we study the robustness over the cardinality variation for the knapsack problem. For the knapsack problem and a positive number α ≤1, we say that a feasible solution is α-robust if, for any positive integer k , it includes an α -approximation of the maximum k -knapsack solution, where a k -knapsack solution is a feasible solution that consists of at most k items. In this paper, we show that, for any ε 0, the problem of deciding whether the knapsack problem admits a (ν +ε )-robust solution is weakly NP-hard, where ν denotes the rank quotient of the corresponding knapsack system. Since the knapsack problem always admits a ν -robust knapsack solution [7], this result provides a sharp border for the complexity of the robust knapsack problem. On the positive side, we show that a max-robust knapsack solution can be computed in pseudo-polynomial time, and present a fully polynomial time approximation scheme (FPTAS) for computing a max-robust knapsack solution.