Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
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SIAM Journal on Discrete Mathematics
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Sequencing problems with an unknown covering or packing constraint appear in various applications, e.g., in real-time computing environments with uncertain run-time availability. A sequence is called α-robust when, for any possible constraint, the maximal or minimal prefix of the sequence that satisfies the constraint is at most a factor α from an optimal packing or covering. It is known that the covering problem always admits a 4-robust solution, and there are instances for which this factor is tight. For the packing variant no such constant robustness factor is possible in general. In this work we address the fact that many problem instances may allow for a much better robustness guarantee than the pathological worst case instances. We aim for more meaningful, instance-sensitive performance guarantees. We present an algorithm that constructs for each instance a solution with a robustness factor arbitrarily close to optimal. This implies nearly optimal solutions for previously studied problems such as the universal knapsack problem and for universal scheduling on an unreliable machine. The crucial ingredient and main result is a nearly exact feasibility test for dual-value sequencing with a given target function. We show that deciding exact feasibility is strongly NP-hard, and thus, our test is best possible, unless P=NP. We hope that the idea of instance-sensitive performance guarantees inspires to revisit other optimization problems and design algorithm tailored to perform well for each individual instance.