Abstract
We study the approximability of covering problems when the set of items chosen to satisfy the covering constraints must form an ideal of a given partial order. We examine the general case with multiplicity constraints, where item i can be chosen up to \(d_i\) times. For the basic Precedence-Constrained Knapsack problem (PCKP) we answer an open question of McCormick et al. [10] and show the existence of approximation algorithms with strongly-polynomial bounds. PCKP is a special case, with a single covering constraint, of a Precedence-Constrained Covering Integer Program (PCCP). For a general PCCP where the number of covering constraints is \(m \ge 1,\) we show that an algorithm of Pritchard and Chakrabarty [11] for Covering Integer Programs can be extended to yield an f-approximation, where f is the maximum number of variables with nonzero coefficients in a covering constraint. This is nearly-optimal under standard complexity-theoretic assumptions and surprisingly matches the bound achieved for the problem without precedence constraints.
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Kolliopoulos, S.G., Skarlatos, A. (2021). Precedence-Constrained Covering Problems with Multiplicity Constraints. In: Koenemann, J., Peis, B. (eds) Approximation and Online Algorithms. WAOA 2021. Lecture Notes in Computer Science(), vol 12982. Springer, Cham. https://doi.org/10.1007/978-3-030-92702-8_15
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DOI: https://doi.org/10.1007/978-3-030-92702-8_15
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