Abstract
A covering problem is an integer linear program of type \(\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \text{ integral }\}\) where \(A\in \mathbb {Z}^{m\times n}_+\), \(D\in \mathbb {Z}_+^m\), and \(c,d\in \mathbb {Z}_+^n\). In this paper, we study covering problems with additional precedence constraints \(\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}\), where \(\mathcal {P}=([n], \preceq )\) is some arbitrary, but fixed partial order on the items represented by the column-indices of \(A\). Such precedence constrained covering problems (PCCP) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, i.e., where \(m=1\) and \(d\equiv 1\).
Our main result is a strongly-polynomial primal-dual approximation algorithm for PCCP with \(d\equiv 1\). Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of \(\mathcal {P}\), i.e., by the size of a maximum antichain in \(\mathcal {P}\). Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets \(\mathcal {P}\). For the general case with , we present pseudo-polynomial algorithms.
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References
Boland, N., Bley, A., Fricke, C., Froyland, G., Sotirov, R.: Clique-based facets for the precedence constrained knapsack problem. Math. Program. 133(1–2), 481–511 (2012)
Boyd, A.: Polyhedral results for the precedence-constrained knapsack problem. Discret. Appl. Math. 41(3), 185–201 (1993)
Carnes, T., Shmoys, D.B.: Primal-dual schema for capacitated covering problems. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 288–302. Springer, Heidelberg (2008)
Carr, R., Fleischer, L., Leung, V., Phillips, C.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 106–115. Society for Industrial and Applied Mathematics (2000)
Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34(5), 1129–1146 (2005)
Fujito, T., Yabuta, T.: Submodular integer cover and its application to production planning. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 154–166. Springer, Heidelberg (2005)
Hajiaghayi, M., Jain, K., Konwar, K., Lau, L., Mandoiu, I., Russell, A., Shvartsman, A., Vazirani, V.: The minimum k-colored subgraph problem in haplotyping and dna primer selection. In: Proceedings of the International Workshop on Bioinformatics Research and Applications (IWBRA) (2006)
Ibarra, O.H., Kim, C.E.: Fast approximation algorithms for the knapsack and sum of subset problems. J. ACM (JACM) 22(4), 463–468 (1975)
Johnson, D., Niemi, K.: On knapsacks, partitions, and a new dynamic programming technique for trees. Math. Oper. Res. 8(1), 1–14 (1983)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)
Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-\(\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)
Kolliopoulos, S.G., Steiner, G.: Partially-ordered knapsack and applications to scheduling. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 612–624. Springer, Heidelberg (2002)
Kolliopoulos, S.G., Young, N.E.: Tight approximation results for general covering integer programs. In: Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, 2001, pp. 522–528. IEEE (2001)
Koufogiannakis, C., Young, N.E.: Greedy \(\delta \)-approximation algorithm for covering with arbitrary constraints and submodular cost. Algorithmica 66(1), 113–152 (2013)
Van de Leensel, R., van Hoesel, C., van de Klundert, J.: Lifting valid inequalities for the precedence constrained knapsack problem. Math. program. 86(1), 161–185 (1999)
Park, K., Park, S.: Lifting cover inequalities for the precedence-constrained knapsack problem. Discret. Appl. Math. 72(3), 219–241 (1997)
Pritchard, D., Chakrabarty, D.: Approximability of sparse integer programs. Algorithmica 61(1), 75–93 (2011)
Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing, pp. 453–461. ACM (2001)
Woeginger, G.: On the approximability of average completion time scheduling under precedence constraints. Discret. Appl. Math. 131(1), 237–252 (2003)
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Wierz, A., Peis, B., McCormick, S.T. (2015). Primal-Dual Algorithms for Precedence Constrained Covering Problems. In: Bampis, E., Svensson, O. (eds) Approximation and Online Algorithms. WAOA 2014. Lecture Notes in Computer Science(), vol 8952. Springer, Cham. https://doi.org/10.1007/978-3-319-18263-6_22
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