Abstract
We consider Integer Linear Programs (ILPs), where each variable corresponds to an integral point within a polytope \(\mathcal {P}\subseteq \mathbb {R}^{d}\), i. e., ILPs of the form \(\min \{c^{\top }x\mid \sum _{p\in \mathcal {P}\cap \mathbb {Z}^d} x_p p = b, x\in \mathbb {Z}^{|\mathcal {P}\cap \mathbb {Z}^d|}_{\ge 0}\}\). The distance between an optimal fractional solution and an optimal integral solution (called the proximity) is an important measure. A classical result by Cook et al. (Math. Program., 1986) shows that it is at most \(\varDelta ^{\varTheta (d)}\) where \(\varDelta =\Vert \mathcal {P}\cap \mathbb {Z}^{d} \Vert _{\infty }\) is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the right-hand side b is replaced by another right-hand side \(b'\). The distance between an optimal solution x w.r.t. b and an optimal solution \(x'\) w.r.t. \(b'\) (called the sensitivity) is similarly bounded, i. e., \(\Vert b-b' \Vert _{1}\cdot \varDelta ^{\varTheta (d)}\), also shown by Cook et al. (Math. Program., 1986).
Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of \(\varDelta \), require negative entries in the constraint matrix, or have fractional right-hand sides. Hence, these lower bounds often do not correspond to instances from algorithmic problems. This work presents for each \(\varDelta > 0\) and each \(d > 0\) ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least \(\varDelta ^{\varTheta (d)}\). Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the configuration ILP. We thereby show that the results of Cook et al. are indeed tight, even for instances arising naturally from problems in combinatorial optimization.
This work was supported by DFG project JA 612/20-1.
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References
Akiyama, J., Kano, M.: Matchings and 1-factors. In: Factors and Factorizations of Graphs, pp. 15–67. Springer (2011). https://doi.org/10.1007/978-3-642-21919-1_2
Aliev, I., Henk, M., Oertel, T.: Distances to lattice points in knapsack polyhedra. Math. Program. 175–198 (2019). https://doi.org/10.1007/s10107-019-01392-1
Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation schemes for scheduling on parallel machines. J. Sched. 1(1), 55–66 (1998)
Caprara, A., Dell’Amico, M., Díaz, J.C.D., Iori, M., Rizzi, R.: Friendly bin packing instances without integer round-up property. Math. Program. 150(1), 5–17 (2015)
Cook, W.J., Gerards, A.M.H., Schrijver, A., Tardos, É.: Sensitivity theorems in integer linear programming. Math. Program. 34(3), 251–264 (1986)
Cslovjecsek, J., Eisenbrand, F., Weismantel, R.: N-fold integer programming via LP rounding. CoRR abs/2002.07745 (2020)
Eisenbrand, F., Weismantel, R.: Proximity results and faster algorithms for integer programming using the steinitz lemma, 16(1), 5:1–5:14. ACM (2020)
Epstein, L., Levin, A.: A robust APTAS for the classical bin packing problem. Math. Program. 119(1), 33–49 (2009)
Epstein, L., Levin, A.: Robust approximation schemes for cube packing. SIAM J. Optim. 23(2), 1310–1343 (2013)
Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting-stock problem. Oper. Res. 9(6), 849–859 (1961)
Goemans, M.X., Rothvoß, T.: Polynomiality for bin packing with a constant number of item types. In: SODA, pp. 830–839. SIAM (2014)
Hadamard, J.: Resolution d’une question relative aux determinants. Bull. des Sci. Math. 2, 240–246 (1893)
Hochbaum, D.S.: Monotonizing linear programs with up to two nonzeroes per column. Oper. Res. Lett. 32(1), 49–58 (2004)
Hochbaum, D.S., Shanthikumar, J.G.: Convex separable optimization is not much harder than linear optimization, 37(4), 843–862. ACM (1990)
Holton, D.A., Sheehan, J.: The Petersen Graph. Cambridge University Press, Cambridge (1993)
Jansen, K., Klein, K.-M.: A robust AFPTAS for online bin packing with polynomial migration. SIAM J. Discret. Math. 33(4), 2062–2091 (2019)
Jansen, K., Klein, K.-M., Maack, M., Rau, M.: Empowering the configuration-ip - new PTAS results for scheduling with setups times. In: ITCS, volume 124 of LIPIcs, pp. 44:1–44:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
Jansen, K., Klein, K.-M., Verschae, J.: Closing the gap for makespan scheduling via sparsification techniques. In: ICALP, volume 55 of LIPIcs, pp. 72:1–72:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
Jansen, K., Lassota, A., Maack, M.: Approximation algorithms for scheduling with class constraints. In: SPAA (in print) (2020)
Jansen, K., Lassota, A., Rohwedder, L.: Near-linear time algorithm for n-fold ILPS via color coding. In: ICALP, volume 132 of LIPIcs, pp. 75:1–75:13 (2019)
Jansen, K., Robenek, C.: Scheduling jobs on identical and uniform processors revisited. In: Solis-Oba, R., Persiano, G. (eds.) WAOA 2011. LNCS, vol. 7164, pp. 109–122. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29116-6_10
Kempe, A.B.: A memoir in the theory of mathematical form. Philos. Trans. R. Soc. Lond. 177, 1–70 (1886)
Lee, J., Paat, J., Stallknecht, I., Xu, L.: Improving proximity bounds using sparsity. CoRR abs/2001.04659 (2020)
Paat, J., Weismantel, R., Weltge, S.: Distances between optimal solutions of mixed-integer programs. Math. Program. 455–468 (2018). https://doi.org/10.1007/s10107-018-1323-z
Petersen, J.: Sur le théorèm de tait. L’Intermédiare des Mathématiciens 5(1), 225–227 (1898)
Sanders, P., Sivadasan, N., Skutella, M.: Online scheduling with bounded migration. Math. Oper. Res. 34(2), 481–498 (2009)
Schrijver, A.: Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization. Wiley (1999)
Skutella, M., Verschae, J.: Robust polynomial-time approximation schemes for parallel machine scheduling with job arrivals and departures. Math. Oper. Res. 41(3), 991–1021 (2016)
Subramani, K.: On deciding the non-emptiness of 2sat polytopes with respect to first order queries. Math. Log. Q. 50(3), 281–292 (2004)
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The authors want to thank Lars Rohwedder for enjoyable and fruitful discussions at the beginning of this project.
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Berndt, S., Jansen, K., Lassota, A. (2021). Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs. In: Bureš, T., et al. SOFSEM 2021: Theory and Practice of Computer Science. SOFSEM 2021. Lecture Notes in Computer Science(), vol 12607. Springer, Cham. https://doi.org/10.1007/978-3-030-67731-2_25
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