Abstract
Sherali and Adams (SIAM J Discrete Math 3:411–430, 1990) and Lovász and Schrijver (SIAM J Optim 1:166–190, 1991) developed systematic procedures to construct the hierarchies of relaxations known as lift-and-project methods. They have been proven to be a strong tool for developing approximation algorithms, matching the best relaxations known for problems like Max-Cut and Sparsest-Cut. In this work we provide lower bounds for these hierarchies when applied over the configuration LP for the problem of scheduling identical machines to minimize the makespan. First we show that the configuration LP has an integrality gap of at least 1024/1023 by providing a family of instances with 15 different job sizes. Then we show that for any integer n there is an instance with n jobs in this family such that after \(\varOmega (n)\) rounds of the Sherali–Adams (\(\text {SA}\)) or the Lovász–Schrijver (\(\text {LS}_+\)) hierarchy the integrality gap remains at least 1024/1023.
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Notes
The description of the projection of the configuration LP onto the assignment space corresponds to the assignment LP stranghtened with additional constraints [29].
This definition is slightly different from the one in Sherali and Adams [27]; for simplicity we give a definition that, in the case of equality constraints, is equivalent. Indeed, note that if for every variable \(x_k\) there exist a set of variables’ indices \(S_k\), \(k \in S_k\) and a constant \(c_k\ge 1\) such that the constraint \(\sum _{\ell \in S_k} x_\ell = c_k\) is valid for the starting polytope P, then for every subsets \(I, K \subseteq [n]\), \(|I \dot{\cup }K| \le r\) and every constraint of P, \(\sum a_j x_j -b \ge 0\) the generated SA constraint \(\mathbb {L}\left( (\sum a_j x_j - b) \prod _{i \in I} x_i \prod _{k \in K}(1-x_k)\right) \ge 0\) can be replaced with \(\mathbb {L}\left( (\sum a_j x_j - b) \prod _{i \in I} x_i \prod _{k \in K}(\sum _{\ell \in S_k \setminus \{k\}} x_\ell +c_k-1) \right) \ge 0\). The operator \(\mathbb {L}\) corresponds to the linearization of the polynomial products.
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Supported by the Swiss National Science Foundation project 200020-144491/1 “Approximation Algorithms for Machine Scheduling Through Theory and Experiments,” by Sciex Project 12.311, by DFG Grant MO 2889/1-1, and by CONICYT-PCHA/Doctorado Nacional/2014-21140930.
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Kurpisz, A., Mastrolilli, M., Mathieu, C. et al. Semidefinite and linear programming integrality gaps for scheduling identical machines. Math. Program. 172, 231–248 (2018). https://doi.org/10.1007/s10107-017-1152-5
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DOI: https://doi.org/10.1007/s10107-017-1152-5