Abstract
We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of “well-behaved” univariate polynomial inequalities.
We illustrate the technique on two problems, one unconstrained and the other with constraints. More precisely, we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. We also show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.
Supported by the Swiss National Science Foundation project 200020-144491/1 “Approximation Algorithms for Machine Scheduling Through Theory and Experiments” and by Sciex Project 12.311.
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Notes
- 1.
- 2.
In order to keep the notation simple, we do not emphasize the parameter q as the dimension of the vectors should be clear from the context.
- 3.
We define the set-valued permutation by \(\pi (I) = \left\{ \pi (i)~|~i \in I \right\} \).
- 4.
It can be shown that the roots \(r_i\) can be assumed to be real numbers.
- 5.
Recall that at level n the integrality gap vanishes.
- 6.
The full version of the paper is available at http://arxiv.org/abs/1407.1746.
- 7.
Denote by \({x}^{\underline{m}} = x(x-1)\cdots (x-m+1)\) the falling factorial (with the convention that \({x}^{\underline{0}} = 1\)).
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Acknowledgements
The authors would like to express their gratitude to Ola Svensson for helpful discussions and ideas regarding this paper.
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Kurpisz, A., Leppänen, S., Mastrolilli, M. (2016). Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_30
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