Breaking symmetries to rescue Sum of Squares: The case of makespan scheduling

Victor Verdugo, José Verschae

The Sum of Squares (SoS) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of the hierarchy give integrality gaps matching the best known approximation algorithm. In many other, however, ad-hoc techniques give significantly better approximation ratios. Notably, the lower bounds instances, in many cases, are invariant under the action of a large permutation group. The main purpose of this paper is to study how the presence of symmetries on a formulation degrades the performance of the relaxation obtained by the SoS hierarchy. We do so for the special case of the minimum makespan problem on identical machines. Our first result is to show that a linear number of rounds of SoS applied over the configuration linear program yields an integrality gap of at least $1.0009$. This improves on the recent work by Kurpisz et al. [Math. Program. '18] that shows an analogous result for the weaker LS$_+$ and SA hierarchies. Then, we consider the weaker assignment linear program and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying the SoS hierarchy for $O_\varepsilon(1)$ rounds to this linear program reduces the integrality gap to $(1+\varepsilon)$. Our results suggest that for this classical problem the symmetries of the natural assignment linear program were the main barrier preventing the SoS hierarchy to give relaxations with integrality gap $(1+\varepsilon)$ after a constant number of rounds. We leave as an open question whether this phenomenon occurs for different problems where the SoS hierarchy yields weak relaxations.

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