Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling

Abstract

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 this hierarchy give integrality gaps matching the best known approximation algorithms. For many other problems, however, ad-hoc techniques give better approximation ratios than \sos{} in the worst case, as shown by corresponding lower bound instances. Notably, in many cases these instances are invariant under the action of a large permutation group. This yields the question how symmetries in a formulation degrade the performance of the relaxation obtained by the \sos{} hierarchy. In this paper, we study this for the case of the minimum makespan problem on identical machines. Our first result is to show that $\Omega(n)$ rounds of \sos{} applied over the \emph{configuration linear program} yields an integrality gap of at least $1.0009$, where $n$ is the number of jobs. Our result is based on tools from representation theory of symmetric groups. Then, we consider the weaker \emph{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 $2^{\tilde{O}(1/\varepsilon^2)}$ rounds of the SA hierarchy to this stronger linear program reduces the integrality gap to $1+\varepsilon$, which yields a linear programming based polynomial time approximation scheme. Our results suggest that for this classical problem, symmetries were the main barrier preventing the \sos{}/ SA hierarchies to give relaxations of polynomial complexity with an integrality gap of~$1+\varepsilon$. We leave as an open question whether this phenomenon occurs for other symmetric problems.