SOS Lower Bounds with Hard Constraints: Think Global, Act Local
Pravesh K. Kotharia, Ryan O’Donnellb, Tselil Schrammc
- aDepartment of Computer Science, Princeton University and Institute for Advanced Study, Princeton, USA
- bDepartment of Computer Science, Carnegie Mellon University, Pittsburgh, USA
- cDepartment of Computer Science, Harvard and MIT, Cambridge, USA
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Abstract
Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value beta, it did not necessarily actually "satisfy" the constraint "objective = beta". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating global constraints into local constraints. Using these ideas, we also show that degree-Omega(sqrt{n}) SOS does not provide a (4/3 - epsilon)-approximation for Min-Bisection, and degree-Omega(n) SOS does not provide a (11/12 + epsilon)-approximation for Max-Bisection or a (5/4 - epsilon)-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.