Linear Level Lasserre Lower Bounds for Certain k-CSPs

We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random k-CSP instance over any predicate type implied by k-XOR constraints, for example k-SAT or k-XOR. (One constant is said to imply another if the latter is true whenever the former is. For example k-XOR constraints imply k-CNF constraints.) As a result the Ω(n) level Lasserre relaxation fails to approximate such CSPs better than the trivial, random algorithm. As corollaries, we obtain Ω(n) level integrality gaps for the Lasserre hierarchy of 7 6 − ε for VertexCover, 2 − ε for k-UniformHypergraphVertexCover, and any constant for k-UniformHypergraphIndependentSet. This is the first construction of a Lasserre integrality gap. Our construction is notable for its simplicity. It simplifies, strengthens, and helps to explain several previous results.

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Grant Schoenebeck
• Keywords: FOCS 2008, K-XOR Constraints, Lasserre Hierarchy, Random K-CSP Instance, Theoretical Computer Science