Approximate Graph Colouring and the Hollow Shadow

Abstract

We show that approximate graph colouring is not solved by constantly many levels of the liftand-project hierarchy for the combined basic linear programming and affine integer programming relaxation. The proof involves a construction of tensors whose fixed-dimensional projections are equal up to reflection and satisfy a sparsity condition, which may be of independent interest.

1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise

“…A consequence of this fact is that the universal-algebraic tools that allow generating an infinite set of new identities from a single polymorphic identity fails for PCSPs. This, in turn, stimulated the use of different tools to study PCSP polymorphisms, including Boolean function analysis [BGS23a], topology [KOWŽ23], matrix and tensor theory [CŽ23c,CŽ23a,CŽ23b], and Fourier analysis [HMS23].…”