Approximate Graph Colouring and Crystals

Abstract

We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor.

Approximate Graph Colouring and the Hollow Shadow

“…integer) distributions over the set of partial assignments from portions of the instance of size at most k to A. The former choice results in the Sherali-Adams LP hierarchy [67], which we call the BLP hierarchy; the latter results in the affine integer programming hierarchy [32], which we call the AIP hierarchy. Crucially, the former but not the latter choice ensures local consistency: Each assignment receiving nonzero weight in the BLP hierarchy corresponds to a partial homomorphism, while the same is not true for the AIP hierarchy.…”

“…Then, the family of all (not necessarily oriented) low-dimensional projections of C satisfies the required symmetries. We call such a tensor C a crystal (in accordance with [32]), while the shadow of C is any of its oriented projections. We then reformulate the problem to its final form; the solution of this problem is the main technical result of the paper.…”

“…This is in sharp contrast with the weaker AIP hierarchy, for which a similar acceptance result holds, cf. [32]. Hence, unlike for AIP, one cannot simply take large cliques as fooling instances for BA.…”

“…The next step is to build a (not necessarily hollow) 3-dimensional 2-crystal having shadow U . This can be done using the method developed in [32] as a black box (specifically, we use Proposition 54), and it results in the tensor…”

“…As detailed in [56], this implies (together with [45]) that the sum-of-squares hierarchy does not solve AGC, which implies the same result for the weaker Sherali-Adams and bounded width hierarchies. Regarding the second type of algorithms, very recent work of Ciardo and Živný established that the AIP hierarchy does not solve AGC [32].…”