Simple Graph Density Inequalities with No Sum of Squares Proofs

Abstract

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lovász. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. Finally, we show that our setup is equivalent to existing frameworks by Razborov and Lovász-Szegedy, and thus our results hold in these settings too.

Non-trivial squares and Sidorenko's conjecture

“…The labeled vertices in A give rise to labeled vertices in A 2 while each unlabeled vertex in A gives rise to two unlabeled twin vertices in A 2 . In fact, the labeled vertices correspond to the vertices fixed by the involution of Lemma 2.8 of [BRST20] (discussed earlier). The twin unlabeled vertices each belong to the two separate groups in the partition generated by the involution.…”

“…Many wondered whether the Cauchy-Schwarz method (which is equivalent to the sum of squares method ( [BPT12])) was the right tool to attack this problem (see [GL15] for some context). In [BRST20], the authors showed that there are instances of Sidorenko's conjecture that can not be verified with sums of squares. In particular, they showed that it cannot be verified in the case of trivial squares (defined below).…”

“…From [BRST20], we know that if H is not a trivial square, then there must be an involution ϕ of H that fixes a non-empty proper subset of V (H) such that the vertices v that are not fixed by ϕ can be partitioned into two groups, each group consisting of one vertex from each pair {v, ϕ(v)}, such that there are no edges between vertices in the two groups. This is Lemma 2.8 in [BRST20]. The proof relies on the effect squaring has on labeled and unlabeled vertices; see the reference for details.…”

“…From [BRST20], we also know that if H is a trivial square, then H − |E(H)| is not an sos. This is the case for example when H is a path of odd length, the so-called Blakley-Roy inequalities.…”

Symmetry reduction in AM/GM-based optimization

“…Note that because the action is orthogonal, the dual action on the exponent vectors is the same. Optimization problems invariant under this action can arise in different contexts naturally, for example, in the context of graph homomorphisms ( [3]). This action is very natural, and the theory of representation of the symmetric group is very well understood, and affords strong connections with combinatorics.…”