Symmetric sums of squares over k-subset hypercubes

Abstract

Abstract. We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the k-element subsets of [n]. For simplicity, we focus on the case k = 2, but our results extend naturally to all values of k ≥ 2. We develop a variant of the Gatermann-Parrilo symmetryreduction method tailored to our setting that allows for several simplifications and a connection to flag algebras.We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.

Breaking Symmetries to Rescue Sum of Squares: The Case of Makespan Scheduling

“…We show how to prove Theorem 4 following the lines in the work of Raymond et al [59]. We need a few intermediate results, and the symmetry reduction theorem from Gaterman & Parrillo, stated in our setting [16].…”

“…Remarkable in this line is the work of Gatermann & Parrillo, that studied how to obtain reduced sums of squares certificates of non-negativity when the polynomial is invariant under the action of a group, using tools from representation theory [16]. Recently, Raymond et al developed on the Gatermann & Parrillo method to construct symmetry-reduced sum of squares certificates for polynomials over k-subset hypercubes [59]. Furthermore, the authors make an interesting connection with the Razborov method and flag algebras [60,61].…”

“…The following lemma is a variant of [59,Theorem 2] for the action of the symmetric group in our setting. Together with previous lemma and the theorem of Gatermann & Parrillo we can conclude Theorem 4.…”

“…Naturally, since the configuration linear program is stronger than the assignment linear program, our result holds if we apply Ω(n) rounds of SoS over the assignment linear program. The proof of the lower bound relies on tools from representation theory of symmetric groups over polynomials rings and it is inspired on the recent work by Raymond et al for symmetric sums of squares in hypercubes [59]. The lower bound comes by constructing high-degree pseudoexpectations on one hand, and by obtaining symmetry-reduced decompositions of the polynomial ideal defined by the configuration liner program, on the other hand.…”

Simple Graph Density Inequalities with No Sum of Squares Proofs

“…Therefore Cauchy-Schwarz proofs in the flag algebras are equivalent to sos proofs in the gluing algebra. We refer to [12] and [13] for more details.…”