Sum-of-squares hierarchies for binary polynomial optimization

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube Bn={0,1}n. This hierarchy provides for each integer r∈N a lower bound f(r) on the minimum fmin of f, given by the largest scalar λ for which the polynomial f−λ is a sum-of-squares on Bn with degree at most 2r. We analyze the quality of these bounds by estimating the worst-case error fmin−f(r) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t∈[0,1/2], we can show that this worst-case error in the regime r≈t⋅n is of the order 1/2−t(1−t)−−−−−−√ as n tends to ∞. Our proof combines classical Fourier analysis on Bn with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f(r) and another hierarchy of upper bounds f(r), for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube (Z/qZ)n

A recursive theta body for hypergraphs

The theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is a tractable relaxation of the independent-set polytope derived from the Lovász theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel’s theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube

Sum-of-squares hierarchies for binary polynomial optimization

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube Bn= { 0, 1 } n. This hierarchy provides for each integer r∈ N a lower bound f( r ) on the minimum fmin of f, given by the largest scalar λ for which the polynomial f- λ is a sum-of-squares on Bn with degree at most 2r. We analyze the quality of these bounds by estimating the worst-case error fmin- f( r ) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t∈ [ 0, 1 / 2 ], we can show that this worst-case error in the regime r≈ t· n is of the order 1/2-t(1-t) as n tends to ∞. Our proof combines classical Fourier analysis on Bn with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f( r ) and another hierarchy of upper bounds f( r ), for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube (Z/ qZ) n