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For given integers $n$ and $d$, both at least 2, we consider a homogeneousmultivariate polynomial $f_d$ of degree $d$ in variables indexed by the edgesof the complete graph on $n$ vertices and coefficients depending oncardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden(arXiv:2111.05777, 2021) asked whether $f_d$, which arises in a model ofjob-occupancy in redundancy scheduling, attains its minimum over the standardsimplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAMJ. Optim. 31 (2021), 2227--2254] proved that $f_d$ is convex over the standardsimplex if $d=2$ and $d=3$, implying the desired result for these $d$. We givea symmetry reduction to show that for fixed $d$, the polynomial is convex overthe standard simplex (for all $n\geq 2$) if a constant number of constantmatrices (with size and coefficients independent of $n$) are positivesemidefinite. This result is then used in combination with a computer-assistedverification to show that the polynomial $f_d$ is convex for $d\leq 9$.

Symmetry reduction to optimize a graph-based polynomial from queueing theory