Better bounds on Grothendieck constants of finite orders

• URL: http://arxiv.org/abs/2409.03739v1
• Date: Thu, 5 Sep 2024 17:53:52 GMT
• Title: Better bounds on Grothendieck constants of finite orders
• Authors: Sébastien Designolle, Tamás Vértesi, Sebastian Pokutta,
• Abstract summary: We exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some Grothendieck constants. The complete proof relies on solving difficult binary quadratic optimisation problems.
• Score: 20.068273625719943
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart from $d=2$, their values are unknown. Here, we exploit a recent Frank-Wolfe approach to provide good candidates for lower bounding some of these constants. The complete proof relies on solving difficult binary quadratic optimisation problems. For $d\in\{3,4,5\}$, we construct specific rectangular instances that we can solve to certify better bounds than those previously known; by monotonicity, our lower bounds improve on the state of the art for $d\leqslant9$. For $d\in\{4,7,8\}$, we exploit elegant structures to build highly symmetric instances achieving even greater bounds; however, we can only solve them heuristically. We also recall the standard relation with violations of Bell inequalities and elaborate on it to interpret generalised Grothendieck constants $K_G(d\mapsto2)$ as the advantage of complex quantum mechanics over real quantum mechanics. Motivated by this connection, we also improve the bounds on $K_G(d\mapsto2)$.

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