This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue modulo certain prime powers. Improving the upper bound for the Paley graph clique number for odd prime powers is an open problem in combinatorics. Moreover, since quadratic residues exhibit pseudorandom properties, Paley graphs are related to the construction of deterministic restricted isometries, an open problem in compressed sensing and sparse recovery. Recent work provides evidence that the current upper bounds can be improved by the sum-of-squares (SOS) relaxations. In particular the bounds given by the SOS relaxations of degree 4 (SOS-4) are asymptotically growing at an order smaller than square root of the prime. However computations of SOS-4 become intractable with respect to large graphs. Gvozdenovic et al. introduced a more computationally efficient block-diagonal hierarchy of SDPs that refines the SOS hierarchy. They computed the values of these SDPs of degrees 2 and 3 (L2 and L3 respectively) for the Paley graph clique numbers associated with primes p less or equal to 809. These values bound from the above the values of the corresponding SOS-4 and SOS-6 relaxations respectively. We revisit these computations and determine the values of the L2 relaxation for larger p's. Our results provide additional numerical evidence that the L2 relaxations, and therefore also the SOS-4 relaxations, are asymptotically growing at an order smaller than the square root of p.

翻译：本工作研究了用于Paley图团数的块对角半定规划（SDP）松弛。图的最大团大小（团数）是一个经典的NP完全问题；Paley图是一类确定性图，其中两个顶点当且仅当它们的差是特定素数幂下的二次剩余时相连。改善奇素数幂下Paley图团数的上界是组合学中的一个未解决问题。此外，由于二次剩余具有伪随机性质，Paley图与确定性受限等距的构造相关，而后者是压缩感知与稀疏恢复中的开放问题。近期研究提供了证据，表明当前上界可通过平方和（SOS）松弛加以改进。特别地，4次SOS松弛（SOS-4）给出的界在渐近意义上以小于素数平方根的阶数增长。然而，SOS-4的计算在大型图上变得不可行。Gvozdenovic等人引入了一种计算效率更高的SDP块对角层次结构，该结构改进了SOS层次结构。他们针对与素数p≤809相关的Paley图团数，计算了2次与3次SDP（分别为L2与L3）的值，这些值分别给出了相应SOS-4与SOS-6松弛的上界。我们重新审视了这些计算，并确定了更大素数p下L2松弛的值。我们的结果提供了额外数值证据，表明L2松弛（从而也包括SOS-4松弛）在渐近意义上以小于p平方根的阶数增长。