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 numerical 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) have been empirically observed to be growing at an order smaller than square root of the prime. However, computations of SOS-4 appear to be 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 above the values of the corresponding SOS-4 and SOS-6 relaxations respectively. We revisit these computations and compute the values of the L2 relaxations 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：本文研究了用于Paley图团数的块对角半定规划（SDP）松弛问题。图的最大团数量（团数）是一个经典的NP完全问题；Paley图是一类确定性图，其中两个顶点相连当且仅当它们的差是特定素数幂的二次剩余。对于奇素数幂，改进Paley图团数的上界是组合数学中的一个开放问题。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性质的构造相关，而这又是压缩感知与稀疏恢复领域的开放问题。近期研究提供了数值证据，表明当前上界可通过平方和（SOS）松弛加以改进。具体而言，实验观察到4次SOS松弛（SOS-4）给出的界以低于素数平方根的阶增长。然而，对于大规模图，SOS-4的计算似乎难以处理。Gvozdenovic等人引入了一种计算效率更高的块对角SDP层次结构，该结构对SOS层次结构进行了细化。他们计算了与素数p≤809相关的Paley图团数的2次和3次SDP（分别记为L2和L3）值，这些值分别给出了相应SOS-4和SOS-6松弛的上界。我们重新审视了这些计算，并计算了更大p值的L2松弛值。我们的结果提供了额外数值证据，表明L2松弛（因此也包括SOS-4松弛）的渐近增长率低于p的平方根。