Revisiting Block-Diagonal SDP Relaxations for the Clique Number of the Paley Graphs

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 (square) in a finite field with the number of elements given by certain primes and prime powers. Improving the upper bound for the Paley graph clique number for prime powers that are non-squares 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. 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 and computed the values of these SDPs of degrees 2 (L2) for the Paley graph clique numbers associated with primes p less or equal to 809, which bound from above the corresponding SOS-4 relaxations. We compute the values of the L2 relaxations for p's between 821 and 997. Our results provide some numerical evidence that these relaxations, and therefore also the SOS-4 relaxations, may be scaling at an order smaller than the square root of p. However, due to the size of the SDPs, we have not been able to compute L2 relaxations for p's greater than 997. Therefore, our scaling estimate is not conclusive and presents an interesting open problem for further study.