Optimization and complexity of inertia-type bounds on the independence and chromatic numbers of graph powers

The inertia bound, introduced by Cvetkovi\'c in 1971, is a fundamental result in spectral graph theory that provides an upper bound for the independence number of a graph in terms of spectral information about a weighted adjacency matrix of the graph. Recently, this bound has been extended to the socalled inertia-type bounds for estimating the independence and chromatic numbers of graph powers ($k$-independence number and distance-$k$ chromatic number of a graph). These bounds have recently found applications in coding theory and quantum information theory. The inertia-type bounds depend on the choice of a polynomial of degree $k$ and on the eigenvalues of the graph. Currently, optimizing these bounds requires solving several MILPs, which quickly becomes computationally demanding as the graph size or $k$ grows. This computational barrier is a major obstacle to the practical use of these bounds. Moreover, we have a limited theoretical understanding of their performance, even for small $k$. In this paper, we investigate their optimization and complexity. In particular, we improve the MILP formulations, reducing their computational burden and significantly decreasing the running time. Furthermore, we show that the optimization problems associated with the bounds are solvable in polynomial time for fixed $k$ and for small $k$.