Comparing the Switch and Curveball Markov Chains for Sampling Binary Matrices with Fixed Marginals

The Curveball algorithm is a variation on well-known switch-based Markov Chain Monte Carlo approaches for the uniform sampling of binary matrices with fixed row and column sums. We give a spectral gap comparison between switch chains and the Curveball chain using a decomposition of the switch chain based on Johnson graphs. In particular, this comparison allows us to prove that the Curveball Markov chain is rapidly mixing whenever one of the switch chains is rapidly mixing. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tetali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest.