Maximal saturated domino covering of a rectangle
Sagemath, 60 bytes
lambda m,n:m*n-len(graphs.GridGraph([m,n]).dominating_set())

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From Saturated Domino Coverings by Buchanan et al:

Corollary 6.3: If \$B\$ is a rectangular \$m \times n\$ board, then \$d(B) = |B| - \gamma(G_{m,n})\$.

Where \$\gamma(\ldots)\$ is the domination number and \$G_{m,n}\$ is a grid graph. We also clearly have that if \$B\$ represents an \$m\times n\$ board then \$|B| = mn\$.