  Illusie has suggested that one should think of the classifying group of
$M_X^{gp}$-torsors on a logarithmically smooth curve $X$ over a standard
logarithmic point as a logarithmic analogue of the Picard group of $X$. This
logarithmic Picard group arises naturally as a quotient of the algebraic Picard
group by lifts of the chip firing relations of the associated dual graph. We
connect this perspective to Baker and Norine's theory of ranks of divisors on a
finite graph, and to Amini and Baker's metrized complexes of curves. Moreover,
we propose a definition of a combinatorial rank for line bundles on $X$ and
prove that an analogue of the Riemann-Roch formula holds for our combinatorial
rank. Our proof proceeds by carefully describing the relationship between the
logarithmic Picard group on a logarithmic curve and the Picard group of the
associated metrized complex. This approach suggests a natural categorical
framework for metrized complexes, namely the category of logarithmic curves.
