We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $\bb $, produces a vector $\xxt$ within relative distance $ε$ of the solution to $A \xx = \bb$ in time $O (m^{1.31} \log (n κ_{f} (A)/ε)^{O (1)})$, where $κ_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (κ_{f} (A)) = O (b \log n)$, where $b$ is the logarithm of the ratio of the largest to smallest non-zero entry of $A$. If the graph of $A$ has genus $m^{2θ}$ or does not have a $K_{m^θ} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 θ$ and $(9/8) (1+θ)$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.