Given a multigraph $G=(V,E)$ with a positive rational weight $w(e)$ on each edge $e$, the weighted density problem (WDP) is to find a subset $U$ of $V$, with $|U|\ge 3$ and odd, that maximizes $2w(U)/(|U|-1)$, where $w(U)$ is the total weight of all edges with both ends in $U$, and the weighted fractional edge-coloring problem can be formulated as the following linear program: minimize ${\bm 1}^T {\bm x}$ subject to $ A{\bm x} = {\bm w}$, ${\bm x} \ge {\bm 0}$, where $A$ is the edge-matching incidence matrix of $G$. These two problems are closely related to the celebrated Goldberg--Seymour conjecture on edge-colorings of multigraphs, and are interesting in their own right. Even when $w(e)=1$ for all edges $e$, determining whether WDP can be solved in polynomial time was posed by Jensen and Toft [Topics in Chromatic Graph Theory, Cambridge University Press, Cambridge, 2015, pp. 327--357] and by Stiebitz et al. [Graph Edge Colouring: Vizing's Theorem and Goldberg's Conjecture, John Wiley, New York, 2012] as an open problem. In this paper we present strongly polynomial-time algorithms for solving them exactly, and develop a novel matching removal technique for multigraph edge-coloring.