Lagrange multipliers for linear programs can be interpreted as shadow prices. Shadow prices typically represent marginal/differential changes in the objective from a marginal loosening of a given constraint, but this does not work for an integer program (like assignment problems) where infinitesimal movements are not available. For instance, consider the following binary integer program: $$ \text{IP:} \hspace{2cm} \begin{align*} &\text{maximize }& &c^\prime x\\ &\text{subject to }& &Ax \leq b\\ &\forall i&&x_i \text{ binary} \end{align*} $$ Suppose also that every entry in $A$ is either $0$ or $1$ and every entry in $b$ is a positive integer. The constraints can be thought of as capacity constraints. To extend the notion of shadow prices to an assignment IP, we could say that the shadow price of constraint $j$ above is the difference between the maximized value of the objective in the problem above and the maximized value of the same problem except taking the $j^\text{th}$ constraint and replacing $b_j$ with $b_j+1$. In words, the shadow price of a capacity constraint would be the value of loosening that constraint by one unit, i.e. adding one unit of capacity (rather than an infinitesimal loosening).
In practice, these will be hard to compute because we'll need to resolve the program to calculate the shadow price on each constraint. This begs the question:
When are Lagrange multipliers of LP relaxations of assignment IPs good approximations for assignment IP shadow prices?
For example, if we take the LP-relaxation of the program above, $$ \text{LP-relaxation of IP:} \hspace{2cm} \begin{align*} &\text{maximize }& &c^\prime x\\ &\text{subject to }& &Ax \leq b\\ &&&0\leq x \leq1, \end{align*} $$ and we find that the Lagrange multiplier on constraint $j$, $\lambda_j$, is greater than that on constraint $k$, $\lambda_k$, under what conditions will this imply that the shadow price (as defined above) of constraint $j$ in the assignment IP will be greater than that of constraint $k$? I suppose it is almost certainly the case if $A$ is totally unimodular so that the solution to the LP-relaxation also solves the IP, but I'm not sure if anything can be said beyond this case.