  The chiral Luttinger liquid model for the edge dynamics of a two-dimensional
electron gas in a strong magnetic field is derived from coarse-graining and a
lowest Landau level projection procedure at arbitrary filling factors $\nu<1$
-- without reference to the quantum Hall effect. Based on this model, we
develop a formalism to calculate the Landauer-B\"uttiker conductances in
generic experimental set-ups including multiple leads and voltage probes. In
the absence of tunneling between the edges the "ideal" Hall conductances
($G_{ij}= \frac{e^2 \nu}{h}$ if lead $j$ is immediately upstream of lead $i$,
and $G_{ij}=0$ otherwise) are recovered. Tunneling of quasiparticles of
fractional charge $e^*$ between different edges is then included as an
additional term in the Hamiltonian. In the limit of weak tunneling we obtain
explicit expressions for the corrections to the ideal conductances. As an
illustration of the formalism we compute the current- and temperature-dependent
resistance $R_{xx}(I,T)$ of a quantum point contact localized at the center of
a gate-induced constriction in a quantum Hall bar. The exponent $\alpha$ in the
low-current relation $R_{xx}(I,0) \sim I^{\alpha -2}$ shows a nontrivial
dependence on the strength of the inter-edge interaction, and its value changes
as $e^*V_H$, where $V_H = \frac{h I}{\nu e^2}$ is the Hall voltage, falls below
a characteristic crossover energy $\frac{\hbar c}{d}$, where $c$ is the edge
wave velocity and $d$ is the length of the constriction. The consequences of
this crossover are discussed vis-a-vis recent experiments in the weak tunneling
regime.
