  We provide a unifying, black-box tool for establishing existence of
approximate equilibria in weighted congestion games and, at the same time,
bounding their Price of Stability. Our framework can handle resources with
general costs--including, in particular, decreasing ones--and is formulated in
terms of a set of parameters which are determined via elementary analytic
properties of the cost functions.
  We demonstrate the power of our tool by applying it to recover the recent
result of Caragiannis and Fanelli [ICALP'19] for polynomial congestion games;
improve upon the bounds for fair cost sharing games by Chen and Roughgarden
[Theory Comput. Syst., 2009]; and derive new bounds for nondecreasing concave
costs. An interesting feature of our framework is that it can be readily
applied to mixtures of different families of cost functions; for example, we
provide bounds for games whose resources are conical combinations of polynomial
and concave costs.
  In the core of our analysis lies the use of a unifying approximate potential
function which is simple and general enough to be applicable to arbitrary
congestion games, but at the same time powerful enough to produce
state-of-the-art bounds across a range of different cost functions.
