  Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any
two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying
the diamond condition for $I$. We introduce an intrinsic notion of deformation
of reduction systems and show that there is an equivalence of deformation
problems between deformations of the associative algebra $A$ and deformations
of the reduction system $R$, the latter being controlled by a natural, explicit
L$_\infty$ algebra. It follows in particular that any formal deformation of the
associative multiplication on $A$ can, up to gauge equivalence, be given by a
combinatorially defined star product, and the approach via reduction systems
can be used to give a concrete and complete description of the deformation
theory of $A$. For the polynomial algebra in a finite number of variables, this
combinatorial star product can be described via bidifferential operators
associated to graphs, which we compare to the graphs appearing in Kontsevich's
universal quantization formula.
  Using the notion of admissible orders on the set of paths of the quiver $Q$,
we give criteria for the existence of algebraizations of formal deformations,
which we also interpret geometrically via algebraic varieties of reduction
systems. In this context the Maurer-Cartan equation of the L$_\infty$ algebra
can be viewed as a generalization of the Braverman-Gaitsgory criterion for
Poincar\'e-Birkhoff-Witt deformations of Koszul algebras.
