Retiming of circuits containing multiplexers

Classical retiming optimization algorithms do not consider circuits containing multiplexers or demultiplexers driven by a clock signal, because the associated retiming equations differ from the special classical form, which make applicable combinatorial algorithms of polynomial order. In order to provide an algorithm for multiplexer circuits it is shown here that retiming, being an integer linear programming problem inherently, can be relaxed to a linear programming formulation with real valued variables. This is due to the unimodularity of the matrices of the retiming formulation. Multiplexer circuits change this property in a way which suggests how to use an integer linear programming problem to derive a polynomial retiming algorithm.