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This paper discusses a mathematical optimization problem with a number of applications in the timing analysis of synchronous and asynchronous circuits. This problem, which we call min-max linear programming (mmLP), involves the solution of linear programs that have min and max functions added to their constraints. Instances of problem mmLP are described, and a simple proof of NPcompleteness is given. Two alternate methods are presented for mmLP solution. The jirst uses a branch-andbound algorithm which is optimized to specifically reduce the number of operations required by the simplex linear programming algorithm. The second uses a transformation to a standard mixed integer linear programming (MILP) formulation. We evaluate both approaches on a variety of problems, including several large previously unsolved optimal clocking problems. 1 mmLP: min-max Linear Programming We define the problem of min-max linear programming (mmLP) as the optimization of a linear objective function subject to a set of linear, tin, and max constraints. Its general form is:

Min-max linear programming and the timing analysis of digital circuits