The statistical properties of two combinatorial optimization problems that arise in the physical design of circuits, circuit partitioning, and floorplanning, are discussed. For the partitioning problem, the solutions generated by the Kernighan-Lin (1970) procedure and those generated by a random search are examined. It is shown that in both cases, the Type 3 (Weibull) extreme value distribution provides an excellent model for the distribution of local minima generated. The location parameter of the Weibull provides an estimate of the minimum cost. Various techniques that were used to estimate the parameters of the Weibull are discussed, and ample empirical evidence to support the hypothesis is provided. For the floorplanning problem, a number of test problems whose optimal value are known are constructed. By using a representation of slicing floorplans developed by D.F. Wong and C.L. Liu (1986), the solution space is randomly sampled and a number of local minima are generated. As with the partitioning problem, it is found that the Weibull distribution provides an excellent model for estimating the minimum cost.>

In this paper we examine some statistical properties exhibited by combinatorial optimization problems. Although the paper focuses on two particular problems that arise in chip design, namely circuit partitioning and floorplanning, the results seem valid for a much larger set of such problems. For the partitioning problem, we examine the solutions generated by the well known Kernighan-Lin procedure [5,10] and solutions generated by random search. We find that in both cases, the Type 3 (Weibull) extreme-value distribution provides an excellent model for the distribution of local minima generated. The location parameter of the Weibull provides an estimate of the minimum cost. For the floorplanning problem, we construct a number of test problems, whose optimal value is known. As with the partitioning problem, we find that the Weibull distribution provides an excellent model for estimating the minimum cost.