THE COMPLEXITY OF OPTIMIZATION PROBLEMS

We consider NP-complete optimization problems at the level of computing their optimal value, and define a class of functions called OptP to capture this level of structure. We show that traveling salesperson and knapsack are complete for OptP, and that clique and coloring are complete for a subclass of OptP. These results show a deeper level of structure in these problems than was previously known. We also show that OptP is closely related to FPSAT, the class of functions computable in polynomial time with an oracle for NP. This allows us to quantify exactly 'how much' NP-completeness is in these problems. In particular, in this measure, we show that traveling salesperson is strictly harder than clique and that clique is strictly harder than bin packing. A further result is that an OptP-completeness result implies certain completeness results, tying several classes closely together.