Classes of discrete optimization problems and their decision problems

摘要

In the earlier papers by Karp and Held and by Ibaraki, the representation of a discrete optimization problem given in the form of a discrete decision process (ddp) by a finite state model called a sequential decision process (sdp) was considered. An sdp is a finite automaton with a cost function associated with each state transition. When the cost function satisfies a certain monotonicity condition, it is called a monotone sdp(msdp). As pointed out by Karp and Held, there is a close relationship between an msdp and the dynamic programming developed by Bellman.These models are further restricted in this paper by assuming that each cost function is a recursive function. The resulting models are called r-ddp, r-sdp, and r-msdp, respectively. Two types of representation theorems and properties of sets of optimal policies are investigated in detail for r-sdp and r-msdp. Various decision problems are also considered, and most of them are proved to be unsolvable. In particular, there exists no algorithm to obtain an optimal policy of an arbitrarily given r-sdp or r-msdp. Since this is quite inconvenient from the view point of practical application, a subclass of r-msdp, r-imsdp, is introduced in the last half of this paper. For an arbitrarily given r-imsdp, there exists an algorithm to obtain an optimal policy if it has at least one optimal policy. Most of other decision problems, however, are proved to be still unsolvable.