Minimal representations of some classes of dynamic programming

It is known that various discrete optimization problems can be represented by finite state models called sequential decision processes (sdp's). A subclass of sdp's, the class of monotone sdp's (msdp's), is particularly important since the method of dynamic programming is applicable to obtain optimal policies. Several subclasses of msdp's have also been introduced from the viewpoint of computational complexity for obtaining optimal policies. For each of these classes of sdp's, optimal policies are usually obtained (if possible at all) in fewer steps if a given optimization problem is represented by a model with fewer states.Thus we are naturally led to the problem of finding a minimal (with the fewest states) representation of a given optimization problem by an sdp of a specified class. This paper investigates the existence or nonexistence of such minimization algorithms (in the sense of the theory of computation) for various classes of sdp's. It is shown that there exist minimization algorithms for some classes of sdp's, but there exist no algorithms for others.The nonuniqueness of a minimal representation is also proved for each class of sdp's

Traveling Salesman Problems with a Capacity Constraint of the Delivery Truck

This paper considers the following delivery route problem. A truck delivers rᵢ unit production resources to cities i＝2, 3, ···, n in some order starting from city 1, and receives wᵢ unit production wastes at cities i＝2, 3, ···n. Let cᵢj (1≤i≤n, 1≤j≤n, i≠j) be the time the delivery truck requires from city i to city j. At city i (i≠1), the production starts upon receiving the production resources and takes tᵢ (≥0) unit time until completion. Moreover, the delivery truck has the carrying capacity ⊿ and starts from city 1 with ren sources of Σ rᵢ units. At each city i, the total of remaining resources and the collected wastes can not exceed ⊿. The problem is to find a delivery route that visits each city i exactly once, and minimizes the completion time of production at all cities i＝2, 3, ···, n. This paper shows that the well known dynamic programming approach for the traveling salesman problem can be generalized to incorporate the capacity constraint

Decision lists and related Boolean functions

AbstractWe consider Boolean functions represented by decision lists, and study their relationships to other classes of Boolean functions. It turns out that the elementary class of 1-decision lists has interesting relationships to independently defined classes such as disguised Horn functions, read-once functions, nested differences of concepts, threshold functions, and 2-monotonic functions. In particular, 1-decision lists coincide with fragments of the mentioned classes. We further investigate the recognition problem for this class, as well as the extension problem in the context of partially defined Boolean functions (pdBfs). We show that finding an extension of a given pdBf in the class of 1-decision lists is possible in linear time. This improves on previous results. Moreover, we present an algorithm for enumerating all such extensions with polynomial delay

Double Horn Functions

AbstractIn this paper, we define double Horn functions, which are the Boolean functionsfsuch that bothfand its complement (i.e., negation)fare Horn, and investigate their semantical and computational properties. Double Horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined Boolean functions (pdBfs), where a pdBf is a pair (T,F) of disjoint setsT,F⊆{0,1}nof true and false vectors, respectively, and an extension of (T,F) is a Boolean functionfthat is compatible withTandF. We derive syntactic and semantic characterizations of double Horn functions, and determine the number of such functions. The characterizations are then exploited to give polynomial time algorithms (i) that recognize double Horn functions from Horn DNFs (disjunctive normal forms), and (ii) that compute the prime DNF from an arbitrary formula, as well as its complement and its dual. Furthermore, we consider the problem of determining a double Horn extension of a given pdBf. We describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double Horn extensions of a pdBf with polynomial delay. However, finding a shortest double Horn extension (in terms of the size of a formulaϕrepresenting it) is shown to be intractable

An Optimal Replacement Problem of A Semi-Markovian Deteriorating System

This paper discusses an optimal replacement problem of a multi-state system when the deterioration of the system state is described by a semi-Markov process. It is assumed that the system has operating costs and replacement costs depending on its states. The problem is to derive a replacement policy which minimizes the expected average cost per unit time over the infinite horizon. Moreover, under some reasonable conditions reflecting the physical and economical meaning of the deterioration, we show that an optimal replacement policy has a monotone structure

On Arbitration for the Bayesian Collective Choice Problem

This paper deals with arbitration for the Bayesian collective choice problem. A similar problem is discussed in Myerson (1979) under the assumption that the arbitrator chooses a bargaining solution, derived from the generalized Nash product of Harsanyi and Selten (1972). This paper, however, asserts that arbitration differs from pure bargaining, because an arbitrator behaves so that the fairness-utility function evaluated by himself is maximized. We argue that the functional form of the fairness-utility function is uniquely determined if the arbitrator acts according to some plausible criteria

Graph Packing over a Rooted Tree

This paper investigates the computational complexity of the graph packing problem over a rooted tree (GPT) as a generalization of the one dimensional bin packing problem, where both the bins and the set of items to be packed are rooted trees. GPT is defined under two problem settings, edge GPT (EPT) and node GPT (NPT). In EPT, the items packed in a bin cannot share any edge but can share some node, while in NPT, the items can share neither node nor edge. We first prove that these problems are in general NP-complete, which strongly suggests that these problems are computationally intractable. However, for the case where the number k of different kinds of items is fixed, we derive a recursive formula of dynamic programming for the minimum number of bins required to pack all the items. This formula can be solved in polynomial time, if the bins and items are all uniform trees and/or comb-shaped trees in which each non-leaf node has the same number of sons. Furthermore, for GPT's with bins of uniform (d, H) trees and only one kind of item, of uniform (d, h) trees, we derive explicit formulas for the number of bins required