On Nondeterministic Unranked Tree Automata with Sibling Constraints
We continue the study of bottom-up unranked tree automata with equality and ::: disequality constraints between direct subtrees. In particular, we show that ::: the emptiness problem for the nondeterministic automata is decidable. In ::: addition, we show that the universality problem, in contrast, is undecidable.
We consider the problem of preemptively scheduling jobs on a single processor with the objective of minimizing the total weighted flowtime. Each job has a weight wi>0 which represents the job priority and a release date ri>=0, i.e., jobs can arrive at different times to the system. The problem is known to be NP-hard and finds practical applications in the areas of manufacturing systems, computer science, communications, logistics, and service industries. Two necessary conditions for the optimality of the problem are given first, and an enumerative algorithm for solving the problem based upon a branch and bound concept is then proposed. The algorithm uses these necessary conditions and a procedure for computing a lower bound at any node in the tree search that helps increase efficiency of the algorithm. Several test problems with different sizes were randomly generated and solved using the proposed algorithm.