Maximally Satisfying Lower Quotas in the Hospitals/Residents Problem with Ties

Motivated by the serious problem that hospitals in rural areas suffer from a shortage of residents, we study the Hospitals/Residents model in which hospitals are associated with lower quotas and the objective is to satisfy them as much as possible. When preference lists are strict, the number of residents assigned to each hospital is the same in any stable matching because of the well-known rural hospitals theorem; thus there is no room for algorithmic interventions. However, when ties are introduced to preference lists, this will no longer apply because the number of residents may vary over stable matchings. In this paper, we formulate an optimization problem to find a stable matching with the maximum total satisfaction ratio for lower quotas. We first investigate how the total satisfaction ratio varies over choices of stable matchings in four natural scenarios and provide the exact values of these maximum gaps. Subsequently, we propose a strategy-proof approximation algorithm for our problem; in one scenario it solves the problem optimally, and in the other three scenarios, which are NP-hard, it yields a better approximation factor than that of a naive tie-breaking method. Finally, we show inapproximability results for the above-mentioned three NP-hard scenarios

Light and electron microscopic immunocytochemistry on the localization of 17α-hydroxylase/C17,20-lyase (P450c17) in the rat placenta

The rat placenta is the primary source of androgens during the second half of pregnancy. Androgens are converted to estrogens in the ovaries and contribute to the maintenance of normal pregnancy. We immunocytochemically characterized the cellular and subcellular localization of cytochrome P450 of 17α-hydroxylase/C 17,20-lyase (P450c17), an enzyme responsible for androgen synthesis, in the rat placenta. We also observed the fine structure of the placenta by electron microscopy. The rat placenta had a different structure from the primate, and contained four zones :labyrinth, basal zone, decidua basalis, and metrial gland. The labyrinth had three trophoblastic layers and fetal endothelium, and P450c17 immunoreactivity was homogeneously localized in the three trophoblastic layers but not in the fetal endothelium. In the basal zone, various types of trophoblasts were observed, and the immunoreaction was localized in small basophilic cells and giant cells. The intensity of staining was heterogeneous among these cells. The decidua basalis showed no immunostaining. Subcellular localization of the enzyme was in the cytoplasm, but not in the nucleus or mitochondria. The present study demonstrated a steroidogenic potency in both the labyrinth and the basal zone, although it was shown only in the basal zone in previous studies