88,397 research outputs found

    Counting Houses of Pareto Optimal Matchings in the House Allocation Problem

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    Let A,BA,B with A=m|A| = m and B=nm|B| = n\ge m be two sets. We assume that every element aAa\in A has a reference list over all elements from BB. We call an injective mapping τ\tau from AA to BB a matching. A blocking coalition of τ\tau is a subset AA' of AA such that there exists a matching τ\tau' that differs from τ\tau only on elements of AA', and every element of AA' improves in τ\tau', compared to τ\tau according to its preference list. If there exists no blocking coalition, we call the matching τ\tau an exchange stable matching (ESM). An element bBb\in B is reachable if there exists an exchange stable matching using bb. The set of all reachable elements is denoted by EE^*. We show Ei=1,,mmi=Θ(mlogm).|E^*| \leq \sum_{i = 1,\ldots, m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m). This is asymptotically tight. A set EBE\subseteq B is reachable (respectively exactly reachable) if there exists an exchange stable matching τ\tau whose image contains EE as a subset (respectively equals EE). We give bounds for the number of exactly reachable sets. We find that our results hold in the more general setting of multi-matchings, when each element aa of AA is matched with a\ell_a elements of BB instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable elements, i.e., elements of BB that are used by all ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise

    Number of Holes in Unavoidable Sets of Partial Words II

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    We are concerned with the complexity of deciding the avoidability of sets of partial words over an arbitrary alphabet. Towards this, we investigate the minimum size of unavoidable sets of partial words with a fixed number of holes. Additionally, we analyze the complexity of variations on the decision problem when placing restrictions on the number of holes and length of the words
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