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- W1750951292 abstract "Let $A,B$ with $|A| = m$ and $|B| = nge m$ be two sets. We assume that every element $ain A$ has a reference list over all elements from $B$. We call an injective mapping $tau$ from $A$ to $B$ a matching. A blocking coalition of $tau$ is a subset $A'$ of $A$ such that there exists a matching $tau'$ that differs from $tau$ only on elements of $A'$, and every element of $A'$ improves in $tau'$, compared to $tau$ according to its preference list. If there exists no blocking coalition, we call the matching $tau$ an exchange stable matching (ESM). An element $bin B$ is reachable if there exists an exchange stable matching using $b$. The set of all reachable elements is denoted by $E^*$. We show [|E^*| leq sum_{i = 1,ldots, m}{leftlfloorfrac{m}{i}rightrfloor} = Theta(mlog m).] This is asymptotically tight. A set $Esubseteq B$ is reachable (respectively exactly reachable) if there exists an exchange stable matching $tau$ whose image contains $E$ as a subset (respectively equals $E$). 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 $a$ of $A$ is matched with $ell_a$ elements of $B$ instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable elements, i.e., elements of $B$ that are used by all ESM's. This yields efficient algorithms to determine all unavoidable elements." @default.
- W1750951292 created "2016-06-24" @default.
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- W1750951292 date "2014-01-21" @default.
- W1750951292 modified "2023-10-01" @default.
- W1750951292 title "Counting Houses of Pareto Optimal Matchings in the House Allocation Problem" @default.
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