Let $X$ and $Y$ denote two sets of $m$ and $n$ points distributed uniformly at random in the unit interval. When $m$ and $n$ are both large, is there a bound for the expected cost of a minimum-weight maximum bipartite matching between them? The case $m=n$ (in which case we get a perfect matching) is well-studied and the cost is proportional to $\sqrt{n}$, and is easily bounded as such by looking at differences of order statistics.
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$\begingroup$ Do you want unit interval or unit square? $\endgroup$Sandeep Silwal– Sandeep Silwal2024-12-19 03:33:10 +00:00Commented Dec 19, 2024 at 3:33
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$\begingroup$ @SandeepSilwal Unit interval. The unit square apparently looks more like $\sqrt{n \log n}$. $\endgroup$Tom Solberg– Tom Solberg2024-12-19 04:55:01 +00:00Commented Dec 19, 2024 at 4:55
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