Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Scaling algorithms for approximate and exact maximum weight matching   The {\em maximum cardinality} and {\em maximum weight matching} problems can
be solved in time $\tilde{O}(m\sqrt{n})$, a bound that has resisted improvement
despite decades of research. (Here $m$ and $n$ are the number of edges and
vertices.) In this article we demonstrate that this "$m\sqrt{n}$ barrier" is
extremely fragile, in the following sense. For any $\epsilon>0$, we give an
algorithm that computes a $(1-\epsilon)$-approximate maximum weight matching in
$O(m\epsilon^{-1}\log\epsilon^{-1})$ time, that is, optimal {\em linear time}
for any fixed $\epsilon$. Our algorithm is dramatically simpler than the best
exact maximum weight matching algorithms on general graphs and should be
appealing in all applications that can tolerate a negligible relative error.
  Our second contribution is a new {\em exact} maximum weight matching
algorithm for integer-weighted bipartite graphs that runs in time
$O(m\sqrt{n}\log N)$. This improves on the $O(Nm\sqrt{n})$-time and
$O(m\sqrt{n}\log(nN))$-time algorithms known since the mid 1980s, for $1\ll
\log N \ll \log n$. Here $N$ is the maximum integer edge weight.

5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
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The categories of this paper are: math-ph
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci