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There are two fast algorithms for maximum matching on general graphs:

  • Micali and Vazirani in $O(E\sqrt{V})$.

  • Mucha and Sankowski in $O(V^{2.376})$.

Can these be also used for maximum weighted matching on general graphs? Note that Edmonds' Blossom algorithm can be used to solve both problems.

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  • $\begingroup$ Have you checked the respective papers? Do they mention anything relevant? We expect you to first attempt to solve your question on your own by looking up the relevant background. $\endgroup$ – Yuval Filmus Feb 13 '19 at 4:31
  • $\begingroup$ I have checked and it seems like they don’t solve the weighted problem. However I don’t understand them well enough, so I decided to ask here. $\endgroup$ – Dmitry Kamenetsky Feb 13 '19 at 9:33
  • $\begingroup$ If they don't claim to solve the weighted problem, then they probably don't solve it. $\endgroup$ – Yuval Filmus Feb 13 '19 at 9:48
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Ran Duan and Seth Pettie survey maximum matching algorithms in their 2014 paper Linear-Time Approximation for Maximum Weight Matching. In particular, Table III in their paper (page 5) lists algorithms for maximum weight matching in general graphs.

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  • $\begingroup$ Great paper thank you! Wow linear time approximation is very impressive. $\endgroup$ – Dmitry Kamenetsky Feb 13 '19 at 10:31

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