Editing Assignment problem (section)

=== Balanced assignment ===
In the balanced assignment problem, both parts of the bipartite graph have the same number of vertices, denoted by ''n''.

One of the first polynomial-time algorithms for balanced assignment was the [[Hungarian algorithm]]. It is a ''global'' algorithm&nbsp;– it is based on improving a matching along augmenting paths (alternating paths between unmatched vertices). Its run-time complexity, when using [[Fibonacci heap]]s, is <math>O(mn + n^2\log n)</math>,<ref>{{Cite journal|last1=Fredman|first1=Michael L.|last2=Tarjan|first2=Robert Endre|date=1987-07-01|title=Fibonacci Heaps and Their Uses in Improved Network Optimization Algorithms|journal=J. ACM|volume=34|issue=3|pages=596–615|doi=10.1145/28869.28874|s2cid=7904683|issn=0004-5411|doi-access=free}}</ref> where ''m'' is a number of edges.  This is currently the fastest run-time of a [[strongly polynomial]] algorithm for this problem. Some variants of the Hungarian algorithm also benefit from parallel computing, including GPU acceleration.<ref>{{Cite journal |last=Kawtikwar |first=Samiran |last2=Nagi |first2=Rakesh |date=2024-05-01 |title=HyLAC: Hybrid linear assignment solver in CUDA |url=https://linkinghub.elsevier.com/retrieve/pii/S0743731524000029 |journal=Journal of Parallel and Distributed Computing |volume=187 |pages=104838 |doi=10.1016/j.jpdc.2024.104838 |issn=0743-7315|doi-access=free }}</ref> If all weights are integers, then the run-time can be improved to  <math>O(mn + n^2\log \log n)</math>, but the resulting algorithm is only weakly-polynomial.<ref>{{Cite journal|last=Thorup|first=Mikkel|date=2004-11-01|title=Integer priority queues with decrease key in constant time and the single source shortest paths problem|journal=Journal of Computer and System Sciences|series=Special Issue on STOC 2003|volume=69|issue=3|pages=330–353|doi=10.1016/j.jcss.2004.04.003|issn=0022-0000|doi-access=free}}</ref> If the weights are integers, and all weights are at most ''C'' (where ''C''>1 is some integer), then the problem can be solved in <math>O(m\sqrt{n} \log(n\cdot C))</math> weakly-polynomial time in a method called ''weight scaling''.<ref>{{Cite journal|last1=Gabow|first1=H.|last2=Tarjan|first2=R.|date=1989-10-01|title=Faster Scaling Algorithms for Network Problems|journal=SIAM Journal on Computing|volume=18|issue=5|pages=1013–1036|doi=10.1137/0218069|issn=0097-5397}}</ref><ref>{{Cite journal|last1=Goldberg|first1=A.|last2=Kennedy|first2=R.|date=1997-11-01|title=Global Price Updates Help|journal=SIAM Journal on Discrete Mathematics|volume=10|issue=4|pages=551–572|doi=10.1137/S0895480194281185|issn=0895-4801}}</ref><ref>{{Cite journal|last1=Orlin|first1=James B.|last2=Ahuja|first2=Ravindra K.|date=1992-02-01|title=New scaling algorithms for the assignment and minimum mean cycle problems|journal=Mathematical Programming|language=en|volume=54|issue=1–3|pages=41–56|doi=10.1007/BF01586040|s2cid=18213947|issn=0025-5610}}</ref>

In addition to the global methods, there are ''local methods'' which are based on finding local updates (rather than full augmenting paths). These methods have worse asymptotic runtime guarantees, but they often work better in practice. These algorithms are called [[auction algorithm]]s, push-relabel algorithms, or preflow-push algorithms. Some of these algorithms were shown to be equivalent.<ref>{{Cite journal |last1=Alfaro |first1=Carlos A. |last2=Perez |first2=Sergio L. |last3=Valencia |first3=Carlos E. |last4=Vargas |first4=Marcos C. |date=2022-06-01 |title=The assignment problem revisited |url=https://doi.org/10.1007/s11590-021-01791-4 |journal=Optimization Letters |language=en |volume=16 |issue=5 |pages=1531–1548 |doi=10.1007/s11590-021-01791-4 |s2cid=238644205 |issn=1862-4480}}</ref>

Some of the local methods assume that the graph admits a ''perfect matching''; if this is not the case, then some of these methods might run forever.<ref name=":0" />{{Rp|3}}  A simple technical way to solve this problem is to extend the input graph to a ''complete bipartite graph,'' by adding artificial edges with very large weights. These weights should exceed the weights of all existing matchings, to prevent appearance of artificial edges in the possible solution.

As shown by Mulmuley, Vazirani and Vazirani,<ref>{{Cite journal|last1=Mulmuley|first1=Ketan|last2=Vazirani|first2=Umesh|author-link2=Umesh Vazirani|last3=Vazirani|first3=Vijay|author-link3=Vijay Vazirani|year=1987|title=Matching is as easy as matrix inversion|journal=Combinatorica|volume=7|issue=1|pages=105–113|doi=10.1007/BF02579206|s2cid=47370049|author-link1=Ketan Mulmuley}}</ref> the problem of minimum weight perfect matching is converted to finding minors in the [[adjacency matrix]] of a graph. Using the [[isolation lemma]], a minimum weight perfect matching in a graph can be found with probability at least {{frac|1|2}}. For a graph with ''n'' vertices, it requires <math> O(\log^2(n)) </math> time.