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{{Short description|Combinatorial optimization problem}}
[[File:hungarian_algorithm_unbalanced_assignment_problem_example.svg|thumb|upright=2|Worked example of assigning tasks to an unequal number of workers using the [[Hungarian method]]]]
The '''assignment problem''' is a fundamental [[combinatorial optimization]] problem. In its most general form, the problem is as follows:
:The problem instance has a number of ''agents'' and a number of ''tasks''. Any agent can be assigned to perform any task, incurring some ''cost'' that may vary depending on the agent-task assignment. It is required to perform as many tasks as possible by assigning at most one agent to each task and at most one task to each agent, in such a way that the ''total cost'' of the assignment is minimized.
Alternatively, describing the problem using graph theory:
:The assignment problem consists of finding, in a [[weighted graph|weighted]] [[bipartite graph]], a [[Matching (graph theory)|matching]] of
If the numbers of agents and tasks are equal, then the problem is called '''balanced assignment''', and the graph-theoretic version is called '''minimum-cost perfect matching'''. Otherwise, it is called '''unbalanced assignment'''.<ref name=":0">{{Cite web|url=https://www.
If the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called '''linear assignment'''. Commonly, when speaking of the ''assignment problem'' without any additional qualification, then the ''linear balanced assignment problem'' is meant. ==Examples==
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The greedy algorithm would assign Task 1 to Alice and Task 2 to George, for a total cost of 9; but the reverse assignment has a total cost of 7.
Fortunately, there are many algorithms for finding the optimal assignment in time [[polynomial time|polynomial]] in ''n''. The assignment problem is a special case of the [[transportation problem]], which is a special case of the [[minimum cost flow problem]], which in turn is a special case of a [[linear program]]. While it is possible to solve any of these problems using the [[simplex algorithm]], or in worst-case polynomial time using the [[ellipsoid method]], each specialization has a smaller solution space and thus more efficient algorithms designed to take advantage of its special structure.
=== 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 – 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|url-access=subscription }}</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.
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=== Other methods and approximation algorithms===
Other approaches for the assignment problem exist and are reviewed by Duan and Pettie<ref>{{Cite journal|last1=Duan|first1=Ran|last2=Pettie|first2=Seth|date=2014-01-01|title=Linear-Time Approximation for Maximum Weight Matching|url= https://
== Many-to-many assignment{{Anchor|mtm}} ==
In the basic assignment problem, each agent is assigned to at most one task and each task is assigned to at most one agent. In the '''many-to-many assignment problem''',<ref>{{Cite journal |last=Zhu |first=Haibin |last2=Liu |first2=Dongning |last3=Zhang |first3=Siqin |last4=Zhu |first4=Yu |last5=Teng |first5=Luyao |last6=Teng |first6=Shaohua |date=2016-03-07 |title=Solving the Many to Many assignment problem by improving the Kuhn–Munkres algorithm with backtracking |url=https://www.sciencedirect.com/science/article/pii/S0304397516000037 |journal=Theoretical Computer Science |volume=618 |pages=30–41 |doi=10.1016/j.tcs.2016.01.002 |issn=0304-3975}}</ref> each agent ''i'' may take up to ''c<sub>i</sub>'' tasks (''c<sub>i</sub>'' is called the agent's ''capacity''), and each task ''j'' may be taken by up to ''d<sub>j</sub>'' agents simultaneously (''d<sub>j</sub>'' is called the task's ''capacity''). If the sums of capacities in both sides are equal (<math>\sum_i c_i = \sum_j d_j</math>), then the problem is ''balanced'', and the goal is to find a perfect matching (assign exactly ''c<sub>i</sub>'' tasks to each agent ''i'' and exactly ''d<sub>j</sub>'' agents to each task ''j'') such that the total cost is as small as possible.
The problem can be solved by reduction to the [[Network flow problem|minimum cost network flow problem]].<ref>{{Cite web |last=D.W. |title=High-multiplicity maximum-weight matching |url=https://cs.stackexchange.com/questions/161149/high-multiplicity-maximum-weight-matching/161151#161151 |access-date=2025-01-15 |website=Computer Science Stack Exchange |language=en}}</ref> Construct a flow network with the following layers:
* Layer 1: One source-node '''s'''.
* Layer 2: a node for each agent. There is an arc from '''s''' to each agent ''i'', with cost 0 and capacity ''c<sub>i</sub>'' .
* Level 3: a node for each task. There is an arc from each agent ''i'' to each task ''j'', with the corresponding cost, and capacity 1.
* Level 4: One sink-node '''t'''. There is an arc from each task to '''t''', with cost 0 and capacity ''d<sub>j</sub>''.
An integral maximum flow of minimum cost can be found in polynomial time; see [[network flow problem]]. Every integral maximum flow in this network corresponds to a matching in which at most ''c<sub>i</sub>'' tasks are assigned to each agent ''i'' and at most ''d<sub>j</sub>'' agents are assigned to each task ''j'' (in the balanced case, exactly ''c<sub>i</sub>'' tasks are assigned to ''i'' and exactly ''d<sub>j</sub>'' agents are assigned to ''j''). A min-cost maximum flow corresponds to a min-cost assignment.
== Generalization ==▼
When phrased as a graph theory problem, the assignment problem can be extended from [[bipartite graph]]s to arbitrary graphs. The corresponding problem, of finding a [[matching (graph theory)|matching]] in a [[weighted graph]] where the sum of weights is maximized, is called the [[maximum weight matching|maximum weight matching problem]].
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*[[Rank-maximal matching]]
*[[Secretary problem]]
*[[Stable
*[[Stable roommates problem]]
*[[Weapon target assignment problem]]
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