:The assignment problem consists of finding, in a [[weighted graph|weighted]] [[bipartite graph]], a [[Matching (graph theory)|matching]] of a given size, in which the sum of weights of the edges is minimum.
If the numbers of agents and tasks are equal, then the problem is called ''balanced assignment''. Otherwise, it is called ''unbalanced assignment''.<ref name=":0">{{Cite web|url=https://www.hpllabs.hphpe.com/techreports/2012/HPL-2012-40R140.pdf|title=On minimum-cost assignments in unbalanced bipartite graphs|last=Lyle Ramshaw, Robert E. Tarjan|date=2012|website=HP research labs}}</ref> 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.
