An assignment problem is not infeasible simply because there are more workers than tasks, or vice versa. First decide which side must be fully matched, then check whether the allowed worker–task pairings can satisfy that requirement. Use dummy assignments only when they represent a real outcome—such as an idle worker or an uncovered task—and price that outcome deliberately.
First decide what “assigned” means
Write down the coverage rule before changing the model. Does every worker need a task? Must every task have a worker? Must both sides be fully matched, or is a maximum-size partial matching acceptable? These are different requirements, and a solver’s meaning of “full” depends on the formulation.
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- Every task covered: each task must receive a worker; extra workers may remain idle.
- Every worker assigned: each worker must receive a task; extra tasks may remain uncovered.
- Both sides fully matched: every worker and task must participate. This requires equal set sizes unless the model includes an explicit alternative outcome.
- Partial matching: assign as many compatible pairs as possible, without requiring complete coverage of either side.
Rectangular assignment can be valid rather than infeasible: SciPy’s linear_sum_assignment supports rectangular cost matrices, and elements on the larger side need not all be assigned (SciPy documentation). Google OR-Tools also illustrates a model with five workers and four tasks in which each task is assigned exactly one worker and a worker can be left unassigned (OR-Tools assignment example).
Handle unequal numbers with an explicit policy
If the requirement truly calls for a square, full assignment, balance the model by adding dummy rows or columns. A dummy match should mean something operationally meaningful: for example, a worker is idle, or a task is left uncovered. Give that outcome a cost reflecting its actual consequence. A zero cost is appropriate only when the outcome is genuinely costless.
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Alternatively, model the coverage rule directly. For example, if each task must be covered but workers may be idle, constrain each task to exactly one worker and each worker to at most one task. This avoids forcing every worker into an artificial match. Choose the formulation that expresses the real policy, not merely one that makes the matrix square.
Represent forbidden pairings as unavailable
If a worker cannot perform a task, that pair is not an expensive option; it is not an option at all. Exclude the edge or choice using the solver’s supported mechanism. OR-Tools demonstrates omitting incompatible assignments, and its example shows that enough restrictions can leave no possible assignment (OR-Tools linear assignment documentation).
A large finite penalty is not the same as a forbidden pair: an optimizer can still select it if every alternative is worse or unavailable. If the solver does not support explicit exclusions, any penalty-based substitute needs justified bounds and checks for numerical behavior and overflow.
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- Restate the coverage requirement. Specify which side must be fully matched and whether unmatched entities are permitted.
- Check dimensions and interpretation. Confirm that rows and columns represent the intended groups, and that a rectangular formulation is not being mistaken for a square one.
- Review compatibility. Verify that forbidden pairs are excluded and that allowed pairs have not been accidentally removed.
- Check for a matching of the required size. Count alone is insufficient: a cluster of workers may all depend on too few tasks, or a cluster of tasks may have too few eligible workers. The allowed pairings must support the required number of distinct matches.
- Choose a policy response. If no such matching exists, decide whether to relax coverage, allow additional pairings, or change the model. Adding dummy rows or columns fixes a size mismatch; it does not create valid real pairings.
- Revisit the solver class if needed. If the rules include dependencies or other logic beyond one-to-one costs, express those constraints in a more general model rather than hiding them in cost values.
Choose a solver that matches the model
For a basic one-to-one cost-minimization problem, a specialized linear assignment solver is a natural fit. Google describes its linear sum assignment solver as specialized for simple assignment instances and notes that it can be faster than MIP or CP-SAT for that setting (OR-Tools documentation). When the model adds logical constraints or otherwise exceeds simple assignment structure, consider MIP or CP-SAT.
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Pay attention to function semantics as well as solver names. SciPy’s sparse min_weight_full_bipartite_matching requests a full matching with cardinality equal to the smaller partition and raises an error if no matching of that size exists (SciPy v1.18.0 documentation). In that phrase, “full” does not mean every vertex on both sides must be matched when the partitions differ. Check the deployed SciPy version and the specific function’s definition before relying on terminology.
Keep algorithm claims in scope
Google’s OR-Tools Hungarian-algorithm reference describes its implementation as having O(n4) complexity and advises using its graph linear assignment implementation, whose complexity is usually smaller (OR-Tools documentation). That bound concerns the referenced implementation; it is not a universal complexity claim for all assignment solvers, nor a measured runtime prediction.
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