Unbalanced Assignment Problem with Constraints

• UAP-C is a family of models where unmatched cardinalities and partial assignments meet additional hard constraints like capacity, budgets, and diversity quotas.
• The formulations extend classical matching by incorporating multidimensional resource limits, eligibility criteria, and conflict or quota-based restrictions across varied applications.
• Recent algorithmic approaches include LP relaxations, reinforcement learning with masking, and MILP techniques, highlighting both NP-hard complexities and tractable subclasses.

Unbalanced Assignment Problem with Constraints (UAP-C) denotes a family of assignment models in which the cardinalities of the two sides need not coincide, not every entity must be matched, and feasibility is governed by additional hard constraints beyond classical one-to-one matching. In the formulations collected under this label, the assignment variables remain binary, but the constraint system may include multidimensional capacities, bin-activation budgets, worker eligibility and time capacities, distributional quotas, type–block diversity caps, or pairwise conflicts among assignments. The resulting models subsume several distinct research directions: generalized budgeted assignment in transportation (Jiang et al., 2022), capacitated assignment with dynamic state and action masking in reinforcement learning (Pathan et al., 2021), assignment mechanisms under distributional constraints (Ashlagi et al., 2018), assignment with type constraints and diversity quotas (Benabbou et al., 2017), and assignment with conflicts solved by CP-SAT (Montemanni et al., 3 Jun 2025).

1. Formal models and the meaning of “unbalanced”

The defining structural feature of UAP-C is that feasibility is not tied to a perfect matching. In the generalized budgeted assignment formulation, there are $L$ bins and $P$ items, each item is assigned to at most one open bin, and items need not all be assigned. The canonical integer program is

$\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$

subject to

$$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$

$\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$

$\sum_{l=1}^L c_l y_l \le B,$

$x_{lp} \le y_l \quad \forall l,\forall p,$

with $x_{lp},y_l \in \{0,1\}$. Here unbalancedness means that $|P|$ may differ from $|L|$ and the item constraints are “$P$0” rather than equality (Jiang et al., 2022).

A second recurrent form is capacitated many-to-one assignment. In the reinforcement-learning formulation, workers $P$1 and tasks $P$2 are linked by binary variables $P$3, with objective

$P$4

worker capacities

$P$5

task coverage

$P$6

and eligibility constraints

$P$7

Although the paper requires every task to be assigned, it uses $P$8 in experiments and many workers remain unused, so the mapping is still unbalanced on the worker side (Pathan et al., 2021).

Distributional and diversity-constrained formulations express unbalancedness differently. In assignment mechanisms under distributional constraints, an outside option $P$9 absorbs unmatched agents, and the objective is to assign as many agents as possible to regular objects subject to lower and upper quotas over types or subsets of types (Ashlagi et al., 2018). In assignment with type constraints, agents are assigned to items or slots with capacities $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$0, each agent receives at most one item, and type–block pairs are capped by quotas $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$1: $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$2 subject to

$\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$3

This directly encodes unmatched agents and unused capacity (Benabbou et al., 2017).

The conflict-constrained assignment formulation starts from the classical complete bipartite setting, but adapts naturally to unbalanced variants by replacing perfect-matching equalities with “at most one” constraints. With feasible assignment edges $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$4 and conflict set $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$5, the unbalanced model uses

$\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$6

plus pairwise disjunctions

$\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$7

and a linear cost objective (Montemanni et al., 3 Jun 2025).

Variant	Optimization core	Distinctive constraints
GBAP	Maximize $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$8	Multidimensional capacities, activation budget, $\max \sum_{l=1}^L \sum_{p=1}^P v_{lp} x_{lp}$9
RL capacitated AP	Minimize $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$0	Worker capacities, task assignment, eligibility
Distributional assignment	Maximize regular-object assignments	Lower/upper quotas by type or category, outside option $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$1
Type-constrained assignment	Maximize $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$2	Item capacities, type–block quotas
Assignment with conflicts	Minimize $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$3	Pairwise conflict inequalities on assignment edges

Taken together, these models suggest that UAP-C is better understood as a constrained assignment umbrella than as a single standardized formulation. The common kernel is binary assignment under nonclassical feasibility structure; the principal variations lie in whether constraints are resource-based, distributional, strategic, or logical.

2. Constraint structure and modeling primitives

The budgeted capacitated model introduces a particularly rich resource structure. Each bin $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$4 has an $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$5-dimensional capacity vector, and each item $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$6 has a binary weight vector $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$7 whose ones, if present, appear in consecutive positions. The consecutive ones property is application-driven—especially for route segments on a line—but it also has algorithmic consequences, because it facilitates single-bin feasibility and separation in the configuration LP (Jiang et al., 2022). A plausible implication is that interval structure is not merely a modeling convenience; it is part of what makes the stronger relaxation computationally usable.

The reinforcement-learning formulation emphasizes operational constraints rather than budget activation. Capacities are worker-side time units $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$8, task efforts are $$\sum_{p=1}^P r_{lp}^{(i)} x_{lp} \le C_l^{(i)} \quad \forall l,\forall i,$$9, and feasibility is filtered by eligibility indicators $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$0. The paper states that “time” in the network input indexes decision stage rather than scheduling start/end times, so true temporal scheduling constraints are not part of the AP experiments (Pathan et al., 2021). This distinction matters because UAP-C is often conflated with scheduling; in this formulation, capacities are cumulative resource limits, not interval scheduling windows.

Distributional formulations replace scalar capacities with quota systems over types or subsets of types. In the distributional-constraints framework, for each object $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$1 and subset $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$2 there are lower and upper quotas,

$\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$3

and related category constraints can aggregate over sets of objects. The objective is not cardinal utility maximization but maximizing the number of agents assigned to regular objects subject to these quota systems (Ashlagi et al., 2018). This shifts UAP-C from pure combinatorial optimization toward constrained allocation and mechanism design.

Type–block diversity constraints impose a simpler but still hard quota family. Agents are partitioned into types, items into blocks, and each type–block pair has an upper quota $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$4. The paper notes that this already makes the problem NP-hard, even though the unconstrained assignment problem is polynomial-time computable (Benabbou et al., 2017). A common misconception is that quota systems are a modest extension of capacity constraints; the hardness results indicate that even one additional quota dimension can fundamentally change the problem class.

Conflict constraints are logically different from capacities and quotas. Here the hard constraints prohibit selecting certain pairs of assignment edges simultaneously. The resulting feasible set can be interpreted as an assignment polytope intersected with a set-packing structure induced by conflicts (Montemanni et al., 3 Jun 2025). This matters because pairwise incompatibility is not reducible to simple capacity accounting: two individually feasible assignments may become jointly infeasible for purely combinatorial reasons.

Several formulations also describe standard adaptations for partial coverage. The reinforcement-learning paper states that if tasks could be left unassigned, one would introduce a dummy worker $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$5 with large cost $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$6 and capacity $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$7, or slack variables with penalties; the conflict-constrained formulation analogously introduces unassigned indicators $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$8 or dummy nodes to balance the bipartition (Pathan et al., 2021, Montemanni et al., 3 Jun 2025). These constructions make explicit that UAP-C includes both “hard coverage” and “soft coverage” regimes.

3. Algorithmic frameworks, relaxations, and guarantees

The generalized budgeted assignment model uses two LP relaxations. The simple relaxation lets $\sum_{l=1}^L x_{lp} \le 1 \quad \forall p,$9 and scales capacities by $\sum_{l=1}^L c_l y_l \le B,$0, but the paper notes that it can be loose. The stronger formulation is a configuration LP with variables $\sum_{l=1}^L c_l y_l \le B,$1 for feasible item sets $\sum_{l=1}^L c_l y_l \le B,$2, where

$\sum_{l=1}^L c_l y_l \le B,$3

The LP can be solved in polynomial time via the ellipsoid method using the single-bin separation oracle, and the consecutive ones structure enables dynamic programming or min-cost flow style separation in polynomial time (Jiang et al., 2022).

Its rounding scheme has two phases. First, for each bin, at most one configuration is sampled independently from the configuration LP solution. Second, budget feasibility is enforced by an online fractional knapsack accept/reject rule, while item-at-most-once is enforced by a generalized $\sum_{l=1}^L c_l y_l \le B,$4-conservative magician with one unit of mana and $\sum_{l=1}^L c_l y_l \le B,$5. Defining $\sum_{l=1}^L c_l y_l \le B,$6, the paper proves that for $\sum_{l=1}^L c_l y_l \le B,$7,

$\sum_{l=1}^L c_l y_l \le B,$8

and for $\sum_{l=1}^L c_l y_l \le B,$9,

$x_{lp} \le y_l \quad \forall l,\forall p,$0

It also states that the algorithm returns a feasible solution with probability $x_{lp} \le y_l \quad \forall l,\forall p,$1, and for $x_{lp} \le y_l \quad \forall l,\forall p,$2 the lower bound exceeds $x_{lp} \le y_l \quad \forall l,\forall p,$3 (Jiang et al., 2022). The contrast drawn with the prior RLPP randomized rounding is specific: RLPP attains ratio $x_{lp} \le y_l \quad \forall l,\forall p,$4 but may violate budget with probability at most $x_{lp} \le y_l \quad \forall l,\forall p,$5.

A very different algorithmic line is reinforcement learning with hard feasibility masks. The capacitated assignment paper formulates the problem as an MDP whose state includes remaining capacities, remaining task efforts, eligibility, the current decision time index, and the last completed task id. The actor outputs a distribution over workers, but infeasible workers are removed by masking, including workers with $x_{lp} \le y_l \quad \forall l,\forall p,$6 or insufficient capacity (Pathan et al., 2021). Because infeasible actions are removed from the action space, the sampled action is feasible by construction.

The RL method uses proximal policy optimization with clipped surrogate loss, value loss, and entropy bonus. The reported hyperparameters are $x_{lp} \le y_l \quad \forall l,\forall p,$7, $x_{lp} \le y_l \quad \forall l,\forall p,$8, learning rate $x_{lp} \le y_l \quad \forall l,\forall p,$9 with decay $x_{lp},y_l \in \{0,1\}$0, batch size $x_{lp},y_l \in \{0,1\}$1, $x_{lp},y_l \in \{0,1\}$2 epochs per episode, and experience buffer length $x_{lp},y_l \in \{0,1\}$3. The feature extractor combines a 1D convolution with $x_{lp},y_l \in \{0,1\}$4 filters over dynamic features, dense layers of $x_{lp},y_l \in \{0,1\}$5 units, and separate actor and critic heads; “current time” and “last task” are included so that an RNN/LSTM decoder is unnecessary (Pathan et al., 2021). The paper reports that masking yields faster training and convergence than penalty-based methods, although it does not provide a formal proof.

For type-constrained assignment with diversity caps, the paper gives a polynomial-time $x_{lp},y_l \in \{0,1\}$6-approximation by reduction to Bounded Color Matching, where colors correspond to type–block pairs and color capacities correspond to the quotas. It also identifies two tractable cases: type-uniform utilities and block-uniform utilities, both reducible to min-cost flow (Benabbou et al., 2017). These results place part of UAP-C in the approximation-and-special-case tradition rather than the LP-rounding or RL traditions.

For conflict-constrained assignment, the computational approach is direct MILP encoding solved with Google OR-Tools CP-SAT. The model uses BoolVar variables, linear assignment constraints, pairwise conflict inequalities, and a linear cost objective. The paper reports solving the MILP directly with CP-SAT version 9.12, a time limit of $x_{lp},y_l \in \{0,1\}$7 seconds per instance, and default settings (Montemanni et al., 3 Jun 2025). This approach is notable because it relies on general-purpose propagation and search rather than a bespoke combinatorial approximation algorithm.

4. Distributional constraints, diversity quotas, and mechanism design

Under distributional constraints, the assignment problem acquires strategic and fairness dimensions that are absent from pure optimization models. The mechanism-design framework introduces agents with publicly known types and private ordinal preferences over objects, plus an outside option $x_{lp},y_l \in \{0,1\}$8. Feasibility is defined through lower and upper quotas over types or subsets of types, and the optimization benchmark is the maximum number of agents assigned to regular objects in LP1 or LP3 (Ashlagi et al., 2018).

The Generalized Serial Dictatorship (GSD) mechanism processes agents sequentially and uses an auxiliary LP, denoted LP2 in the synthesis, to determine whether an agent can be assigned to a preferred school while preserving eventual attainment of at least $x_{lp},y_l \in \{0,1\}$9. The mechanism maintains an incomplete type-assignment vector $|P|$0, relaxation variables $|P|$1, and a set of partially assigned agents. Its formal guarantees are strong but approximate on constraints: it is strategyproof, allocatively efficient in the sense of assigning at least $|P|$2 agents to regular schools, Pareto efficient with respect to the relaxed constraints, and satisfies

$|P|$3

with sharper per-type bounds

$|P|$4

In laminar cases with integer quotas, the paper states that exact integral feasibility is achievable without violations via a reduction to max-flow with lower/upper bounds (Ashlagi et al., 2018).

The Generalized Probabilistic Serial (GPS) mechanism is the fractional counterpart. It maintains extendability: the current fractional vector must remain extendable to an LP3-optimal feasible solution. The output is ordinally efficient and within-type envy-free, and the paper states that every LP3-optimal fractional assignment can be implemented as a lottery over integral allocations that violate any quota by at most $|P|$5 and assign at least $|P|$6 agents to regular schools (Ashlagi et al., 2018).

The central controversy in this line is not empirical but impossibility-theoretic. The paper shows that no mechanism is simultaneously ordinally efficient, within-type envy-free, and weakly strategyproof under distributional constraints (Ashlagi et al., 2018). This result rules out a frequent expectation that fairness, efficiency, and nonmanipulability can all be retained under quota systems.

The type–block diversity framework studies a related but welfare-oriented question: the loss induced by quota-based diversity. It defines the price of diversity

$|P|$7

and proves the quota-only upper bound

$|P|$8

which is stated to be tight. It also proves a disparity-aware bound

$|P|$9

and summarizes them as

$|L|$0

These bounds formalize the trade-off between diversity caps and welfare under hard quota systems (Benabbou et al., 2017).

5. Applications and empirical behavior

The motivating application for generalized budgeted assignment is transit line planning. The paper maps bins to candidate transit lines with operating costs, capacity vectors to per-edge passenger capacities on a line, items to passenger trips, and the binary consecutive-ones vectors $|L|$1 to the edges used by a trip on a line. Rewards quantify welfare gain or coverage, and the global budget is the operating budget (Jiang et al., 2022). This is a particularly clear instance of UAP-C because it combines unbalanced assignment, multidimensional capacities, and activation decisions in one model.

The empirical study reported for that application uses NYC data with candidate lines $|L|$2, $|L|$3 trip requests, bus capacity $|L|$4, and several budgets. The proposed GBAP algorithm is reported to find higher-quality solutions than the prior RLPP rounding algorithm, especially at larger budgets such as $|L|$5 and $|L|$6. The paper also states that for typical $|L|$7 values $|L|$8, $|L|$9, and $P$00, the bound exceeds $P$01, $P$02, and $P$03 respectively (Jiang et al., 2022).

The reinforcement-learning assignment framework targets dynamic many-to-one task allocation and reports experiments on AP10, AP20, AP30, AP40, and AP50, with workers $P$04, worker capacities $P$05, and task efforts sampled in $P$06. The paper states that the RL framework outperforms Google OR-Tools using MIP and CP-SAT solvers on large problem instances in solution quality and computation time. The most specific timing summaries given are: for AP50, Google OR-Tools solution time is approximately $P$07 minutes, while RL inference time is approximately $P$08 minutes; for AP30 with $P$09 tasks updated, OR-Tools requires approximately $P$10 minutes to rebuild, while the RL framework adapts in approximately $P$11 minutes without retraining (Pathan et al., 2021). The same framework is also demonstrated on bin packing and capacitated vehicle routing.

Diversity-constrained assignment is studied empirically in Singapore public housing and Chicago school choice. In the Singapore HDB simulations, the paper uses $P$12 HDB blocks, $P$13 flats, ethnicity quotas $P$14, $P$15, and $P$16, with applicant pools $P$17 and $P$18. It reports that the price of diversity is approximately $P$19 across several utility models, while lottery performance varies: under distance-based utilities, a simple lottery with quotas yields at least about $P$20 of OPT when $P$21, but around $P$22 at $P$23 when $P$24 increases to $P$25; under price-based utilities with high variance, lottery performance can be around $P$26 of OPT at $P$27 when $P$28 (Benabbou et al., 2017).

In Chicago school choice, the paper considers $P$29 schools, approximately $P$30 seats for grade $P$31, four tiers with equal fractional quotas $P$32 per tier per school, and applicant pools $P$33 and $P$34. It reports that the price of diversity decreases with higher $P$35, but the lottery has high welfare loss across settings and performs worse when $P$36 (Benabbou et al., 2017). These observations suggest that welfare loss from quotas and welfare loss from lottery-based implementation are distinct empirical phenomena.

Conflict-constrained assignment is evaluated on literature instances with $P$37 ranging from $P$38 to $P$39, conflict pairs $P$40 ranging from $P$41 to $P$42, and $P$43 usable instances out of the original $P$44. The compared methods are LS, RDS, BIP, B&B, and CP-SAT, each run up to $P$45 seconds. The paper’s summary is that LS is fastest for very quick solutions, CP-SAT is the best option for quickly finding optimal solutions when more computation time is allowed, BIP is generally the most robust and fastest exact method overall, and B&B scales better on the largest instances (Montemanni et al., 3 Jun 2025).

6. Complexity, tractable subclasses, and limitations

Several of the cited models formalize the computational escalation caused by additional constraints. The type-constrained diversity model proves that adding type–block constraints makes the problem NP-complete, and the distributional-constraints work notes that exact feasibility under hard constraints can be NP-complete to decide (Benabbou et al., 2017, Ashlagi et al., 2018). These results clarify that the difficulty in UAP-C often comes less from unbalancedness itself than from quota structure, multiattribute feasibility, or logical incompatibilities.

At the same time, multiple tractable subclasses are identified. For type–block quotas, type-uniform utilities and block-uniform utilities permit polynomial-time min-cost flow formulations (Benabbou et al., 2017). For distributional constraints, laminar quota systems with integer quotas admit exact feasibility via max-flow with lower and upper bounds (Ashlagi et al., 2018). For generalized budgeted assignment, the consecutive ones property enables polynomial-time single-bin separation for the configuration LP (Jiang et al., 2022). These are not generic simplifications; they are structural conditions under which otherwise hard UAP-C variants recover specialized exact methods.

The models also differ in how they treat feasibility. The budgeted assignment rounding explicitly guarantees capacity, item-at-most-once, and budget feasibility with probability $P$46 (Jiang et al., 2022). The reinforcement-learning framework guarantees capacity and eligibility feasibility during decoding only because infeasible actions are masked out; the paper emphasizes that no repair step is needed, but it also notes limitations under very tight capacities or near infeasibility, where a pure mask may leave few actions and increase variance (Pathan et al., 2021). The mechanism-design formulations deliberately allow bounded additive violations of quotas—up to $P$47—in order to preserve strategyproofness, ordinal efficiency, or allocative efficiency (Ashlagi et al., 2018). A common misunderstanding is to compare these guarantees as if they were commensurate; they apply to different objective and feasibility regimes.

Further limitations are model-specific. The reinforcement-learning experiments do not model true time windows, precedence constraints, or explicit scheduling dynamics, and the paper states that richer temporal constraints would require richer states and masks or hybrid RL+OR integration (Pathan et al., 2021). The budgeted assignment guarantees depend on the parameter $P$48, with the bound improving for larger $P$49 and reverting to the universal constant $P$50 for small $P$51 (Jiang et al., 2022). The type-constrained assignment paper states that lower quotas may cause infeasibility when the composition of the agent pool is not guaranteed ex ante (Benabbou et al., 2017). The conflict-constrained paper focuses on pairwise inequality modeling and does not introduce strengthened formulations such as clique cuts within that work (Montemanni et al., 3 Jun 2025).

Open problems are stated explicitly in several sources. The distributional-constraints work asks whether the $P$52 violation bound can be reduced under richer structural constraints, whether GPS rounding can achieve smaller per-realization violations under broader constraints, and how to handle multi-type agents or overlapping categories (Ashlagi et al., 2018). The diversity-quota work asks whether approximation factors better than $P$53 are achievable in polynomial time, how to quantify LP integrality gaps under natural utility distributions, and how to analyze lotteries theoretically under quotas (Benabbou et al., 2017). Collectively, these questions suggest that UAP-C remains a heterogeneous field in which exact optimization, approximation, learning-based control, and mechanism design each address only part of the constraint landscape.