An unassigned worker and an empty position can coexist even when everybody could be placed. The worker left outside the plan may not be able to take the remaining position directly. Yet another worker can move there, releasing a place for someone else and eventually opening a suitable position for the person initially left out. The missing opportunity sits in the relationships between assignments rather than in the headcount.
On 5 February 2026, Kommersant reported increased shift-paid side-work offers in Russia, citing Avito Podrabotka data. Those attributed figures do not identify the platform's assignment method. The fictional example below examines a separate planning problem.
The distinction matters because a roster is more than two lists of equal length. One list names people; another names positions. Neither list, considered alone, shows which combinations are possible. Even a complete list of possible combinations does not yet specify which ones should be used together. A workable plan must avoid assigning the same person twice while also avoiding two people being allocated to one position.
Define the assignments before looking for a better plan
Imagine three anonymous workers, A, B and C, and three positions, X, Y and Z. All positions occur simultaneously. Each needs one person, and each person can take at most one position. There are no successive visits, differing job durations, travel between sites or later arrivals. The objective is simply to fill as many of these three positions as the stated compatibility conditions allow.
Every pairing described as feasible includes the worker's willingness to accept it. The example takes place while a tentative plan is being prepared, before commitments have been made. It does not authorise changing an accepted shift against someone's wishes. Pay, preferences between acceptable positions and contractual obligations are not variables in this exercise. The word feasible is a stipulated boundary, not permission to disregard those matters in practice.
Worker A can take X or Y. Worker B can take X only. Worker C can take Y or Z. These are the complete permitted pairings; nothing else is available. The differences might represent task compatibility and availability in an anonymous planning problem, but no personal characteristics or actual qualification requirements are implied. Letters distinguish the participants without ranking their ability or commercial value.
- A can be paired with X or Y.
- B can be paired with X.
- C can be paired with Y or Z.
- No person may appear in two selected pairs.
- No position may appear in two selected pairs.
It is useful to separate permitted pairings from selected pairings. A permission says that a combination could be used. A selection says that it is used in the current plan. A may have two permissions while receiving only one assignment. Counting both permissions as filled positions would double-count the same person's capacity. Conversely, discarding an unused permission would erase a possible route for revising the plan.
A plausible first pass leaves somebody outside
Suppose the tentative plan assigns A to X and C to Y. Both choices are individually permitted. Neither worker appears twice, and neither position receives two workers. The plan is therefore feasible as far as it goes. It fills two positions, leaving B unassigned and Z vacant. There is no arithmetic mistake or forbidden pair hidden in that starting arrangement.
A planner now asks the obvious local question: can B take Z? The answer is no. B's only permitted position is X. If the planner treats the two existing selections as unchangeable, the process stops. The remaining person does not fit the remaining place. But the conclusion applies to that fixed partial plan, not necessarily to the complete set of permitted combinations.
The initial choices have used flexible people in positions that other people might need. A's ability to take Y is unused, while C's ability to take Z is also unused. Neither unused permission fills Z on its own while every other selection remains fixed. Their value appears when the relationships are considered together. The first pass conceals that connection because its selected pairs look individually satisfactory.
This is a different problem from having fewer people than positions. There are three of each, but equal totals alone neither prove nor disprove that all positions can be filled. The relevant information is which pairs are allowed and whether they can coexist. The initial failure to place B cannot settle that question without considering revisions to the assignments already made.
Follow the chain from the unassigned worker
Start with B, the person outside the plan. B can take X, but X is currently occupied by A. Rather than ending the analysis there, follow the current assignment from X back to A. A can also take Y, which is currently occupied by C. Follow that assignment to C. C can take Z, and Z is currently empty.
The complete chain is B, X, A, Y, C, Z. Between B and X is a permitted pairing not selected in the original plan. Between X and A is an existing selection. Between A and Y is another unused permission; between Y and C is another existing selection; between C and Z is a final unused permission. The chain alternates between unused and used connections.
Its endpoints explain why it can improve the count. It begins with an unassigned person and ends with an empty position. Inside the chain are two people and two positions that are already paired, but those pairings can be replaced. The chain identifies a connected set of changes, not a claim that the first idle worker suddenly becomes suitable for the last empty position.
Replace the two old pairs with three new ones
Remove A–X and C–Y from the tentative plan. Add B–X, A–Y and C–Z. Every new pair is on the permitted list. Each worker now appears exactly once, and each position appears exactly once. The number of selected pairs rises from two to three. No worker is hired, no position is deleted, and no compatibility condition is relaxed.
The revision is considered as one change to the plan before commitments are made. It is not a physical instruction to move people through occupied workplaces in the order of the letters. There is no intermediate working schedule in which someone must cover two simultaneous positions. The chain is a way to discover and explain a replacement set of assignments, whose final feasibility is then checked directly.
A simple table could record the same change by person: A moves from X to Y; B moves from no assignment to X; C moves from Y to Z. Reading by position gives the complementary account: X changes from A to B, Y changes from C to A, and Z changes from empty to C. Both views describe the same three-pair result.
Prove the improvement without claiming more than it establishes
The completed plan is not merely better than the first pass. It fills the maximum possible number of positions in this particular example. There are only three positions, so no permitted plan can fill four. The displayed plan fills all three and therefore reaches that upper bound. This direct argument needs no claim about a real platform's software or the performance of a general-purpose algorithm.
The final assignment can also be derived from the most restricted participant. B can take only X, so any plan assigning all three workers must place B there. A then cannot use X and must take Y. C consequently takes Z. This reasoning confirms that the three-pair solution is consistent with every permission. It also shows why the original allocation of A to X blocked completion when treated as permanent.
Starting with B would have avoided this particular first-pass problem. That observation does not establish a universal rule that always assigning the most restricted person first solves every larger allocation problem. The small example proves its own result. It does not supply a general ranking procedure, handle tied restrictions or establish what should happen when new information arrives after assignments are accepted.
The important lesson is the distinction between a plan that cannot be extended by one direct addition and a plan that cannot be improved by revision. Our starting plan has no permitted pair connecting its unmatched endpoints B and Z. Nevertheless, a connected revision increases the number assigned. Failing the first test is therefore insufficient to establish the second in this example.
One missing connection creates a genuine shortfall
Now change exactly one assumption. Remove C's permission to take Z. Leave everything else unchanged: there are still three workers and three simultaneous positions, A can take X or Y, B can take X, and C can take Y. The headcounts are identical to the original version, but the permitted relationships are different.
In this revised version, no worker can take Z. Every assignment must use X or Y. Because each of those positions can receive only one person, no plan can assign more than two people. A–Y together with B–X achieves two, so two is both attainable and the maximum. C remains outside that plan, but choosing a different worker to leave outside would not fill Z.
The earlier chain now ends at C because the final connection to Z is absent. Repeatedly rearranging A, B and C between X and Y cannot manufacture that missing permission. This is a structural shortfall, not merely an unfortunate first pass. It requires a change in the feasible relationships or the problem itself before a three-position plan can exist.
The comparison is deliberately economical. We did not lower anybody's effort, reduce the number of people or introduce a surprise absence. Removing one permitted pair changed the maximum attainable count. Conversely, restoring that same pair restores the displayed three-person solution. The effect comes from where the connection sits in the network, not from a change in an aggregate staffing total.
Adding another person able to take X alone would not solve the empty-Z problem in this revised example. More names would compete for the same two reachable positions while Z remained unreachable. That does not mean additional recruitment is generally useless. It means that a new person's relevant compatibility must be specified before a larger headcount can be credited with resolving this particular shortfall.
A feasible connection is not the same as an attractive outcome
The objective here counts filled positions equally. It does not assign a monetary value to a position or a preference score to a worker's assignment. Filling three instead of two therefore establishes a change in coverage under the stipulated rules. It does not establish a change in profit, productivity, service quality or anyone's welfare. Those would require different information and, potentially, a different objective.
Likewise, a worker's willingness to accept two positions does not imply indifference between them. The model simply assumes both are acceptable for the plan being considered. If a planner wanted to maximise satisfaction among all fully staffed plans, a record of preferences would become relevant. That question is deliberately not answered by counting connections or by declaring the maximum coverage result universally best.
The consent assumption carries practical weight. A tentative pairing can be revised within the example because all listed pairings remain acceptable before commitment. Once an actual agreement is made, changing it can introduce obligations absent from the model. A diagram of feasible possibilities does not override a promise. The analytical result concerns what can be planned together under its conditions, not authority to impose a reassignment.
Keep the relationship data separate from the roster
A list containing only the current roster would show A–X and C–Y. It would not show A's alternative Y or C's alternative Z, so the improving chain would be invisible. A list containing every permission, on the other hand, would show five possible pairs without saying which two are currently selected. Both records are needed to distinguish opportunity from present allocation.
The same separation prevents double counting during review. A person with two feasible positions remains one person, not two units of available labour. A position with two possible workers remains one position. The new plan increases coverage by choosing a different compatible set, not by adding up all possibilities as if they could be exercised simultaneously.
A useful check follows each identifier through the proposed result. Does anyone appear more than once? Is any position assigned twice? Does each selected pair appear on the permitted list? In our final three-pair plan, all three questions have straightforward answers. This verification is distinct from discovering the chain: an attractive-looking revision still needs to be checked against the original conditions.
The inputs must also refer to the same planning situation. A's permission to take Y on one date cannot silently be combined with C's permission to take Z on another date. Our model stipulates simultaneous positions and a fixed set of feasible pairings. If availability changes, the graph changes. A previously valid chain may then cease to be a valid explanation of the new plan.
The empty place can be several connections away
Looking only at the unassigned person and the vacant position makes the original example appear stuck. Looking at every feasible relationship reveals a way to replace two tentative assignments with three. The extra placement is made possible by a connected revision, even though the original unmatched endpoints cannot be paired directly.
The altered example provides the necessary contrast. When C cannot take Z, no reassignment among the existing relationships fills that position. Equal headcounts persist, but the three-pair solution disappears. Together the cases distinguish a remediable allocation from a genuine compatibility limit. The useful question is not simply how many people and positions are listed, but whether their permitted relationships support the complete plan being proposed.