The study, authored by Claudia Regina de Freitas and José Roberto Dale Luche of São Paulo State University, delivers two linked contributions. The first is a public, reproducible benchmark anchored to an anonymized real instance drawn from a university psychology clinic: 141 students, six rooms, four sequential internship stages, and a weekly grid of 65 hourly periods. Around that reference, the authors built a parametrized generator that produced 60 synthetic instances organized into six families, each independently varying one structural lever of the problem. Every instance is released in three interoperable formats, a relational SQLite database, per-table CSV files, and JSON metadata, so that a researcher can load the data with a database driver, a dataframe library, or a constraint modeling toolkit without writing bespoke parsing code.

The second contribution is an analysis of what actually makes such an instance structurally difficult. The authors introduce the pairing feasibility rate, defined for each internship stage as the fraction of same-stage student pairs whose availability windows intersect. The quantity has an elegant graph-theoretic reading: it is the edge density of a temporal compatibility graph whose vertices are the students of a stage and whose edges join pairs that share at least one available period. Reciprocal pairs correspond to edges of this graph, and candidate triadic cycles correspond to its triangles. Because reciprocal supervision requires two students to be free at the same time, the rate measures the raw supply of possible partnerships before any solver is ever run.

The real instance reveals how sparse and clustered clinical availability really is. Of the 9,165 student-period cells in the availability matrix, only 600 are positive, a density of roughly 6.5 percent, with the average student free in just over four of the 65 weekly periods. Yet aggregate room capacity is far from scarce: the demand-to-slot ratio is only 0.38, meaning the clinic’s six rooms could in principle absorb the full load of 150 pending requirements many times over. The binding scarcity is relational, not physical. Between a third and a half of all potential supervision partnerships are temporally incompatible before any scheduling decision is made, with per-stage pairing feasibility rates ranging from 0.49 to 0.64.

The audit also shows why triadic supervision exists at all. Disjoint pairs can only be formed inside connected components of the compatibility graph, and any component holding an odd number of students always leaves one unpaired. In the reference instance, stage one splits into components of 31 and 9 students, so two students remain uncovered despite the stage’s even total of 40; stages three and four each leave one more. Four requirements in total cannot be met by binary pairing, and a triadic cycle, which needs pairwise compatibility among three students but no period common to all three, is the structural remedy. An isolated student with no compatible peer, by contrast, belongs to no edge and no triangle and remains uncoverable by any configuration.

The synthetic suite turns these observations into controlled experiments. A scale family varies the student count from 30 to 200; an availability-density family sweeps configured density from 0.04 to 0.15; a temporal-clustering family manipulates how tightly same-stage students share dominant day-windows; a curriculum-load family varies the proportion of students carrying two consecutive stages; a period-capacity family reshapes the planning grid; and a triadic-stress family replaces the day-window model with a chain of partially overlapping sub-windows that thins global compatibility while preserving local triangles. Across the suite, the pairing feasibility term shows the largest marginal variation of the three components of the authors’ composite structural difficulty index, and the index correlates with pairing feasibility at r = -0.88, a relationship the authors carefully flag as partly algebraic, since the term enters the index by construction.

The most striking finding comes from the solver-based validation. The authors solved the benchmark’s canonical task, maximizing the number of covered student-stage requirements, to proven optimality on all 60 instances using CPLEX through GAMSPy with a one-hour limit per instance, and an independent PuLP-CBC implementation cross-checked representative cases. Total solving time was 3,905 seconds, with a median of 31.4 seconds. Counterintuitively, structural scarcity and computational effort turned out to be distinct, even opposing, dimensions: solving time correlates positively with pairing feasibility at Spearman +0.63 and negatively with the difficulty index at -0.25. The lowest-compatibility family, triadic stress, solved in about four seconds on average, while the compatibility-rich availability-density family took roughly thirteen times longer. The dominant correlate of effort was simply the size of the candidate set, at Spearman +0.87, because denser compatibility graphs generate far more candidate pairs and triangles and thus much larger binary programs.

Coverage behaves independently of both. Achieved coverage spans a narrow band from 0.85 to 0.94 across families and barely correlates with any structural metric. But a targeted ablation on the largest scale instance exposed a different villain: local room congestion. On that 200-student instance, relaxing the room-capacity constraint lifted optimal coverage from 0.696 to 0.957, while removing the student non-overlap rule raised it only to 0.734. In the optimal 144-session schedule, 20 of the 65 periods were filled to their six-room capacity while 24 held no session at all, with overall occupancy at just 37 percent. Aggregate abundance, in other words, does not preclude local binding when availability is temporally concentrated, echoing the real clinic’s late-afternoon peaks in which 31 students of a single stage are simultaneously free against six rooms.

The authors are explicit about the limits of their claims. The structural difficulty index is a transparent, solver-independent ordering of instances, not a predictor of computational hardness, and the effort findings are specific to their binary formulation and single-thread configuration. The reference instance comes from a single institution and a single planning horizon, and the socio-academic enrichment layer, synthetic attributes calibrated to the real cohort through a Gaussian copula, is deliberately independent of the scheduling core and carries no designed signal for difficulty or coverage. A re-identification audit found every one of the 141 real students unique on their attribute combination, so no real micro-data are distributed; the released attributes reproduce only aggregate distributions and selected correlations.

What makes the work resonate beyond psychology clinics is the framing itself. The compatibility graph formulation treats reciprocity-coupled scheduling as a first-class problem, applicable wherever two actors must jointly occupy a session in complementary roles, from medical residencies to clinical placements of any kind. Because the generator exposes its levers explicitly and the entire 60-instance suite can be regenerated bit for bit from a single configuration file and seed, researchers can extend the benchmark with new families, probe exact, heuristic, and learning-based methods on equal footing, and eventually test whether structural descriptors like pairing feasibility can predict outcomes such as non-allocation risk. For the coordinators still wrestling with spreadsheets, and for the algorithm designers who never knew their problem existed, the benchmark finally gives both sides a shared, measurable substrate.

Subject of Research: Synthetic benchmark generation and instance difficulty analysis for educational clinical internship scheduling with reciprocal supervision

Article Title: A parametrized synthetic benchmark and instance difficulty analysis for educational clinical internship scheduling with reciprocal supervision

Article References: de Freitas, C. R., & Luche, J. R. D. (2026). A parametrized synthetic benchmark and instance difficulty analysis for educational clinical internship scheduling with reciprocal supervision. International Journal of Data Science and Analytics, 22(1), Article 318. https://doi.org/10.1007/s41060-026-01316-1

Image Credits: AI Generated

DOI: 10.1007/s41060-026-01316-1

Keywords: clinical internship scheduling, reciprocal supervision, educational timetabling, synthetic benchmark, instance difficulty, temporal compatibility graph, pairing feasibility rate, operational research, combinatorial optimization, reproducibility, constraint programming, psychology clinics