OPDiv: Optimal Selection of Top-K High-Scoring, Diverse Compounds

1. Introduction

• Machine-learning and generative methods may exploit locally promising clusters and get stuck in local optima or reinforcement-learning-induced modal collapse.Limited diversity and combinatorial explosion may also challenge ultra-large chemical spaces and DNA-encoded libraries.Source: §1 · p. 1
• The paper asks how to select compounds for high scores and diversity optimally, and how to evaluate those selections.Source: §1 · p. 1
• Figure 1 depicts the score–diversity trade-off: stricter minimum-distance thresholds require greater separation and can make a size-k selection infeasible.The caption also frames OPDiv as covering the achievable Pareto frontier.Source: Figure 1 · p. 1
• Virtual-screening pipelines typically rank and filter candidates, then use a separate diversity-selection step such as clustering, scaffold deduplication, or sphere exclusion.The paper proposes fair evaluation of scoring pipelines based on their ability to find high-scoring compounds while maintaining diversity.Source: §1 · pp. 1–2
• OPDiv makes compound selection optimal by finding the highest-scoring set that meets a chosen size and diversity requirement.Its mean score also provides a common criterion for evaluating the candidate pool.Source: §1 · p. 2

2.1. Diversity Selection Background

• OPDiv maximizes the mean score of a fixed-size subset under explicit diversity constraints; its score–diversity curve describes the best attainable score at each requirement.The interpretation concerns supplied scores and molecular representations, not a direct estimate of experimental success or financial-style risk.Source: §2.1 · p. 2

2.2. Optimal and greedy selection

• A molecular portfolio is a set of N distinct compounds selected jointly for experimental follow-up, with each compound contributing equally to its mean score.The default formulation constrains which compounds may be selected together to meet diversity requirements.Source: §2.2 · p. 3
• GPDiv scans compounds from highest to lowest score and accepts each sufficiently dissimilar candidate until N are selected or the pool is exhausted.When GPDiv returns N compounds, its mean score cannot exceed OPDiv; otherwise, no size-N GPDiv value is assigned.Source: §2.2 · p. 3

2.3. Allowing several analogs per series

• CPDiv maximizes mean score for N compounds under upper limits on fixed, disjoint clusters, replacing pairwise similarity constraints.With those group limits alone, greedy selection is exact; overlapping groups, minimum quotas, or added pairwise exclusions require a different optimization method.Source: §2.3 · p. 3
• Cluster-capacity selection can still include similar compounds across different clusters, and reclustering a growing pool can change the selection rule.A limit of one selects the best-scoring representative from each selected group.Source: §2.3 · p. 3

2.4. Fixed chemotype weights

• Fixed chemotype weights can replace molecule scores in the objective, preserving its linear form and existing selection constraints.For example, inverse reference-collection group size gives less weight to members of abundant groups; weights and group assignments must remain fixed across compared pools.Source: §2.4 · pp. 3–4

3.1. Credit for the best available shortlist

• OPDiv evaluates a pool by the strongest feasible size-N shortlist, while comparisons of search methods require matched acquisition budgets.The criterion does not remove effects of pool composition or score error, and claims about experimental hit or chemotype recovery require independent activity labels.Source: §3.1 · p. 4
• The ordinary top-N mean minus OPDiv is the cost of requiring diversity; OPDiv minus another allowed selection's mean is score lost by that procedure.The latter loss could be recovered from the same pool without weakening the diversity requirement; unresolved optimization bounds make the differences ranges.Source: §3.1 · p. 4

3.2. The score–diversity trade-off

• A score–diversity curve gives the best achievable mean at each diversity requirement, showing the score cost of greater separation and whether pool comparisons depend on the cutoff.Stricter separation can only lower the optimum or make the requested set infeasible; crossing curves mean relative value depends on the required diversity.Source: §3.2 · p. 4
• A ranked diversity spectrum sorts a selected set's scores to reveal whether a high mean is sustained across the set or conceals weak members.Each requested size is optimized separately, so the first ten members of an optimal 50-compound set need not be the optimal ten-compound set.Source: §3.2 · p. 4

3.3. An enrichment factor based on random acquisition

• The proposed random-acquisition factor asks how many randomly acquired compounds are needed to supply a portfolio matching the candidate pool's feasible OPDiv value.It compares sampling effort rather than score values; in the finite-library definition, the budget counts distinct eligible compounds acquired and scored.Source: §3.3 · p. 4
• The factor depends on the reference library, acquisition policy, portfolio size, and diversity requirement, and is proposed but not evaluated in the experiments.Unreachable targets are reported as such, and other acquisition-cost conventions require matched random baselines.Source: §3.3 · p. 5

4. Implementation and optimization guarantees

• The implementation removes duplicate and ineligible records, then identifies pairs too similar to select together using fingerprint, embedding, or three-dimensional comparisons.Identity handling must specify salts, tautomers, stereochemistry, and preparation states; asymmetric comparisons need symmetrization, while explicit 3D comparisons need conformer, protonation, and alignment policies.Source: §4 · p. 5
• Binary selection variables and constraints fix portfolio size and exclude forbidden pairs in the integer optimization.The output includes selected identities, original scores, and an interval L ≤ QN ≤ U; the gap between the best validated mean and upper bound indicates possible remaining improvement.Source: §4 · p. 5
• Computation can focus on the highest-scoring part of a pool while optimistic placeholders bound the full pool, and selected identities, counts, scores, and pairs are checked independently.At the time limit, solver optimality may remain unresolved or feasibility may not be established.Source: §4 · p. 5

5.1. Data and comparison protocol

• The comparison uses 10,000 unique acquisitions per run for DRD2, GSK3, and JNK3, selecting 20 compounds from a shared pool without a minimum-score filter.The full experiment covers 17 method families and 324,616 parameter-and-seed combinations; computational budgets vary across methods.Source: §5.1 · p. 5

5.2. How much score can better selection recover?

• Several selectors approach the best attainable mean over parts of the distance range, particularly for GSK3, while other gaps show score available from a better combination in the same pool.The shaded gap above the OPDiv reference indicates further possible improvement; comparisons use each method's highest mean among saved selections meeting the same distance requirement.Source: §5.2 · p. 5
• 0.013 (DRD2), 0.004 (GSK3), and 0.018 (JNK3): at minimum Tanimoto distance 0.70, optimized selection improves mean score over GPDiv across four repeats.Gains over Butina centers are 0.124, 0.049, and 0.106, respectively; ordinary top-N fails this distance requirement in all 12 pools. Some compared methods optimize criteria other than minimum distance.Source: §5.2 · pp. 5–6, Figure 2 · p. 6

5.3. Scores within selected sets

• For 50-compound DRD2 selections, stricter limits lower optimal means from 0.714 to 0.636 for Tanimoto, 0.720 to 0.654 for shape, and 0.720 to 0.645 for electrostatics.The lowest selected scores also fall from 0.670 to 0.550, 0.680 to 0.577, and 0.670 to 0.560, respectively, revealing weaker members needed to fill more separated sets.Source: §5.3 · p. 6
• Under matched budgets and selection rules, high scores sustained deeper into the selected-set profile indicate a stronger supply of promising candidates.The DRD2 profiles use original scores as the selection objective and compare Morgan Tanimoto with learned shape and electrostatic similarities.Source: §5.3 · p. 6

6.1. Matching the selection to the experimental decision

• OPDiv selects the highest-scoring set meeting the chosen size and diversity requirements; the appropriate rule depends on whether follow-up advances distinct series or allows several analogs per series.For disjoint groups with fixed allowances, CPDiv provides an exact, inexpensive selector; OPDiv ranks feasible sets by mean score.Source: §6.1 · p. 6
• The formulation assumes equal follow-up costs; unequal prices, shared synthetic routes, or batch chemistry require additional constraints or a different budget objective.Source: §6.1 · p. 6

6.2. Evaluating and guiding molecular search

• A generator with many close analogs can still provide an excellent diverse shortlist, so best-set quality at fixed acquisition budgets separates finding useful candidates from assembling the final selection.The acquisition budget accounts for effort spent obtaining candidates; score–diversity curves add context about separation.Source: §6.2 · p. 7
• At fixed N, OPDiv can plateau when a pool already contains an excellent set; optimal means cannot rise with larger follow-up sizes, although optimal sets need not be nested.Coverage measures describe explored chemical space, while diverse-hit counts measure separated candidates above a score cutoff.Source: §6.2 · p. 7
• Best-set improvement could guide candidate acquisition or reinforcement learning, but such a reward would need to handle sparse rewards, repeated optimization costs, and solver-approximation score changes.Source: §6.2 · p. 7

6.3. Library selection and design

• OPDiv can select scored hits from DNA-encoded or make-on-demand libraries, but designing a whole combinatorial library requires explicit treatment of shared reagents, feasibility, cost, and coverage.That extension would link product scores to building-block and reaction choices.Source: §6.3 · p. 7

6.4. Robustness and limitations

• Exact OPDiv values are independent of input order and duplicate records after deduplication, and adding candidates cannot worsen the optimum if existing scores and pairwise relationships stay fixed.Generators may still exploit score errors, representation weaknesses, or a similarity boundary.Source: §6.4 · p. 7
• Comparisons require scores with a common meaning and scale, while fingerprints, embeddings, clustering policies, and thresholds remain modeling choices.Nonlinear score transformations can change preferred combinations even if individual rankings stay unchanged.Source: §6.4 · pp. 7–8

• OPDiv finds the highest-scoring set of a given size that meets an explicit diversity requirement, providing both a compound shortlist and an evaluation of its source pool.Source: §7 · p. 8
• GPDiv supplies a fast greedy reference, while CPDiv gives exact selection for fixed disjoint groups with upper limits.Source: §7 · p. 8
• Score–diversity curves, selected-set profiles, feasibility checks, and optimization bounds help distinguish the cost of requiring diversity from score unnecessarily lost during selection.Source: §7 · p. 8

• Source code and package links are provided on the paper’s first page.Source: Software and data availability · p. 8
• The molecules and scores used in the paper come from the public diverse-hit benchmark.Source: Software and data availability · p. 8
• The authors plan to release reproduction code, selected molecules, parameter settings, checkpoint hashes, embeddings, and solver bounds separately; Appendix A records supporting definitions and protocols.Source: Software and data availability · p. 8

A.1. Related measures and selection rules

• #Circles is the maximum number of mutually separated molecules at a chosen distance threshold.A size-N OPDiv selection is feasible exactly when this maximum is at least N under the same boundary convention; heuristic packing gives only a lower bound.Source: §A.1 · p. 10
• Sphere-exclusion diversity is the fraction retained by sequential acceptance and neighbor exclusion, and its result depends on input order, tie handling, deduplication, and sampling.Internal diversity averages pairwise dissimilarity (Benhenda, 2017), while cluster counts depend on the chosen representation, algorithm, and threshold.Source: §A.1 · p. 10
• Diverse-hit counts apply a score cutoff before counting separated qualifying molecules; a heuristic count is only a lower bound on the maximum.A maximum count of at least N guarantees an allowed size-N set whose members all meet the cutoff, unlike an OPDiv mean above the cutoff, which may include lower-scoring members.Source: §A.1 · p. 10
• MaxMin and MaxSum are greedy distance-based rules, while OptiSim samples candidates after excluding nearby molecules and selects the sampled candidate farthest from its nearest selected neighbor (Borodin et al., 2012).These selectors stop at N or exhaustion and keep any separation threshold fixed; OptiSim with k = 1 gives random-order sphere exclusion.Source: §A.1 · p. 10

A.2. Selector settings and matched comparisons

• MaxMin, MaxSum, and OptiSim are compared on score-filtered prefixes, while GPDiv, Butina, Renz-style hit packing, and Score Erosion use specified threshold or factor grids.MaxMin and MaxSum use 30 random starts plus a highest-score start; OptiSim uses 30 seeds with k values 1, 10, and 100.Source: §A.2 · p. 10

A.3. Shape and electrostatic score profiles

• Shape and electrostatic profiles use separately loaded CHEESE encoders to produce normalized 256-dimensional embeddings, with cosine-similarity limits set for each representation.Allowed pairs satisfy s(i, j) ≤ τ + 10−12, and scores, identities, sizes, selected pairs, and artifacts are checked or retained.Source: §A.3 · p. 10

A.4. Integer optimization and full-pool bounds

• The integer formulation represents molecules and forbidden pairs as a conflict graph, so feasible selections are independent sets; conflict-list union and intersection encode different rules.Deduplication must specify treatment of repeated measurements because keeping the highest of noisy repeated scores can inflate the score.Source: §A.4 · p. 11
• When a top-M prefix omits molecules, adding N universally compatible placeholders at the highest omitted score yields a relaxed problem whose optimum upper-bounds the full-pool optimum.Final selections contain only original-pool molecules.Source: §A.4 · p. 11
• The original-score upper bound accounts for integer-score rounding errors, while the lower bound is the best validated allowed size-N selection from optimization or a comparator.If solver status is unknown, the method retains the top-N upper bound and known feasible selections; a result is certified within 10−6 when U − L ≤ 10−6 in original score units.Source: §A.4 · p. 11
• Bounds require conflicts to be represented correctly: missed conflicts or extra restrictions that remove allowed sets need separate justification before they can certify the intended problem.Pair constraints are compressed into clique constraints with every conflict edge covered, preserving allowed sets.Source: §A.4 · p. 11

B.1. Connection to cluster-average enrichment

• Cluster-average enrichment is expressible as an additive weighted utility: actives contribute inverse active-group counts, while inactives contribute zero.The algebraic connection to the established weighting scheme is not a prospective selection rule using unknown outcomes; validation labels are withheld during selection and used afterward.Source: §B.1 · p. 11
• For uniformly sampled size-N subsets, expected cluster-average enrichment is 1, but diversity-constrained random selection need not have that expectation.Inverse frequencies of all molecules in structural groups are not the same correction as inverse active counts.Source: §B.1 · p. 11
• With nonuniform weights, Equation 2 uses a fixed denominator N, so adding a constant to all scores can change the selected set when total selected weight varies.Scores and weights therefore need a declared utility convention.Source: §B.1 · p. 11

B.2. Estimating the random matching budget

• The random matching-budget estimator samples independent permutations and locates the first prefix reaching the target; monotone success permits binary search after full-library reachability is established.Each check tests feasible selection under the score threshold, and the sample mean and empirical quantiles estimate the budget and its distribution.Source: §B.2 · pp. 11–12

B.3. Interpretation and reporting

• Matching-budget comparisons are conditional on the candidate pool, target, and declared acquisition policy; nonuniform sampling, repeated or ineligible proposals, and unequal costs require explicit accounting.For CPDiv with fixed disjoint groups and upper capacities alone, each prefix optimum is obtained exactly by greedy selection, but obtaining scores and group assignments or pairwise relationships adds costs.Source: §B.3 · p. 12, §B.2 · p. 12
• The random-equivalent factor is an acquisition-budget ratio, not a significance test or biological-enrichment estimate, and weighted and random-equivalent extensions need separate empirical evaluation.It has no guaranteed expectation of one when the candidate pool is itself randomly generated.Source: §B.3 · p. 12