Min-k: Parameterized Concepts in Research

• Min-k is a parameterized notation whose value of k governs distinct constraints in applications ranging from beyond-planar graph drawing to language-model decoding and privacy auditing.
• In graph drawing, Min-k ensures that for each crossing pair one edge has bounded crossings, while in clustering and cut problems it specifies the number of partitions for balanced solutions.
• Across domains, Min-k regulates optimization and inference by exposing varying degrees of structure and uncertainty, guiding both algorithm design and performance guarantees.

Min-k is a parameterized label that appears in several technically unrelated research literatures. In the works considered here, it denotes families of constructions indexed by an integer $k$: a local crossing constraint in beyond-planar graph drawing, a number of parts or clusters in partitioning and cut problems, a dynamic truncation boundary in large-language-model decoding, a bottom-$k\%$ scoring rule in membership inference and pre-training data detection, a weighted similarity kernel for nonnegative vectors, and a fixed-$k$ query or encoding parameter in succinct data structures (Binucci et al., 2023, Lokshtanov et al., 2020, Ding et al., 13 Apr 2026, Zhang et al., 2024, Li, 2015).

1. Parameterized semantics of the term

The same string “Min-k” therefore has no single field-independent definition; its meaning is supplied by the surrounding formalism.

Domain	Object called “Min-k”	Meaning of $k$
Beyond-planar graph drawing	min-$k$-planarity	Crossing budget on one edge of every crossing pair
Graph partitioning and clustering	Min $k$-Cut, Minmax $k$-cut, Min-$k$-Partition	Number of parts or clusters
LLM decoding	Min-$k$ Sampling	Dynamically chosen candidate-set size
Privacy auditing of LMs	min-k, Min-K%, Min-K%++	Lowest $k\%$ tokens or windows used for scoring
Kernel methods	“Min-k” shorthand for min-max kernel	Not a count parameter; a shorthand name
Succinct range queries	range top-$k\%$0, range selection, range min-max	Query parameter or encoded answer size

In graph drawing, the parameter controls a local condition: for every crossing pair, at least one edge must have crossing count at most $k\%$1. In graph partitioning and clustering, $k\%$2 is the number of blocks, parts, or clusters. In LLM inference, Min-$k\%$3 can be a dynamic truncation rule defined from sorted logits, while in membership inference it refers to averaging the least likely $k\%$4 of tokens or windows. In similarity learning, “Min-k” is a shorthand for the min-max kernel rather than a literal optimization over $k\%$5 (Binucci et al., 2023, Qiu et al., 12 Jun 2025, Ding et al., 13 Apr 2026, Arkhmammadova et al., 2 Aug 2025, Li, 2015). This suggests that “Min-k” is best understood as a notational family rather than a single concept.

2. Min-$k\%$6-planarity in beyond-planar graph drawing

In graph drawing, a drawing $k\%$7 is min-$k\%$8-planar if for every pair of crossing edges $k\%$9,

$k$0

Equivalently, whenever two edges cross, at least one of them is “light,” meaning it has at most $k$1 crossings. The framework is defined for simple drawings, where adjacent edges do not cross, any two independent edges cross at most once, and no three edges meet at one crossing point. Every $k$2-planar drawing is min-$k$3-planar, but not conversely; a min-1-planar drawing may contain an edge with an unbounded number of crossings, provided each of its crossings is paired with an edge of crossing number at most $k$4 (Binucci et al., 2023).

This relaxation still permits strong extremal results. For min-$k$5-planar simple graphs with $k$6,

$k$7

The bounds sharpen for small $k$8: $k$9 for $k$0, $k$1 for $k$2, and $k$3 for $k$4, with the $k$5 bounds tight. The class relations are also nontrivial: $k$6, $k$7, and for every $k$8, min-$k$9-planar graphs are contained in both $k$0-gap-planar and $k$1-quasi-planar graphs, while for fan-planarity the classes are incomparable for any $k$2 (Binucci et al., 2023).

Recognition is already hard at $k$3. Min-1-Planarity is NP-complete, with NP-hardness proved by reduction from 3-Partition. A central gadget is an uncrossable-edge construction built from three copies of $k$4, spoke paths, and ten length-2 $k$5-$k$6 paths; in every simple min-1-planar drawing it forces a $k$7-$k$8 connection that does not cross any edge outside the gadget. The proof also exploits the fact that for min-1-planarity, recognition is equivalent to asking for a simple min-1-planar drawing, a property that fails in general for fixed $k$9. The complexity of testing min-$k$0-planarity for fixed $k$1 is left open (Okada, 14 May 2026).

3. Partitioning, balanced clustering, and Min-$k$2-Partition

In clustering, one use of Min-$k$3 concerns explicitly balanced partitions. “Balanced k-Means and Min-Cut Clustering” augments both k-means and graph min-cut with an exclusive-lasso regularizer that penalizes concentration of assignments into a few clusters. For hard assignment matrices $k$4, the term

$k$5

reduces to the sum of squared cluster sizes, $k$6, which is minimized when all cluster sizes are equal. The balanced k-means objective is

$k$7

with centroid update

$k$8

followed by row-wise updates of $k$9. The balanced min-cut objective similarly becomes

$k$0

Both alternating procedures monotonically improve their objectives and converge, and on nine benchmark datasets the balanced variants outperform the corresponding classical baselines on clustering accuracy and NMI when the data are naturally balanced (Chang et al., 2014).

A different Min-$k$1-Partition formulation appears in PCI assignment for 5G networks. The general problem is

$k$2

which seeks a $k$3-way partition minimizing weighted within-cluster interference. For PCI assignment, the problem is decomposed through the Chinese Remainder Theorem into Min-3-Partition and Min-10-Partition, corresponding to mod-$k$4 and mod-$k$5 interference components. The paper then gives a relaxation-free continuous reformulation: a label is represented by a one-hot vector $k$6, and the one-hot constraint is characterized exactly through the norm-equality condition $k$7 for $k$8. A penalized formulation is exact once $k$9, and it is optimized by penalized mirror descent with KL geometry. On real-world datasets, the method reports $k$0 collisions and $k$1 confusions, about $k$2 reduction in mod-$k$3 interference, about $k$4 reduction in mod-$k$5 interference, and roughly a $k$6 runtime speedup relative to state-of-the-art methods (Qiu et al., 12 Jun 2025).

Min-$k$7 clustering also appears in metric optimization with a bounded number of clusters. For Min-Sum-Radii (MSR), the objective is to cover a metric space with at most $k$8 balls minimizing $k$9; for Min-Sum-Diameters (MSD), the objective is to partition into at most $k$0 clusters minimizing $k$1. The paper gives an exact MSD algorithm with runtime $k$2, improving the previous $k$3 bound, and deterministic $k$4-approximation algorithms for MSR, MSD, and $k$5-center in doubling metrics with runtime

$k$6

The framework extends to $k$7-powered objectives, outliers, fairness constraints, and mergeable clustering, with ETH-based lower bounds showing that several of the exact upper bounds are essentially tight (Banerjee et al., 29 Jan 2025).

4. Graph cuts, min-max objectives, and robust optimization

In combinatorial optimization, Min $k$8-Cut denotes the problem of partitioning the vertex set of an edge-weighted graph into exactly $k$9 nonempty parts so as to minimize the total weight of edges crossing between parts. A parameterized approximation scheme gives, for every $k\%$0, a randomized $k\%$1-approximation in time

$k\%$2

Its main ingredients are a simple sparsification procedure, a polynomial-time decomposition into highly connected parts, and an exact algorithm for unweighted multigraphs with runtime $k\%$3, where $k\%$4 is the optimum cut value. The method also proves a decomposition theorem in which every bag is $k\%$5-edge-unbreakable (Lokshtanov et al., 2020).

A related but different problem is Minmax $k\%$6-cut, where the objective is

$k\%$7

For $k\%$8, the min-sum and min-max objectives coincide, but for $k\%$9 they diverge sharply, and Minmax $k\%$00-cut can have optimal solutions with disconnected parts. The problem is strongly NP-hard and W[1]-hard when parameterized by $k\%$01. The paper gives an exact algorithm for the unweighted case running in time

$k\%$02

where $k\%$03 is the optimum value, and a parameterized approximation scheme with runtime

$k\%$04

The dynamic programming framework also extends to minimizing $k\%$05-norm measures of $k\%$06-partitioning for every $k\%$07 (Chandrasekaran et al., 2020).

In robust optimization, “Min-$k\%$08” arises through the min-max-min problem

$k\%$09

where $k\%$10 feasible solutions are prepared in advance and the best one is selected after the cost scenario is revealed. The value is monotone in $k\%$11,

$k\%$12

and for $k\%$13 the problem is equivalent to robust optimization over $k\%$14. The paper develops approximation algorithms for the intermediate regime $k\%$15: one solves the $k\%$16 case, recovers a convex combination, and returns the $k\%$17 solutions with largest weights. The guarantees are additive and multiplicative, governed by the quantities $k\%$18 and $k\%$19, and become especially favorable when $k\%$20 or $k\%$21. The same approximation logic extends to robust $k\%$22-adaptability under objective uncertainty (Kurtz, 2021).

5. Language-model decoding and inference attacks

In LLM decoding, Min-$k\%$23 Sampling is a logit-space truncation method that chooses the candidate set size $k\%$24 dynamically from the shape of the sorted logits. Let the logits be sorted as $k\%$25, with range $k\%$26. The weighted relative decay at position $k\%$27 is

$k\%$28

and the truncation boundary is the maximizer $k\%$29, interpreted as the “semantic cliff” separating a high-confidence head from a noisy tail. A fallback

$k\%$30

is used for nearly flat distributions, and the final candidate size is $k\%$31. Because the method depends on relative logit geometry, the selected candidate set is strictly temperature-invariant. Empirically, it remains stable at temperatures up to $k\%$32, reaches a $k\%$33 win rate at $k\%$34 in one creative-writing setting, and shows low hyperparameter sensitivity, with GSM8K performance staying in a band of roughly $k\%$35–$k\%$36 over a $k\%$37-temperature grid (Ding et al., 13 Apr 2026).

A separate use of min-k appears in membership inference attacks on LLMs. Given a sample $k\%$38, the min-k score is the average log probability of the $k\%$39 least likely tokens: $k\%$40 More negative values indicate stronger evidence of non-membership. The method is motivated by the claim that non-members are more likely to contain a few atypical tokens with very low probability, whereas members are less likely to include such extreme outliers. On small LLMs, however, token-level log probabilities become noisier and high-variance, which weakens min-k. Win-k generalizes the method from token-level scoring to window-level scoring: sliding windows $k\%$41 are scored by their average log probability, and the bottom $k\%$42 of window scores are aggregated. On three datasets and eight SLMs, win-k has the best AUROC in $k\%$43 of $k\%$44 cases and the best TPR @ $k\%$45 FPR in $k\%$46 of $k\%$47 cases; the paper also reports variance reductions from $k\%$48 to $k\%$49 on one member example and from $k\%$50 to $k\%$51 on one non-member example (Arkhmammadova et al., 2 Aug 2025).

Pre-training data detection uses a closely related notation. The original Min-K% baseline averages the lowest $k\%$52 token log probabilities. Min-K%++ keeps the same bottom-$k\%$53 aggregation rule but replaces the raw token score with a standardized score: the observed token’s log probability is centered by the mean log probability over the full vocabulary and scaled by the corresponding standard deviation. The theoretical motivation is that maximum-likelihood training tends to make training samples local maxima of the modeled distribution, leading to a discrete analogue of curvature-based detection. On WikiMIA, Min-K%++ improves AUROC over the runner-up by $k\%$54, $k\%$55, and $k\%$56 for input lengths $k\%$57, $k\%$58, and $k\%$59, respectively; on MIMIR it is the best reference-free method and improves over prior Min-K by up to $k\%$60 at $k\%$61B parameters (Zhang et al., 2024).

6. Kernels, succinct encodings, and complete-instance CSPs

In machine learning on nonnegative data, “Min-k” is the shorthand used in “Min-Max Kernels” for the min-max kernel

$k\%$62

This is a weighted generalization of resemblance or Jaccard similarity. Consistent weighted sampling (CWS) produces samples $k\%$63 with collision probability exactly equal to $k\%$64. The paper’s main practical contribution is the “0-bit” CWS scheme, which discards the theoretically unbounded $k\%$65 and keeps only $k\%$66. Empirically, this does not lose essential information, allowing the nonlinear kernel to be approximated with hashed features and then used with linear SVM or logistic regression at scale (Li, 2015).

In succinct data structures, min and max reappear in range-query encodings. Range min-max asks for both the minimum and maximum indices in a query interval and admits an encoding using $k\%$67 bits, with lower bound $k\%$68. For range top-$k\%$69 and range selection, the exact asymptotic space for $k\%$70 is

$k\%$71

where $k\%$72 is the binary entropy function. The $k\%$73 specialization recovers the classical $k\%$74 bound for range maximum/minimum encodings, while range min-max corresponds to a larger combinatorial class linked to Baxter permutations (Gawrychowski et al., 2014).

In constraint satisfaction, Min-$k\%$75-CSP on complete instances means that for fixed arity $k\%$76, there is one nontrivial constraint on every $k\%$77-subset of variables, and the objective is to minimize the number of unsatisfied constraints. The paper gives an $k\%$78-approximation for Min-2-SAT on complete graphs and an $k\%$79-time algorithm deciding satisfiability for every Boolean complete $k\%$80-CSP. A key structural fact is that the number of satisfying assignments is only $k\%$81, derived via VC-dimension and Sauer–Shelah. For larger alphabets, the quasi-polynomial-time tractable pairs are exactly $k\%$82 for every $k\%$83 and $k\%$84 (Anand et al., 2024).

Across these literatures, Min-k repeatedly marks a controlled relaxation indexed by $k\%$85: a local crossing witness instead of a global per-edge bound, a bounded number of parts instead of unrestricted decomposition, a bottom-$k\%$86 score instead of full-sequence likelihood, or a dynamic candidate frontier rather than fixed truncation. This suggests a recurring methodological role for the notation: $k\%$87 specifies how much combinatorial structure, uncertainty, or tail behavior is exposed to optimization or inference, while “Min” identifies the objective being regularized, compressed, or robustified.