k-Visits: Recurrence Across Multiple Domains

• k-Visits is a term used in various fields to denote exact repetition counts, threshold-based selection, or prescribed visitation frequencies.
• In urban mobility and network analysis, k-Visits employs greedy covering and percolation method techniques to extract principal destinations and assess connectivity.
• Applications range from scheduling and combinatorial optimization to stochastic visit-count laws, guiding both practical urban planning and algorithmic design.

Searching arXiv for the cited k-Visits-related papers to ground the article in current records. In recent arXiv literature, the expression k-Visits appears in several technically distinct senses. In urban mobility, it denotes either the $k$ most-frequently visited destinations retained from an origin–destination matrix or the K-Visitation framework that compares habitual destination choice with proximity-based amenity cover. In stochastic-process theory, it denotes the event that a state or site is visited exactly $k$ times. In combinatorial optimization, it appears as the Many Visits TSP or k-Visits TSP, where each city must be visited a prescribed number of times. In scheduling, it names a finite version of Pinwheel Scheduling in which each task appears exactly $k$ times. This suggests that the shared core of the term is not a single unified formalism but a recurring concern with recurrence, truncation, or exact repetition under an integer visitation parameter (Zhang et al., 31 Aug 2025, Kowalik et al., 2020, Kanellopoulos et al., 15 Jul 2025).

1. Principal meanings and formal objects

The main research usages can be organized by the mathematical object to which the visitation count is attached.

Domain	Core object	Representative formulation
Urban mobility	Minimal place cover or top-$k$ destinations	$K_{\text{freq}}$, $K_{\text{dist}}$, $A^{(k)}$
Stochastic processes	Exact visit-count law	$\mathbb{P}(N_1=k\mid N)$, $P_N(X,K\mid Z)$
Combinatorial optimization	Tour with prescribed multiplicities	visit function $k(v)$ or $k$0
Scheduling	Finite schedule with repeated tasks	each task appears exactly $k$1 times
Data structures	Thresholded aggregate visit counting	$k$2-CLAV

Two broad families recur. One family studies exact-count phenomena: the number of times a Markov state, lattice site, city, or task is visited must equal a specified integer. The other studies top-$k$3 or thresholded visitation structure: a mobility backbone is extracted by keeping the $k$4 most-visited destinations, or a query asks whether aggregated visitation time exceeds a threshold.

The term is therefore domain-dependent. In the mobility papers, $k$5 indexes a retained destination set or a minimal cover size; in the TSP and scheduling papers, it is a hard cardinality requirement on repeated appearance; in the probability papers, it is the random variable being counted; and in the data-structural paper, it is a threshold on aggregate time rather than a frequency count.

2. K-Visitation in proximity-centred urban mobility

The K-Visitation framework was introduced for a behaviourally grounded assessment of proximity-centred planning and the 15-Minute City ideal (Zhang et al., 31 Aug 2025). Let $k$6 be the set of all distinct places, represented as H3 level-10 cells visited by a user excluding the home cell; let $k$7 be the set of required amenity categories; let $k$8 be the subset of $k$9 provided by place $k$0; let $k$1 be empirical visitation frequency; and let $k$2 be distance from home. Two orderings are defined: $k$3, sorted in descending order of $k$4, and $k$5, sorted in ascending order of $k$6. The framework then applies a greedy covering algorithm:

$k$7

$k$8

In words, $k$9 is the smallest number of most-frequently-visited places needed to cover every amenity category, whereas $k$0 is the smallest number of nearest places needed to cover every category.

The algorithmic pipeline begins from mobile-phone pings clustered into stay locations with the InfoStop algorithm using dwell time $k$1–$k$2 and radius $k$3. Stays are aggregated to H3 level-10 hexagons of approximately $k$4. Home and work cells are inferred by time-of-day and time-of-week rules and removed from the candidate set $k$5. Each visited cell is annotated with amenity categories within a $k$6 buffer using a standardized POI taxonomy of ten daily categories. The greedy covering procedure is then run twice, including and excluding work, to isolate the structuring role of employment. The framework assumes that all amenity categories are equally essential, and it exploits the submodularity of set cover so that greedy selection provides a logarithmic approximation to the optimum.

Its central theoretical distinction is between behaviour and potential. $k$7 captures realised habitual behaviour, whereas $k$8 captures idealized local potential under proximity alone. The framework also defines an alignment coefficient $k$9 as the Jaccard similarity between the two place-sets,

$K_{\text{freq}}$0

with $K_{\text{freq}}$1 indicating strong concordance and $K_{\text{freq}}$2 indicating that habitual travel systematically escapes the local area. A null model $K_{\text{freq}}$3 randomizes the greedy order under a global distance-frequency decay $K_{\text{freq}}$4 in order to test whether observed ordering exceeds what distance decay alone would predict.

Applied to $K_{\text{freq}}$5 months of anonymised mobility data from Finland, the framework reports systematic misalignment between proximity and behaviour. Median $K_{\text{freq}}$6 across Finland is approximately $K_{\text{freq}}$7–$K_{\text{freq}}$8, well above a distance-decay null of about $K_{\text{freq}}$9, indicating genuine local anchoring. At the same time, central districts such as Helsinki CBD record lower $K_{\text{dist}}$0 than suburbs: residents “over-shoot” despite abundant nearby amenities, whereas peripheral areas show higher $K_{\text{dist}}$1, likely out of necessity rather than choice. Under a 15-minute public-transport benchmark, approximately $K_{\text{dist}}$2 of Helsinki residents could in principle reach all $K_{\text{dist}}$3 places within 15 minutes, but only about $K_{\text{dist}}$4 actually reach their $K_{\text{dist}}$5 places within 15 minutes. An XGBoost classifier with ROC-AUC $K_{\text{dist}}$6 identifies travel time as the strongest predictor of frequent non-proximate visits, followed by POI diversity and experienced segregation at the destination; younger and higher-income home areas also contribute.

Amenity effects are heterogeneous. Everyday essentials such as Groceries, Transport, Services, and Education dominate the $K_{\text{dist}}$7 set and exhibit minimal distance overshoot in $K_{\text{dist}}$8. Specialized or discretionary categories such as Culture, Retail Specialties, and Civic/Religious are over-represented in $K_{\text{dist}}$9 and have large distance differentials relative to the nearest option. The social consequences of localism are likewise spatially contingent: the elasticity of experienced income segregation $A^{(k)}$0 is negative in the historic city centre and positive in affluent suburbs and deprived peripheries. The paper’s planning conclusion is correspondingly narrow and conditional: proximity is necessary but not sufficient, and implementation should be behaviourally informed and place-sensitive.

3. k-most-frequent destination networks and percolation structure

A second urban-mobility meaning of k-visits is the retention of the $A^{(k)}$1 most-frequent destinations from each origin in an origin–destination matrix (Zhang et al., 2024). Let $A^{(k)}$2 be the monthly aggregated flow matrix, with $A^{(k)}$3 equal to the total number of visits from area $A^{(k)}$4 to area $A^{(k)}$5. For each origin $A^{(k)}$6, destinations are sorted in descending order of $A^{(k)}$7, and the k most-visited destinations are $A^{(k)}$8. This defines an unweighted adjacency matrix

$A^{(k)}$9

or equivalently $\mathbb{P}(N_1=k\mid N)$0. A weighted version $\mathbb{P}(N_1=k\mid N)$1 retains only the top-$\mathbb{P}(N_1=k\mid N)$2 links per origin. For percolation analysis the graph is symmetrized by $\mathbb{P}(N_1=k\mid N)$3, producing an undirected network $\mathbb{P}(N_1=k\mid N)$4.

The principal observable is the relative size

$\mathbb{P}(N_1=k\mid N)$5

of the largest connected component. The percolation threshold is

$\mathbb{P}(N_1=k\mid N)$6

with $\mathbb{P}(N_1=k\mid N)$7 used as an empirical tolerance if needed. Across eight major U.S. cities and a 48-month study period, the reported threshold is $\mathbb{P}(N_1=k\mid N)$8, with remarkably little variance across cities, geographies, and time. The interpretation offered is that approximately $\mathbb{P}(N_1=k\mid N)$9 principal destinations per origin suffice to tie the entire city into one connected component; smaller $P_N(X,K\mid Z)$0 leaves at least one Census Block Group isolated, whereas larger $P_N(X,K\mid Z)$1 adds redundant links.

Two additional network diagnostics characterize the transition. The average clustering coefficient $P_N(X,K\mid Z)$2 first decreases as $P_N(X,K\mid Z)$3 grows to approximately $P_N(X,K\mid Z)$4, showing that new nodes join through single-link attachments, and then rises sharply beyond approximately $P_N(X,K\mid Z)$5 as triangle-forming motifs become prevalent. A hubness index $P_N(X,K\mid Z)$6, defined as the mean Kleinberg-HITS hub score of the ten most hub-like nodes, remains roughly flat for $P_N(X,K\mid Z)$7 but spikes at $P_N(X,K\mid Z)$8, indicating the emergence of a small set of highly connected hubs. Degree distributions at $P_N(X,K\mid Z)$9, $k(v)$0, and $k(v)$1 follow a power law over several decades, though at $k(v)$2 a heavier tail of low-degree nodes appears, consistent with the inclusion of more random exploratory links.

The same paper links the percolation backbone to urban inequality through the Proportion of Principal Destinations metric,

$k(v)$3

At $k(v)$4, population-weighted correlations are reported as $k(v)$5 with median household income, $k(v)$6 with percentage population without health insurance, $k(v)$7 with percentage high-school graduates, $k(v)$8 with percentage college enrollment, $k(v)$9 with percentage Master’s degree holders, and $k$00 with percentage residential land use. The authors interpret this as showing that wealthier and more highly educated neighbourhoods tend to have higher PPD, whereas disadvantaged areas exhibit lower PPD.

4. Exact visit-count laws in Markov chains, random walks, and mixing dynamics

In probability theory, k-visits refers to the exact or asymptotic law of how many times a process visits a specified state or set. For a two-state Markov chain with states $k$01 and $k$02, transition probabilities $k$03, and initial distribution $k$04, the problem studied in (Shah, 5 Feb 2025) is to compute

$k$05

where $k$06 is the number of visits to $k$07 in $k$08 successive transitions, counting the initial state if it is $k$09. The derivation conditions on the initial state and uses weak-composition counting of blocks separated by down-crossings $k$10 and up-crossings $k$11. The result is a closed-form piece-wise formula involving two summation terms for each initial state and index limits $k$12, $k$13, and $k$14. The note’s stated contribution is to restore a missing summation term and correct earlier incomplete results; Monte Carlo simulation over $k$15 trials shows full agreement to machine precision.

For the one-dimensional symmetric nearest-neighbour walk on $k$16, the object is the joint law

$k$17

where $k$18 counts visits to a specified site $k$19 by time $k$20 (Percus et al., 2016). The paper derives a bivariate generating function and an exact “one-step-reduced” formula expressing $k$21 in terms of the simple walk distribution $k$22. It also provides the marginal law $k$23, a diffusion-scaling limit with $k$24, $k$25, and $k$26, and explicit expressions for the mean, variance, and higher moments of $k$27. In the special case $k$28, the formulas simplify to differences of shifted binomial terms.

For $k$29-mixing dynamical systems, the same theme appears in a rare-event limit regime (Gallo et al., 2021). If $k$30 is a measurable target set of small measure and $k$31, then the rare-event count is

$k$32

Under right $k$33-mixing with $k$34, the law of $k$35 is approximated in total variation by a compound-Poisson law constructed from local cluster sizes $k$36 via the Stein–Chen method. For shrinking nested cylinders $k$37, the asymptotic limit is compound-Poisson with intensity $k$38 and cluster-size mass function $k$39. Periodic targets yield the Pólya–Aeppli law, while aperiodic points yield the Poisson law. The paper also gives examples for House-of-Cards chains, cylinder neighbourhoods in $k$40-mixing shifts, and temporal synchronisation in $k$41-measures.

These probabilistic usages differ sharply from the urban-mobility ones. Here, $k$42 is a random event count in a stochastic law, not a truncation level or a minimal amenity-cover size.

5. Many-Visits TSP and connected-flow formulations

In combinatorial optimization, Many Visits TSP asks for an optimal tour that visits each city exactly a prescribed number of times (Kowalik et al., 2020). Formally, given a set $k$43 of $k$44 cities, a visit-requirement function $k$45, and an integer distance matrix $k$46, a tour is a cyclic sequence of length $k$47 that visits each $k$48 exactly $k$49 times. Equivalently, one may use a multiplicity function $k$50 satisfying

$k$51

with weakly connected support, and cost

$k$52

Algorithmically, the paper proves three principal results: a randomized polynomial-space exact algorithm with running time $k$53, where $k$54 is the maximum finite distance; a Monte Carlo $k$55-approximation in time $k$56 by scaling and rounding; a deterministic exponential-space dynamic program with running time and space $k$57; and a deterministic polynomial-space exact algorithm with running time $k$58. The exact $k$59 algorithm reduces the problem to a fixed-degree connected subgraph decision problem, then to connected perfect matching in a bipartite clone graph, and finally uses an algebraic Cut-and-Count style detection with determinants of Tutte-type matrices. The paper also states a barrier result: any improvement of the base $k$60 in the $k$61 algorithm would yield an improved algorithm for Directed Hamiltonian Cycle, described there as a fifty-year-old open problem.

A closely related earlier approach gives a deterministic $k$62-time, polynomial-space algorithm (Berger et al., 2018). Its structural idea is to decompose an optimal tour into a directed spanning tree plus a cheapest transportation completion satisfying residual degree demands. The recursion is built on a centroid-based separator lemma for directed spanning trees, yielding a recurrence

$k$63

and a solution $k$64 for the tree subproblem. After enumerating feasible degree profiles, the total time becomes $k$65.

The same optimization family admits a connected-flow generalization and parameterized analysis (Mannens et al., 2021). In that setting, the input is a directed graph $k$66, a demand function $k$67, and a cost function on edges. A feasible solution is an integer flow $k$68 satisfying Euler-balance, demand satisfaction, and connectivity of the support. The paper proves $k$69-completeness already for $k$70 with unit demands and capacities and no edge costs, fixed-parameter tractability without capacities, an $k$71-time algorithm for the general case when parameterized by vertex-cover size $k$72, and a polynomial kernel of size $k$73 for Many Visits TSP.

Within this literature, “k-visits” is a degree or demand prescription on an Eulerian connected multigraph, not a stochastic visit count or a mobility backbone.

6. Finite Pinwheel Scheduling: the k-Visits problem

A different and more literal usage appears in finite scheduling (Kanellopoulos et al., 15 Jul 2025). Given $k$74 tasks with integer deadlines $k$75, the k-Visits problem asks whether there exists a schedule

$k$76

such that each task $k$77 appears exactly $k$78 times and, if its positions are $k$79, then

$k$80

The paper presents this as a finite version of Pinwheel Scheduling, where each task in the infinite problem must appear at least once in every $k$81 time slots.

Its main complexity result is that 2-Visits is strongly NP-complete. The hardness proof starts from Restricted Numerical 3-Dimensional Matching, reduces to an inequality version called INTDM, then to a sum-matching problem SM with constraints $k$82 and $k$83, and finally to 2-Visits. An important intermediate construct is the discretized sequence

$k$84

which supports a normalization theorem: if a 2-Visits instance is feasible, then it admits a schedule in which each primary visit of task $k$85 occurs at position $k$86, while secondary visits occupy the gap positions in order of non-decreasing induced deadline $k$87.

The same paper identifies unusually sharp tractable cases. It observes that the 1-Visit problem is trivial, and it proves that 2-Visits can be solved in linear time if all deadlines are distinct. In that case $k$88, the sum-matching instance collapses to a unique matching, and feasibility reduces to checking $k$89 for each $k$90. The paper also gives an FPT algorithm for 2-Visits parameterized by the maximum cluster size

$k$91

where $k$92 is the collection of clusters of the discretized sequence, with running time $k$93.

The hardness extends beyond the basic model. For every fixed $k$94, Variable $k$95-Visits remains strongly NP-complete even when the first two deadlines coincide for every task and all later deadlines are $k$96. A related Threshold Pinwheel Scheduling generalization is likewise strongly NP-hard for explicit input. This section of the literature is notable for a dichotomy explicitly emphasized in the paper: the problem is in P if the deadlines are a set, but NP-complete if the deadlines are a multiset.

7. Thresholded aggregated visitation and related query models

A neighboring but distinct research direction studies efficient counting of long aggregated visits in large mobility datasets (Afshani et al., 14 Jan 2026). In the Counting Long Aggregated Visits problem, one is given users $k$97, regions $k$98, triplets $k$99 recording how long user $k$00 spent in region $k$01, a threshold $k$02, and a query subset $k$03 of size $k$04. The required answer is

$k$05

This is denoted $k$06-CLAV. The geometric variant places $k$07 in $k$08 and takes $k$09 to be an axis-aligned hyperrectangle.

For exact queries, the paper gives a space–time trade-off using a large/small decomposition of regions by a parameter $k$10. Regions with support larger than $k$11 are “large”; the rest are “small”. The resulting data structure uses

$k$12

words, can be built in $k$13 time, and answers queries in

$k$14

time. Under the Strong $k$15-SetDisjointness conjecture, any exact $k$16-CLAV structure with space $k$17 and query time $k$18 must satisfy

$k$19

Approximate schemes are also developed. A sampling estimator based on

$k$20

is unbiased and, with $k$21 samples, achieves additive error at most $k$22 with probability at least $k$23, using $k$24 space and query time $k$25. A sketch-based scheme built from an FM $k$26 Count-Min sketch yields, with high probability,

$k$27

using $k$28 words and $k$29 query time.

The geometric setting exhibits both exact upper bounds and lower bounds. Any Geometric-CLAV structure, regardless of query time, requires at least

$k$30

bits of space. In one dimension, minimal intervals can be reduced to 2D colored dominance counting, giving space $k$31 and query time $k$32. In $k$33, tabulation across boundary choices yields

$k$34

space with the same $k$35 query time.

These models are adjacent to, but not identical with, the other k-Visits literatures. They do not ask for exact frequency $k$36, nor for top-$k$37 destinations, nor for a schedule or tour with exactly $k$38 repetitions. Instead they threshold aggregated time spent over a query region family. The coexistence of these meanings is a useful caution: in current research, “k-Visits” is a shared lexical label for several formally different notions of repeated visitation.