SplitMe: Diverse Splitting Techniques

• SplitMe is a polysemous term referring to various splitting methods across domains, characterizing problems from route optimization to statistical modeling.
• In vehicle routing, SplitMe specifically denotes an exact, linear-time Split decoder that partitions a giant tour into optimal depot-delimited routes under capacity constraints.
• Beyond routing, SplitMe covers diverse applications including fair multiterminal source coding, minimal perfect hashing via RecSplit, and data partitioning in statistical and machine learning contexts.

Searching arXiv for papers relevant to the term “SplitMe” and its established usages across domains. “SplitMe” is not the canonical title of a single research method. In the literature represented here, it functions as an informal or context-dependent label for several distinct “split” constructions: most prominently the Split decoding algorithm for giant-tour vehicle routing heuristics (Vidal, 2015), but also the SPLIT algorithm for fair multiterminal source coding (Ding et al., 2018), the RecSplit method for minimal perfect hashing (Esposito et al., 2019), statistical split models for context-specific independence (Hojsgaard, 2013), geometry-based data partitioning methods derived from SPlit (Vakayil et al., 2021), and other domain-specific notions of splitting in optimization, combinatorics, topology, amplitude theory, and NLP (Tsukagoshi et al., 2024, Florez-Revuelta, 2021, Guan et al., 2024, Cachazo et al., 2021, Vandelli et al., 21 Mar 2025, Axenovich et al., 2020, Chun et al., 2017, Albert et al., 2016, Cha et al., 2013). The common theme is decomposition by partitioning—of tours, rates, keys, contexts, datasets, amplitudes, or combinatorial objects—but the mathematical role of “splitting” differs substantially by field. This suggests that “SplitMe” is best treated as a family resemblance term rather than a uniquely defined method.

1. SplitMe in vehicle routing: giant-tour decoding by optimal route partitioning

In vehicle routing, “SplitMe” most directly denotes the Split decoding algorithm that converts a giant tour—a permutation of customers with no depot occurrences—into an optimal sequence of depot-delimited routes for the capacitated vehicle routing problem (CVRP) (Vidal, 2015). In this setting, the giant tour is reindexed as $(1,\dots,n)$, and a Split solution partitions this sequence into consecutive segments $(i+1,\dots,j)$, each interpreted as a route $(0,i+1,\dots,j,0)$. With cumulative travel and demand arrays

$$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$

the route cost is

$c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$

and feasibility in the hard-capacity case is

$Q[j]-Q[i]\le Q.$

The classical formulation builds a directed acyclic graph $\mathcal G=(\mathcal V,\mathcal A)$ with nodes $\{0,\dots,n\}$, where arc $(i,j)$ exists iff the block $i+1,\dots,j$ is feasible; Split is then a shortest-path problem from $(i+1,\dots,j)$0 to $(i+1,\dots,j)$1 (Vidal, 2015).

The contribution of Vidal’s “Technical Note: Split Algorithm in O(n) for the Capacitated Vehicle Routing Problem” is to replace the standard $(i+1,\dots,j)$2 Bellman propagation by a linear-time algorithm based on a stronger structural property than the usual Monge relation (Vidal, 2015). For predecessor $(i+1,\dots,j)$3, the transformed label

$(i+1,\dots,j)$4

encodes, respectively, a fixed cost offset and cumulative demand. The key observation is that, for feasible extensions, $(i+1,\dots,j)$5 is constant in $(i+1,\dots,j)$6, so a predecessor that is no better in either dimension can be discarded permanently. This yields a deque-based algorithm maintaining mutually nondominated predecessors ordered by increasing $(i+1,\dots,j)$7 and nondecreasing $(i+1,\dots,j)$8, with amortized $(i+1,\dots,j)$9 updates and total $(0,i+1,\dots,j,0)$0 runtime (Vidal, 2015).

The same paper extends the method to a limited fleet and to soft capacity constraints. With $(0,i+1,\dots,j,0)$1 vehicles, the dynamic program is replicated by vehicle count, producing complexity $(0,i+1,\dots,j,0)$2. With linear overload penalty coefficient $(0,i+1,\dots,j,0)$3, the route cost becomes

$(0,i+1,\dots,j,0)$4

and the transformed predecessor label becomes piecewise linear in cumulative demand, yet the algorithm remains linear because every node still enters and leaves the deque only a constant number of times (Vidal, 2015). Computationally, the paper reports speedups ranging from about $(0,i+1,\dots,j,0)$5 to $(0,i+1,\dots,j,0)$6 for hard capacities, from $(0,i+1,\dots,j,0)$7 to $(0,i+1,\dots,j,0)$8 with a fleet limit, and up to $(0,i+1,\dots,j,0)$9 for soft capacities against unrestricted Bellman propagation (Vidal, 2015).

A later extension generalizes the linear Split paradigm from CVRP to the vehicle routing problem with simultaneous pickup and delivery and time windows (VRPSPDTW) (Gibbons et al., 24 Jan 2026). There, hard feasibility depends on both peak load and service timing. For simultaneous pickup and delivery, the route-load profile is tracked through

$$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$0

and for time windows the service-start recursion is reduced to constant-time evaluation via cumulative service-travel arrays and a last-wait index $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$1 (Gibbons et al., 24 Jan 2026). The paper proves a $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$2 Split for hard VRPSPDTW, a $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$3 soft Split for VRPSPD with peak-load penalty, and a $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$4 soft Split for VRPTW with capacity penalty and time warp penalty, under triangle inequality assumptions for travel times in the hard time-window case (Gibbons et al., 24 Jan 2026). In this branch of the literature, “SplitMe” therefore denotes an exact decoder embedded inside route-first, cluster-second heuristics, hybrid genetic search, large neighborhood search, and other giant-tour-based metaheuristics (Vidal, 2015, Gibbons et al., 24 Jan 2026).

2. SplitMe in information theory: fairness by recursive splitting of the Slepian–Wolf region

In multiterminal lossless data compression, “SplitMe” can reasonably refer to the SPLIT algorithm introduced for computing the weighted egalitarian solution in the Slepian–Wolf region (Ding et al., 2018). The setting is a set of terminals $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$5 observing correlated sources $$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$6, with feasible rate vectors characterized by the Slepian–Wolf constraints

$$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$7

and the sum-rate equality

$$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$8

Equivalently, the feasible set is the base polyhedron

$$D[i]=\sum_{k=1}^{i-1} d_{k,k+1}, \qquad Q[i]=\sum_{k=1}^{i} q_k,$$9

where

$c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$0

Because entropy is submodular, the problem is geometrically a minimum-norm point problem over a submodular base polyhedron (Ding et al., 2018).

The paper formulates fairness as the weighted quadratic program

$c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$1

with strictly positive weights $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$2. The minimizer is the weighted egalitarian solution, which also coincides with weighted min-max and max-min fairness (Ding et al., 2018). The algorithmic primitive is parametric submodular minimization: $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$3 If $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$4, the solution on $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$5 is simply proportional,

$c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$6

Otherwise, $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$7 is split into $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$8 and $c(i,j)=d_{0,i+1}+D[j]-D[i+1]+d_{j0},$9, the first part is solved recursively under the original submodular function, and the second under a residual submodular function

$Q[j]-Q[i]\le Q.$0

The full solution is then assembled as

$Q[j]-Q[i]\le Q.$1

This recursive decomposition mirrors the chain of maximal minimizers in parametric submodular minimization and provides a strongly polynomial algorithm with complexity

$Q[j]-Q[i]\le Q.$2

where $Q[j]-Q[i]\le Q.$3 denotes submodular function minimization (Ding et al., 2018).

The conceptual role of “splitting” here is not heuristic partitioning but exact decomposition of a fairness optimization problem along the lexicographically optimal chain of a base polyhedron (Ding et al., 2018). The paper also stresses that once $Q[j]-Q[i]\le Q.$4 is identified, the two recursive branches are independent and can be computed in parallel or in distributed fashion, which is relevant for wireless sensor networks and other distributed systems (Ding et al., 2018). A common misconception would be to equate this SPLIT with divide-and-conquer in the generic algorithmic sense; the paper explicitly emphasizes that the partition is determined by the maximal minimizer of a precise parametric submodular minimization problem, not by an arbitrary heuristic rule (Ding et al., 2018).

3. SplitMe in data structures and optimization: recursive splitting, partitioned quadratic programming, and exact decomposition

A different use of the split motif appears in RecSplit, a method for constructing minimal perfect hash functions by recursive splitting (Esposito et al., 2019). Here the object being split is not a route or feasible region but a bucket of keys $Q[j]-Q[i]\le Q.$5. A successful split is defined by an enumerated family of random functions $Q[j]-Q[i]\le Q.$6 and prescribed part sizes $Q[j]-Q[i]\le Q.$7: a seed $Q[j]-Q[i]\le Q.$8 is successful when exactly $Q[j]-Q[i]\le Q.$9 keys fall into the $\mathcal G=(\mathcal V,\mathcal A)$0-th range. The bucket is recursively partitioned until each leaf has size at most $\mathcal G=(\mathcal V,\mathcal A)$1, at which point the method brute-forces the first seed producing a bijection on that leaf (Esposito et al., 2019). Only the successful seed indices are stored, with optimal Golomb–Rice coding, which lets the representation approach the information-theoretic lower bound of about $\mathcal G=(\mathcal V,\mathcal A)$2 bits per key. The paper reports practical structures as small as $\mathcal G=(\mathcal V,\mathcal A)$3 bits per key, within $\mathcal G=(\mathcal V,\mathcal A)$4 of the lower bound, with expected linear construction time and expected constant lookup time (Esposito et al., 2019).

In large-scale quadratic programming, the 2025 paper “Parallel splitting method for large-scale quadratic programs” introduces SPLIT, expanded as Subproblem ParalleL Iterative Technique (Vandelli et al., 21 Mar 2025). The optimization problem is written as

$\mathcal G=(\mathcal V,\mathcal A)$5

over a graph $\mathcal G=(\mathcal V,\mathcal A)$6. The variables are partitioned into subgraphs $\mathcal G=(\mathcal V,\mathcal A)$7, and cross-partition couplings are summarized by local fields

$\mathcal G=(\mathcal V,\mathcal A)$8

Each subproblem then minimizes

$\mathcal G=(\mathcal V,\mathcal A)$9

The algorithm alternates between recomputing these fields from the current global iterate, solving subproblems in parallel, concatenating the solutions, and applying a sweep update; the paper describes the framework as a solver-agnostic, quantum-inspired decomposition method that retains cross-interaction information typically neglected by simpler partitioning schemes (Vandelli et al., 21 Mar 2025). Empirically, it is evaluated on MaxCut and antenna placement instances up to 20,000 variables (Vandelli et al., 21 Mar 2025).

These examples show that “SplitMe” can denote very different decomposition logics. In RecSplit, the split is recursive search-space partitioning guided by random hash seeds (Esposito et al., 2019). In SPLIT for quadratic programs, it is parallel graph partitioning with iterative field exchange (Vandelli et al., 21 Mar 2025). In the VRP Split literature, it is exact shortest-path decoding of a fixed order (Vidal, 2015, Gibbons et al., 24 Jan 2026). The shared terminology masks substantial differences in objective, correctness guarantees, and mathematical structure.

4. SplitMe in statistical modeling: context-specific independence, multivariate count splitting, and dataset partitioning

In statistics, the term points to at least two distinct traditions. The first is the theory of split models for contingency tables and context-specific independence (Hojsgaard, 2013). A context-specific independence is written

$\{0,\dots,n\}$0

when

$\{0,\dots,n\}$1

CSI models are specified by context-indexed generators $\{0,\dots,n\}$2 in a generating class $\{0,\dots,n\}$3, with model function

$\{0,\dots,n\}$4

A split model is a graphical subclass of CSI models in which a graph is recursively refined into context graphs by splitting on variable values. The package YGGDRASIL implements estimation, testing, model search, and graphical instantiation for such models (Hojsgaard, 2013). Here the split is a representation of heterogeneous interaction structure across contexts, not an optimization primitive.

The second tradition is the paper “Splitting models for multivariate count data,” where a multivariate count vector $\{0,\dots,n\}$5 is generated by first drawing a total $\{0,\dots,n\}$6 from a univariate count law and then splitting that total according to a singular allocation distribution on the simplex $\{0,\dots,n\}$7 (Fernique et al., 2018). Formally,

$\{0,\dots,n\}$8

and the resulting law is denoted

$\{0,\dots,n\}$9

This framework encompasses the non-singular multinomial, negative multinomial, multivariate logarithmic series, non-singular Dirichlet multinomial, and multivariate generalized Waring constructions, among others (Fernique et al., 2018). In this line of work, “splitting” refers to allocation of a random total across categories, and the compound structure simplifies moment calculations, marginalization, conditioning, and regression extensions (Fernique et al., 2018).

A third statistical/data-analytic use appears in SPlit and Twinning for dataset partitioning (Vakayil et al., 2021). Twinning seeks two disjoint subsets $(i,j)$0 and $(i,j)$1 of a dataset $(i,j)$2 that are as statistically similar as possible by minimizing the empirical energy distance

$(i,j)$3

The paper proves that minimizing the distance between the two twin subsets is equivalent, up to a constant factor, to minimizing the distance between the smaller subset and the full dataset, which is the objective of SPlit (Vakayil et al., 2021). Twinning then provides a greedy, nearly linearithmic alternative using kd-tree nearest-neighbor search, with complexity

$(i,j)$4

compared with SPlit’s DC-NN implementation whose average complexity contains a quadratic term in $(i,j)$5 (Vakayil et al., 2021). This suggests a different “SplitMe” meaning: partition my data into statistically matched subsets, rather than optimize routes or infer contexts.

5. SplitMe in machine learning, NLP, and data splitting practice

In machine learning evaluation and preprocessing, “split” methods often focus on preserving distributional structure under partition. EvoSplit addresses this for multi-label datasets, where a split must preserve both single-label and label-pair distributions while respecting fold sizes (Florez-Revuelta, 2021). A split is encoded as an assignment vector over $(i,j)$6 subsets, with exact subset-size constraints treated as hard constraints. The key metrics are Label Distribution

$(i,j)$7

Label Pair Distribution

$(i,j)$8

and Examples Distribution

$(i,j)$9

The paper proposes both single-objective evolutionary optimization and a multi-objective NSGA-II formulation, reporting exact fold-size preservation $i+1,\dots,j$0 by design and improvements over iterative stratification on several measures (Florez-Revuelta, 2021). In this setting, “SplitMe” implies optimize my train/test or fold partition under multi-label constraints.

In NLP, WikiSplit++ concerns Split and Rephrase, where a complex sentence $i+1,\dots,j$1 is transformed into simpler sentences $i+1,\dots,j$2 with preserved meaning (Tsukagoshi et al., 2024). The paper does not introduce a method named SplitMe, but it is directly relevant to any sentence-splitting system. Its two data-refinement steps are: (i) removing WikiSplit instances where the complex sentence does not entail at least one of the simple sentences, using an NLI classifier such as DeBERTa-v2 XXL fine-tuned on MNLI; and (ii) reversing the order of the reference simple sentences during training to discourage copying and under-splitting (Tsukagoshi et al., 2024). The refined dataset shrinks from 994,481 to 630,433 instances, a reduction of about $i+1,\dots,j$3, yet improves entailment ratio and the number of splits in the reported experiments (Tsukagoshi et al., 2024). Here “split” means segment a sentence into multiple faithful simpler sentences, and the paper’s main message is that data refinement can materially improve faithfulness and splitting behavior without architectural change (Tsukagoshi et al., 2024).

These usages are sometimes conflated with generic dataset splitting. The supplied literature suggests a more nuanced taxonomy. Twinning and EvoSplit optimize representativeness across partitions (Vakayil et al., 2021, Florez-Revuelta, 2021). WikiSplit++ improves linguistic decomposition of sentences (Tsukagoshi et al., 2024). Split models in YGGDRASIL encode context-specific dependence patterns (Hojsgaard, 2013). Treating all of them as instances of one algorithmic family would therefore be misleading.

6. SplitMe in mathematics and physics: structural splitting, factorization, and splitter theorems

Several supplied papers use “split” in a mathematically structural sense rather than as an algorithmic procedure. In knot theory, the splitting number $i+1,\dots,j$4 of a link $i+1,\dots,j$5 is the minimum number of crossing changes between distinct components required to convert $i+1,\dots,j$6 into a split link (Cha et al., 2013). The paper establishes lower bounds such as

$i+1,\dots,j$7

and

$i+1,\dots,j$8

and introduces covering-link and Alexander-polynomial techniques to compute splitting numbers, including a complete determination for links with 9 or fewer crossings (Cha et al., 2013). The split is topological separation of components, not decomposition of a computation.

In QCD, splitting amplitudes are the universal process-independent objects governing the collinear limit of scattering amplitudes (Guan et al., 2024). For two collinear partons $i+1,\dots,j$9, the amplitude factorizes as

$(i+1,\dots,j)$00

The paper computes the universal two-parton QCD splitting amplitudes through three loops for $(i+1,\dots,j)$01, $(i+1,\dots,j)$02, and $(i+1,\dots,j)$03, distinguishing fully universal time-like splitting from the more subtle space-like case where strict collinear factorization is violated in general (Guan et al., 2024). In this domain, “split” means collinear factorization of amplitudes, not partitioning.

The amplitude paper “Smoothly Splitting Amplitudes and Semi-Locality” introduces yet another notion: a smooth 3-split, where an amplitude remains finite on a special kinematic locus yet becomes a product of exactly three amputated Berends–Giele currents (Cachazo et al., 2021). For the planar biadjoint amplitude,

$(i+1,\dots,j)$04

The authors call this semi-locality because any two currents share one external particle (Cachazo et al., 2021). This is explicitly contrasted with ordinary pole factorization and is presented as a new tree-level phenomenon.

In combinatorics, splittability of permutation classes and the related splitter theorems of multimatroid theory are again structurally different. The paper on separable permutations defines a permutation class $(i+1,\dots,j)$05 to be splittable if

$(i+1,\dots,j)$06

for proper subclasses $(i+1,\dots,j)$07, where $(i+1,\dots,j)$08 denotes merge (Albert et al., 2016). It characterizes the unsplittable proper subclasses of the separable permutations as exactly the representable classes (Albert et al., 2016). In multimatroid theory, the splitter theorem for connected tight multimatroids says that, for a connected tight multimatroid $(i+1,\dots,j)$09 and connected minor $(i+1,\dots,j)$10, either $(i+1,\dots,j)$11 is connected for $(i+1,\dots,j)$12, or every other element in the same skew class gives a connected one-element minor still containing $(i+1,\dots,j)$13 as a minor (Chun et al., 2017). These are “splitter theorems” in the Tutte–Brylawski–Seymour sense, not data or route partitioning.

A final combinatorial usage appears in “Splits with forbidden subgraphs,” where $(i+1,\dots,j)$14 is the minimum $(i+1,\dots,j)$15 such that there exists an $(i+1,\dots,j)$16-graph—$(i+1,\dots,j)$17 blobs of size at most $(i+1,\dots,j)$18, with at least one edge between every pair of blobs—that is $(i+1,\dots,j)$19-free (Axenovich et al., 2020). The paper proves $(i+1,\dots,j)$20 for non-bipartite $(i+1,\dots,j)$21, and for bipartite $(i+1,\dots,j)$22 with $(i+1,\dots,j)$23,

$(i+1,\dots,j)$24

as stated in the abstract, with the sharper result

$(i+1,\dots,j)$25

for fixed $(i+1,\dots,j)$26 (Axenovich et al., 2020). Here “splitting” means replacing each vertex of a clique by a bounded-size blob while avoiding a forbidden subgraph.

A plausible implication is that “SplitMe” acquires its strongest ambiguity precisely in mathematically mature areas, where “split” can mean factorization, decomposition, merge obstruction, topological separation, or constrained blow-up, all within rigorous but non-overlapping frameworks.

7. Interpretation, ambiguity, and usage across domains

Across the supplied literature, “SplitMe” is best interpreted as a polysemous research shorthand whose meaning is fixed by context rather than by a universal definition. In routing, it denotes the giant-tour Split decoder and its linear-time descendants (Vidal, 2015, Gibbons et al., 24 Jan 2026). In multiterminal information theory, it denotes a recursive SPLIT algorithm on submodular base polyhedra (Ding et al., 2018). In hashing, the intended reference is almost certainly RecSplit (Esposito et al., 2019). In multivariate statistics, it may point either to split models for context-specific independence (Hojsgaard, 2013) or to splitting distributions for multivariate counts (Fernique et al., 2018). In machine learning practice, it may mean statistically matched partitioning of datasets via Twinning/SPlit (Vakayil et al., 2021) or distribution-preserving multi-label partitioning via EvoSplit (Florez-Revuelta, 2021). In NLP, it can describe sentence Split and Rephrase refinement, as in WikiSplit++ (Tsukagoshi et al., 2024). In theoretical physics and combinatorics, it refers to several precise but unrelated formal notions (Guan et al., 2024, Cachazo et al., 2021, Albert et al., 2016, Chun et al., 2017, Axenovich et al., 2020, Cha et al., 2013).

A common misconception is that the shared word “split” indicates methodological kinship. The supplied papers suggest otherwise. The vehicle-routing Split is a shortest-path decoder over a DAG with dominance pruning (Vidal, 2015, Gibbons et al., 24 Jan 2026). The multiterminal SPLIT algorithm is recursive parametric submodular minimization (Ding et al., 2018). RecSplit is randomized recursive bucket partitioning with coded seed indices (Esposito et al., 2019). Twinning is energy-distance-based geometric partitioning (Vakayil et al., 2021). EvoSplit is evolutionary assignment optimization over folds (Florez-Revuelta, 2021). WikiSplit++ is data refinement for seq2seq training (Tsukagoshi et al., 2024). Splitting amplitudes in QCD are universal collinear building blocks (Guan et al., 2024). Smooth 3-splits are a non-singular kinematic factorization phenomenon (Cachazo et al., 2021). These are linked only by the abstract idea of decomposition.

The most stable encyclopedia-level definition is therefore contextual: “SplitMe” denotes an informal family of split-based methods or structures whose specific meaning depends on the research area. If the context is vehicle routing, the canonical reference is Vidal’s linear-time Split and its extensions (Vidal, 2015, Gibbons et al., 24 Jan 2026). If the context is multiterminal source coding, it is the fairness-oriented SPLIT algorithm (Ding et al., 2018). If the context is minimal perfect hashing, it is RecSplit (Esposito et al., 2019). In other domains, the term should be disambiguated explicitly rather than assumed to name a single established technique.