<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Quantum-PG-HIVE: Schema Discovery for Property Graphs Using Quantum Computing</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Emmanouil Limnaios</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Sophia Sideri</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff2">2</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Haridimos Kondylakis</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>CSD, University of Crete</institution>
          ,
          <addr-line>Heraklion</addr-line>
          ,
          <country country="GR">Greece</country>
        </aff>
        <aff id="aff1">
          <label>1</label>
          <institution>FORTH-ICS</institution>
          ,
          <addr-line>Heraklion</addr-line>
          ,
          <country country="GR">Greece</country>
        </aff>
        <aff id="aff2">
          <label>2</label>
          <institution>LIPADE, Université Paris Cité</institution>
          ,
          <addr-line>Paris</addr-line>
          ,
          <country country="FR">France</country>
        </aff>
      </contrib-group>
      <pub-date>
        <year>2026</year>
      </pub-date>
      <abstract>
        <p>Property graphs (PGs) are widely adopted across domains due to their flexible, schema-free nature. However, the absence of explicit schemas complicates understanding, integration, validation, and analytics. PG-HIVE introduced a hybrid, incremental framework for schema discovery based on semantic embeddings and Locality-Sensitive Hashing (LSH) clustering. While efective and scalable, LSH remains inherently probabilistic, occasionally producing fragmented clusters and requiring non-trivial post-processing. In this demonstration, we present Quantum-PG-HIVE, an optimization-based extension of PG-HIVE that replaces the LSH clustering step with a Quadratic Unconstrained Binary Optimization (QUBO) formulation. Schema discovery is reformulated as a balanced minimum cut problem over a sparse similarity graph derived from pattern embeddings. The resulting QUBO energy function is minimized via simulated annealing and is directly compatible with quantum annealing hardware.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1. Introduction</title>
      <p>
        The rapid adoption of property graph databases across domains such as social networks, neuroscience,
and knowledge graphs has intensified the need for automated and scalable schema discovery techniques.
Property graphs are inherently flexible and often schema-less, enabling rapid data ingestion and
evolution but complicating data understanding, query formulation, and downstream analytics [
        <xref ref-type="bibr" rid="ref1 ref2">1, 2</xref>
        ].
As graphs scale to millions of nodes and edges with heterogeneous labels and properties, manually
engineered schemas become infeasible, motivating principled, data-driven approaches to infer latent
structural regularities [
        <xref ref-type="bibr" rid="ref3">3</xref>
        ].
      </p>
      <p>
        PG-HIVE [
        <xref ref-type="bibr" rid="ref4">4</xref>
        ] is a state-of-the-art framework for hybrid incremental schema discovery in property
graphs, combining pattern extraction, embedding, clustering, and schema inference into a scalable
pipeline. At its core, PG-HIVE relies on Locality-Sensitive Hashing (LSH) to cluster node and edge
patterns in an embedding space, enabling approximate similarity grouping at scale.
The problem. While LSH ofers attractive computational properties, it is fundamentally probabilistic:
similar patterns may be separated into diferent buckets, while dissimilar patterns may collide. In
practice, this leads to fragmented clusters, spurious schema types, and non-trivial post-processing
overhead—especially in graphs with complex, skewed, or noisy structures.
      </p>
      <p>
        The solution. This work introduces Quantum-PG-HIVE, a quantum-inspired clustering module that
replaces the LSH-based clustering step in PG-HIVE with an optimization-based approach grounded in
Quadratic Unconstrained Binary Optimization (QUBO) [
        <xref ref-type="bibr" rid="ref5">5</xref>
        ]. A QUBO is a combinatorial optimization
formulation in which the objective is to minimize a quadratic function of binary decision variables,
making it equivalent to finding the ground state of an Ising model [
        <xref ref-type="bibr" rid="ref6">6</xref>
        ] and directly compatible with
both classical heuristic solvers and quantum annealing hardware. Instead of approximating similarity
neighborhoods via hashing, Quantum-PG-HIVE formulates schema discovery as a balanced minimum
cut problem on a sparse similarity graph derived from pattern embeddings. By explicitly optimizing
a global objective function, Quantum-PG-HIVE produces cleaner, more interpretable schemas while
maintaining scalability and compatibility with existing PG-HIVE infrastructure.
      </p>
      <p>In this demonstration, we showcase the full interactive experience of Quantum-PG-HIVE. Users can
load a property graph from a backend store (e.g., Neo4j), explore clustering options (manual or adaptive),
inspect intermediate and final schema elements, visualize discovered node and edge types, inspect
property constraints, and compare schema extraction results under diferent parameter settings or
noise levels, and baseline approaches. The demo highlights Quantum-PG-HIVE’s ability to (i) discover
types in challenging scenarios, (ii) incrementally update schemas, and (iii) give users full control and
transparency over the inference process through a lightweight, intuitive web interface, going beyond
classical clustering approaches in both eficiency and efectiveness.</p>
      <p>The rest of this paper is structured as follows: Section 2 presents the architecture of the system ans
Sectionr˜efsec:scenario presents a demonstration scenario. Finally Section 4 concludes this paper and
presents directions for future work.</p>
    </sec>
    <sec id="sec-2">
      <title>2. Architecture</title>
      <p>Quantum-PG-HIVE adopts a modular and extensible architecture that augments the original PG-HIVE
pipeline with an optimization-based clustering backend, while preserving the preprocessing, embedding,
and schema inference components. The system is designed to maintain architectural continuity with
PG-HIVE, enabling controlled comparisons between probabilistic LSH-based clustering and global
QUBO-based optimization under identical upstream and downstream conditions. The architecture
follows a layered processing model in which graph data flows from storage to representation, from
representation to similarity modeling, from similarity modeling to clustering, and finally to schema
inference and interactive exploration. The overall architecture of the system is shown in Figure 1.</p>
      <sec id="sec-2-1">
        <title>2.1. Architectural Overview</title>
        <p>At a conceptual level, the system consists of interconnected layers responsible for data acquisition,
feature construction, similarity modeling, clustering, and schema synthesis. These layers are loosely
coupled, allowing the clustering backend to be replaced without afecting the rest of the pipeline. The
primary architectural enhancement introduced by Quantum-PG-HIVE is the insertion of a QUBO-based
optimization engine between similarity modeling and schema inference.</p>
        <p>The overall workflow proceeds as follows. The system first connects to a property graph backend and
extracts structural information. This information is transformed into pattern-based representations. The
patterns are embedded into a hybrid vector space that captures both semantic and structural information.
A sparse similarity graph is then constructed over these embeddings. At this stage, either the LSH-based
clustering module or the QUBO-based optimization module can be invoked. The clustering output is
subsequently passed to the schema inference engine, which generates formal node and edge types,
property constraints, and cardinalities. Finally, results are rendered through an interactive web interface
that exposes both intermediate artifacts and final schema representations.</p>
      </sec>
      <sec id="sec-2-2">
        <title>2.2. Graph Storage and Data Access Layer</title>
        <p>The architecture interfaces with a property graph storage system, typically Neo4j, although the design
remains backend-agnostic provided the store supports labeled property graph semantics and
querybased extraction. Upon user authentication, the system issues structured queries that retrieve nodes,
edges, labels, and associated properties in a uniform format. Extraction is performed in a way that
ensures that all structural information relevant to schema discovery is captured consistently across
datasets of varying size and complexity.</p>
        <p>To support scalability and interactive exploration, the data access layer incorporates batch processing
and optional sampling mechanisms. For very large graphs, users may specify sampling limits that
preserve type diversity while reducing memory consumption. Sampling occurs before embedding and
clustering, ensuring that subsequent computations remain tractable without compromising schema
quality.</p>
      </sec>
      <sec id="sec-2-3">
        <title>2.3. Pattern Extraction and Hybrid Feature Construction</title>
        <p>Following extraction, raw graph elements are transformed into abstract patterns. A node pattern is
defined by its combination of labels and the set of properties it contains. An edge pattern is defined
by its relationship type, the label combinations of its source and target nodes, and its own property
signature. This abstraction decouples schema discovery from individual graph instances and shifts the
focus toward recurring structural motifs.</p>
        <p>Each pattern is then encoded into a hybrid feature vector. The semantic component of this vector
is generated using a Word2Vec model trained on label and property co-occurrence contexts. This
embedding captures latent semantic proximity between patterns, even when their surface-level labels
difer. The structural component is represented as a binary or sparse vector indicating the presence or
absence of properties. The concatenation of semantic embeddings and structural signatures yields a
hybrid representation capable of capturing both contextual similarity and structural distinctiveness.</p>
        <p>This hybrid representation plays a crucial role in enabling the recovery of latent types in scenarios
with incomplete labeling or heterogeneous property distributions. Importantly, this stage remains
identical regardless of the clustering backend selected, ensuring that diferences in final schema quality
are attributable solely to clustering strategy.</p>
      </sec>
      <sec id="sec-2-4">
        <title>2.4. Sparse Similarity Graph Construction</title>
        <p>Instead of directly applying clustering algorithms to the embedding vectors, Quantum-PG-HIVE
constructs an intermediate similarity graph. For each pattern group, pairwise cosine similarities between
hybrid embeddings are computed. To prevent quadratic growth in similarity computations, the
architecture employs a Top- sparsification strategy. For each pattern, only its  most similar neighbors
are retained, forming a sparse weighted graph  = (, , ) where vertices correspond to patterns
and edges are weighted by similarity scores.</p>
        <p>This sparsification serves two purposes. First, it drastically reduces the number of quadratic
interactions that must be encoded in the QUBO formulation, improving computational eficiency. Second,
it preserves the most informative similarity relationships, ensuring that optimization focuses on
semantically meaningful connections. The value of  is configurable and can be tuned based on dataset
density and desired clustering granularity.</p>
      </sec>
      <sec id="sec-2-5">
        <title>2.5. Clustering Layer</title>
        <p>The clustering layer is the central architectural diferentiation between PG-HIVE and
QuantumPG-HIVE. The system supports two interchangeable backends: the baseline LSH module and the
optimization-based QUBO module.</p>
        <p>When the LSH backend is selected, hybrid vectors are grouped into buckets using Euclidean or
MinHash Locality-Sensitive Hashing. Adaptive parameter selection estimates bucket width and the
number of hash tables based on sample statistics of embedding distances and property sparsity. This
approach ofers expected linear-time performance and enables scalable approximate similarity grouping.
However, its probabilistic nature may lead to fragmentation or collisions.</p>
        <p>When the QUBO backend is selected, clustering is reformulated as a balanced minimum cut
optimization problem on the sparse similarity graph. Binary decision variables are introduced to represent
partition assignments. The objective function combines a cut term that penalizes separation of highly
similar patterns and a balance term that discourages trivial partitions. The resulting quadratic energy
function is minimized using simulated annealing. The architecture supports both a high-performance
Scala-based implementation integrated into the Spark pipeline and a Python reference implementation
compatible with D-Wave’s Ocean SDK.</p>
        <p>To obtain multiple clusters, recursive bipartitioning is applied. After each optimization step, the
graph is partitioned into two subsets, and the process is repeated until stopping criteria based on cut
quality or minimum cluster size are satisfied. This hierarchical decomposition yields a dendrogram-like
clustering structure. An adaptive balancing mechanism can be enabled to estimate optimal partition
ratios in datasets exhibiting skewed type distributions, using spectral heuristics to guide the balance
term.</p>
      </sec>
      <sec id="sec-2-6">
        <title>2.6. Schema Inference Engine</title>
        <p>Once clustering is completed, the resulting pattern groups are passed to the schema inference engine.
This component consolidates clusters into formal schema types. Clusters sharing identical label sets
are merged directly, while unlabeled clusters are compared to labeled ones using structural similarity
metrics. If similarity exceeds predefined thresholds, clusters are merged; otherwise, abstract types are
created.</p>
        <p>The engine then infers mandatory and optional properties based on frequency analysis, determines
property datatypes through lightweight inspection of observed values, and computes relationship
cardinalities by analyzing in-degree and out-degree distributions. The final schema is serialized into
PGSchema representations in both STRICT and LOOSE variants, ensuring compatibility with downstream
tools and validation frameworks.</p>
        <p>In incremental mode, newly discovered clusters are merged with existing schema elements without
recomputing the entire dataset. This supports monotonic schema evolution as graphs grow over time.</p>
      </sec>
      <sec id="sec-2-7">
        <title>2.7. Interactive Exploration Layer</title>
        <p>The final architectural layer consists of a browser-based graphical interface. The interface exposes
intermediate artifacts such as similarity graphs, cluster assignments, energy convergence plots in the
QUBO mode, and final schema graphs. Users can dynamically switch between clustering backends,
adjust hyperparameters such as ,  , or simulated annealing iterations, and immediately observe
changes in schema fragmentation, merge overhead, and inferred cardinalities.</p>
        <p>This tight integration between backend computation and frontend visualization ensures transparency
of the optimization process and enables interactive experimentation. The architecture thus supports
both research-oriented evaluation and real-world deployment scenarios.</p>
      </sec>
      <sec id="sec-2-8">
        <title>2.8. Architectural Properties</title>
        <p>The modularity of Quantum-PG-HIVE ensures extensibility and experimental flexibility. Because
clustering is encapsulated within a dedicated layer, additional optimization solvers or hybrid approaches
can be incorporated without altering preprocessing or schema inference logic. The system leverages
distributed data processing through Apache Spark for scalability, while optimization subproblems
remain suficiently localized to maintain tractable runtimes. The architecture is therefore both scalable
and future-proof, providing immediate compatibility with classical simulated annealing solvers and a
direct pathway toward quantum hardware acceleration as it becomes practically viable.</p>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>3. Demonstration Scenario</title>
      <p>The demonstration presents an interactive walkthrough of Quantum-PG-HIVE, guiding participants
through the complete workflow of optimization-enhanced schema discovery and enabling direct
comparison between the baseline LSH clustering and the QUBO-based clustering backend. The scenario is
structured into the following steps.</p>
      <p>1. Dataset Selection and Configuration. The user connects to a Neo4j property graph instance
and selects a dataset. The interface displays basic statistics, including node and edge counts, distinct
labels, and relationship types. The user then chooses the clustering backend (LSH or Quantum) and
configures execution parameters. For LSH, this includes hash tables and bucket settings; for Quantum,
this includes Top- sparsification, balance penalty  , target partition ratio, and simulated annealing
iterations. Execution can be performed in static or incremental mode.</p>
      <p>2. Pattern Extraction and Embedding. The system extracts node and edge patterns based on labels
and property signatures. Each pattern is encoded into a hybrid embedding that combines semantic
Word2Vec representations with structural property indicators. The interface reports the number of
discovered patterns and summarizes structural diversity. This stage is identical for both clustering
backends to ensure fair comparison.</p>
      <p>3. Similarity Graph Construction. A sparse similarity graph is built by computing cosine
similarities between embeddings and retaining only the Top- nearest neighbors per pattern. The interface
presents sparsity metrics and similarity distributions, ofering insight into structural skew and clustering
dificulty.</p>
      <p>4. Clustering Execution. In LSH mode, hybrid vectors are grouped into hash buckets to produce
candidate clusters. In Quantum mode, clustering is formulated as a balanced minimum cut encoded as a
QUBO energy function and solved via simulated annealing with recursive bipartitioning. The interface
displays cluster counts, balance diagnostics, and (in Quantum mode) energy convergence information.</p>
      <p>5. Schema Inference and Comparison. The resulting clusters are consolidated into schema types
through label-based merging and structural similarity checks. The system infers mandatory and optional
properties, datatypes, and relationship cardinalities, and renders the final schema in PG-Schema format.
A side-by-side comparison view highlights diferences in raw cluster counts, merge overhead, and final
node and edge types between LSH and Quantum modes.</p>
      <p>6. Robustness and Incremental Exploration (Optional). Participants can simulate noise or
partial labeling to evaluate robustness and observe how clustering quality evolves under perturbation.
In incremental mode, new graph batches are introduced, and the system updates the schema without
recomputing from scratch, illustrating monotonic schema evolution.</p>
      <p>Through these steps, the demonstration highlights how optimization-based clustering enhances
schema cleanliness, reduces fragmentation, and maintains competitive performance while preserving
the interactive and incremental capabilities of the original PG-HIVE framework.</p>
    </sec>
    <sec id="sec-4">
      <title>4. Conclusion</title>
      <p>In this demonstration, we presented Quantum-PG-HIVE, an optimization-based extension of the
PGHIVE framework for schema discovery in property graphs. By reformulating the clustering phase as a
balanced minimum cut problem expressed through a Quadratic Unconstrained Binary Optimization
(QUBO) model, the system replaces probabilistic hashing with a principled global optimization strategy.
This shift enables more coherent partitioning of pattern embeddings, reduces fragmentation artifacts,
and lowers post-processing overhead, particularly for relationship types where hash-based approaches
are more susceptible to instability.</p>
      <p>The architecture preserves the hybrid embedding and schema inference pipeline of PG-HIVE while
introducing a modular clustering backend that supports both classical simulated annealing and
compatibility with quantum annealing hardware. Through the interactive interface, users can inspect
intermediate similarity graphs, monitor optimization behavior, tune hyperparameters, and directly
compare LSH-based and QUBO-based clustering outcomes under identical conditions. The demonstration
highlights improvements in edge-type cleanliness, robustness under noisy or skewed data distributions,
and competitive runtime performance at practical graph scales.</p>
      <p>Overall, Quantum-PG-HIVE illustrates how global optimization techniques can enhance schema
discovery in flexible graph data models without sacrificing scalability or interpretability. By bridging
property graph data management with QUBO-based optimization, the system not only improves current
clustering quality but also provides a forward-looking pathway toward quantum-enabled graph analytics
as quantum hardware matures.</p>
    </sec>
    <sec id="sec-5">
      <title>Declaration on Generative AI</title>
      <p>The authors have not employed any Generative AI tools.</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <ref id="ref1">
        <mixed-citation>
          [1]
          <string-name>
            <given-names>A.</given-names>
            <surname>Bonifati</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Dumbrava</surname>
          </string-name>
          ,
          <string-name>
            <given-names>H.</given-names>
            <surname>Kondylakis</surname>
          </string-name>
          , G. Troullinou, G. Vassiliou,
          <article-title>Progressive querying on knowledge graphs</article-title>
          , in: A.
          <string-name>
            <surname>Simitsis</surname>
            ,
            <given-names>B.</given-names>
          </string-name>
          <string-name>
            <surname>Kemme</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          <string-name>
            <surname>Queralt</surname>
            ,
            <given-names>O.</given-names>
          </string-name>
          <string-name>
            <surname>Romero</surname>
          </string-name>
          , P. Jovanovic (Eds.),
          <source>Proceedings 28th International Conference on Extending Database Technology, EDBT</source>
          <year>2025</year>
          , Barcelona, Spain, March
          <volume>25</volume>
          -28,
          <year>2025</year>
          , OpenProceedings.org,
          <year>2025</year>
          , pp.
          <fpage>106</fpage>
          -
          <lpage>118</lpage>
          . URL: https://doi.org/10.48786/edbt.
          <year>2025</year>
          .
          <volume>09</volume>
          . doi:
          <volume>10</volume>
          .48786/EDBT.
          <year>2025</year>
          .
          <volume>09</volume>
          .
        </mixed-citation>
      </ref>
      <ref id="ref2">
        <mixed-citation>
          [2]
          <string-name>
            <given-names>G.</given-names>
            <surname>Troullinou</surname>
          </string-name>
          , G. Agathangelos,
          <string-name>
            <given-names>H.</given-names>
            <surname>Kondylakis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>K.</given-names>
            <surname>Stefanidis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Plexousakis</surname>
          </string-name>
          ,
          <source>DIAERESIS: RDF data partitioning and query processing on SPARK, Semantic Web</source>
          <volume>15</volume>
          (
          <year>2024</year>
          )
          <fpage>1763</fpage>
          -
          <lpage>1789</lpage>
          . URL: https://doi.org/10.3233/SW-243554. doi:
          <volume>10</volume>
          .3233/SW-243554.
        </mixed-citation>
      </ref>
      <ref id="ref3">
        <mixed-citation>
          [3]
          <string-name>
            <given-names>H.</given-names>
            <surname>Kondylakis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>S.</given-names>
            <surname>Dumbrava</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            <surname>Lissandrini</surname>
          </string-name>
          ,
          <string-name>
            <given-names>N.</given-names>
            <surname>Yakovets</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            <surname>Bonifati</surname>
          </string-name>
          ,
          <string-name>
            <given-names>V.</given-names>
            <surname>Efthymiou</surname>
          </string-name>
          ,
          <string-name>
            <given-names>G.</given-names>
            <surname>Fletcher</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Plexousakis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Tommasini</surname>
          </string-name>
          ,
          <string-name>
            <given-names>G.</given-names>
            <surname>Troullinou</surname>
          </string-name>
          , E. Ymeralli,
          <article-title>Property graph standards: State of the art &amp; open challenges</article-title>
          ,
          <source>Proc. VLDB Endow</source>
          .
          <volume>18</volume>
          (
          <year>2025</year>
          )
          <fpage>5477</fpage>
          -
          <lpage>5481</lpage>
          . URL: https://www.vldb.org/pvldb/ vol18/p5477-kondylakis.pdf.
          <source>doi:10.14778/3750601</source>
          .3750698.
        </mixed-citation>
      </ref>
      <ref id="ref4">
        <mixed-citation>
          [4]
          <string-name>
            <given-names>S.</given-names>
            <surname>Sideri</surname>
          </string-name>
          ,
          <string-name>
            <given-names>G.</given-names>
            <surname>Troullinou</surname>
          </string-name>
          ,
          <string-name>
            <given-names>E.</given-names>
            <surname>Ymeralli</surname>
          </string-name>
          ,
          <string-name>
            <given-names>V.</given-names>
            <surname>Efthymiou</surname>
          </string-name>
          ,
          <string-name>
            <given-names>D.</given-names>
            <surname>Plexousakis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>H.</given-names>
            <surname>Kondylakis</surname>
          </string-name>
          ,
          <article-title>Pg-hive: Hybrid incremental schema discovery for property graphs</article-title>
          ,
          <source>EDBT</source>
          ,
          <year>2026</year>
          , p. (to appear).
        </mixed-citation>
      </ref>
      <ref id="ref5">
        <mixed-citation>
          [5]
          <string-name>
            <given-names>F. W.</given-names>
            <surname>Glover</surname>
          </string-name>
          ,
          <string-name>
            <given-names>G. A.</given-names>
            <surname>Kochenberger</surname>
          </string-name>
          ,
          <string-name>
            <given-names>R.</given-names>
            <surname>Hennig</surname>
          </string-name>
          ,
          <string-name>
            <given-names>Y.</given-names>
            <surname>Du</surname>
          </string-name>
          ,
          <article-title>Quantum bridge analytics I: a tutorial on formulating and using QUBO models</article-title>
          ,
          <source>Ann. Oper. Res</source>
          .
          <volume>314</volume>
          (
          <year>2022</year>
          )
          <fpage>141</fpage>
          -
          <lpage>183</lpage>
          . URL: https://doi.org/10. 1007/s10479-022-04634-2. doi:
          <volume>10</volume>
          .1007/S10479-022-04634-2.
        </mixed-citation>
      </ref>
      <ref id="ref6">
        <mixed-citation>
          [6]
          <string-name>
            <given-names>A.</given-names>
            <surname>Lucas</surname>
          </string-name>
          ,
          <article-title>Ising formulations of many NP problems</article-title>
          ,
          <source>CoRR abs/1302</source>
          .5843 (
          <year>2013</year>
          ). URL: http://arxiv. org/abs/1302.5843. arXiv:
          <volume>1302</volume>
          .
          <fpage>5843</fpage>
          .
        </mixed-citation>
      </ref>
    </ref-list>
  </back>
</article>