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Mathematical Advances in Combinatorial Optimization: Theory, Methods, and Applications
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Mathematical Advances in Combinatorial Optimization: Theory, Methods, and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D2: Operations Research and Fuzzy Decision Making".

Deadline for manuscript submissions: 31 March 2027 | Viewed by 1292

Editor

Dr. Pablo Adasme

E-Mail Website
Guest Editor
Electrical Engineering Department, Universidad de Santiago de Chile, Santiago 9170124, Chile
Interests: operations research; combinatorial optimization; telecommunications; graph optimization; convex conic optimization; stochastich and robust optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are pleased to invite you to contribute to the Special Issue ‘Mathematical Advances in Combinatorial Optimization: Theory, Methods, and Applications’ in MDPI Mathematics.

Combinatorial optimization is a fundamental and rapidly evolving field of applied mathematics, with broad impact across operations research, computer science, engineering, economics, and data-driven decision making. Its methods are essential for addressing complex discrete problems arising in network design, scheduling, routing, resource allocation, logistics, energy systems, telecommunications, healthcare, manufacturing, and many other application areas.

This Special Issue aims to gather original and high-quality contributions that advance the mathematical foundations, algorithmic developments, and practical applications of combinatorial optimization. We welcome submissions on topics such as theoretical analysis, complexity results, exact methods, heuristics and metaheuristics, approximation algorithms, decomposition techniques, polyhedral approaches, relaxation methods, robust and stochastic optimization, and innovative formulations for classical or emerging problems.

We particularly encourage potential authors and co-authors to submit research that combines mathematical rigor with computational effectiveness and practical relevance. Through this Special Issue, we hope to provide a valuable forum for the dissemination of new ideas, recent advances, and interdisciplinary perspectives that can further enrich this dynamic research area.

We look forward to receiving your contributions.

Dr. Pablo Adasme
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • combinatorial optimization
  • discrete optimization
  • integer programming
  • mixed-integer optimization
  • graph optimization
  • network optimization
  • exact algorithms
  • approximation algorithms
  • heuristics
  • metaheuristics
  • decomposition methods
  • polyhedral analysis
  • valid inequalities
  • relaxation techniques
  • robust optimization
  • stochastic optimization
  • scheduling
  • routing
  • resource allocation
  • optimization applications

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Published Papers (2 papers)

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Research

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36 pages, 1793 KB  
Article
When Does Domination Matter: A Structural and Computational Study of Spanning and Dominating Trees in Geometric Networks
by Pablo Adasme
Mathematics 2026, 14(10), 1605; https://doi.org/10.3390/math14101605 - 9 May 2026
Viewed by 341
Abstract
In geometric communication networks, a backbone is useful only if it is inexpensive to build and, at the same time, close enough to the demand points it must serve. This paper studies a backbone design problem in geometric communication networks that explicitly captures [...] Read more.
In geometric communication networks, a backbone is useful only if it is inexpensive to build and, at the same time, close enough to the demand points it must serve. This paper studies a backbone design problem in geometric communication networks that explicitly captures this trade-off between connectivity and user coverage. Two classical combinatorial optimization paradigms—the minimum spanning tree (MST), which promotes low-cost connectivity, and the dominating tree (DT), which additionally enforces that every node either belongs to the backbone or is adjacent to an active backbone node—are considered. To compare both paradigms within a common framework, this paper proposes a unified mixed-integer optimization model that balances backbone-construction and user-assignment costs. Three classes of exact formulations, namely MTZ, single-flow, and cut-set formulations, are developed. In particular, the single-flow model with valid inequalities and root-aware connectivity cuts is strengthened. For larger instances, the exact approach is complemented with a local branching matheuristic. Finally, theoretical results on computational complexity, formulation structure, and dominance relations between the MST and DT models are provided. Computational experiments show that the single-flow formulation achieves the best scalability. Furthermore, a sensitivity analysis with respect to the communication radius and the weighting parameter α reveals a structural transition: as the network becomes denser or the objective becomes more coverage-oriented, MST and DT solutions tend to converge. The results give a concrete way to identify when domination constraints are worth imposing and when a simpler spanning tree design already captures the relevant structure. Full article
(This article belongs to the Special Issue Mathematical Advances in Combinatorial Optimization: Theory, Methods, and Applications)
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Review

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46 pages, 2303 KB  
Review
Deep Reinforcement Learning for Combinatorial Optimization Problems: A Challenge-Driven Methodology and Systematic Review
by Shengyun Wei, Chuibing Huang, Zhenyi Wang, Yang Wang, Dekang Kong, Haibo Mi and Zhaolong Sun
Mathematics 2026, 14(14), 2538; https://doi.org/10.3390/math14142538 - 14 Jul 2026
Viewed by 604
Abstract
Combinatorial optimization problems (COPs) offer essential mathematical frameworks and algorithmic foundations for modeling complex real-world decision-making tasks. Recent advances in deep reinforcement learning (DRL) have shown promising results for solving COPs, offering the potential to reduce dependence on domain-specific expertise and improve generalization [...] Read more.
Combinatorial optimization problems (COPs) offer essential mathematical frameworks and algorithmic foundations for modeling complex real-world decision-making tasks. Recent advances in deep reinforcement learning (DRL) have shown promising results for solving COPs, offering the potential to reduce dependence on domain-specific expertise and improve generalization across problem instances. These developments have accelerated research in the field and spurred the emergence of numerous innovative methods. Nevertheless, significant theoretical and practical challenges remain. A systematic synthesis of these challenges and their corresponding solutions is critical to guiding the future development of DRL-based approaches. To address this need, we propose a unified challenge-driven framework consisting of four core components: an environment, a state–action–reward mechanism, a solver, and an evaluation module. Using this framework, we conduct a systematic review of approximately 300 recent studies, mapping the evolution of challenges and the progress made in addressing them. We provide a multidimensional analysis of solver designs, training paradigms, and state-of-the-art (SOTA) performance, while documenting publicly available code repositories. Finally, we identify key open problems within the proposed framework to stimulate novel research directions. Full article
(This article belongs to the Special Issue Mathematical Advances in Combinatorial Optimization: Theory, Methods, and Applications)
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