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arxiv: 2311.10670 · v1 · submitted 2023-11-17 · 🧮 math.OC

Target-based Distributionally Robust Minimum Spanning Tree Problem

Yang Xu , Lianmin Zhang This is my paper

Pith reviewed 2026-05-24 05:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributionally robust optimizationminimum spanning treestochastic graphsBenders decompositionPrim algorithmambiguity setrobust optimizationnetwork optimization
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The pith

A target-based distributionally robust model for minimum spanning trees yields exact solutions via Benders decomposition and a modified Prim algorithm when only partial edge-weight statistics are known.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a distributionally robust optimization framework for the minimum spanning tree problem on stochastic graphs whose edge-weight distributions are unknown. Partial statistical information is used to define a target-based ambiguity set that avoids the over-conservatism of classical robust models while remaining tractable. Two exact solution methods are developed: a Benders decomposition approach and a modification of the classical Prim greedy algorithm. The authors claim these methods deliver both computational tractability on large instances and robustness that improves on earlier stochastic and robust spanning-tree formulations.

Core claim

The target-based distributionally robust minimum spanning tree problem can be solved exactly by Benders decomposition and a modified Prim algorithm, providing satisfactory algorithmic performance and robustness when faced with uncertainty in edge weights, using only partial statistical information to define the ambiguity set.

What carries the argument

Target-based ambiguity set built from partial statistical information on edge weights, allowing worst-case distributions to be optimized exactly inside the Benders and modified-Prim procedures.

If this is right

  • The model remains solvable exactly even when edge-weight distributions are only partially characterized.
  • Solutions avoid the conservatism of standard robust MST formulations while retaining distribution-free guarantees.
  • The approach scales to large networks because both algorithms exploit the combinatorial structure of spanning trees.
  • No assumption on the specific family of edge-weight distributions is required beyond the supplied statistical moments or bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same target-based construction could be tested on other network design problems such as shortest paths or facility location under distributional uncertainty.
  • Empirical comparison on transportation or communication networks with real partial data would reveal whether the claimed robustness gain materializes in practice.
  • If the ambiguity set is defined by moment constraints, the worst-case distributions might admit closed-form characterizations that further simplify the subproblems.

Load-bearing premise

Partial statistical information on edge weights is sufficient to construct a non-trivial ambiguity set whose worst-case distributions can be handled exactly by the proposed algorithms without excessive conservatism or computational intractability.

What would settle it

Solving a large-scale instance with the Benders or modified-Prim procedure and finding either that solution times grow exponentially or that the resulting tree performs worse than a classical robust solution under sampled realizations would falsify the tractability and robustness claims.

read the original abstract

Due to its broad applications in practice, the minimum spanning tree problem and its all kinds of variations have been studied extensively during the last decades, for which a host of efficient exact and heuristic algorithms have been proposed. Meanwhile, motivated by realistic applications, the minimum spanning tree problem in stochastic network has attracted considerable attention of researchers, with respect to which stochastic and robust spanning tree models and related algorithms have been continuingly developed. However, all of them would be either too restricted by the types of the edge weight random variables or computationally intractable, especially in large-scale networks. In this paper, we introduce a target-based distributionally robust optimization framework to solve the minimum spanning tree problem in stochastic graphs where the probability distribution function of the edge weight is unknown but some statistical information could be utilized to prevent the optimal solution from being too conservative. We propose two exact algorithms to solve it, based on Benders decomposition framework and a modified classical greedy algorithm of MST problem (Prim algorithm),respectively. Compared with the NP-hard stochastic and robust spanning tree problems,The proposed target-based distributionally robust minimum spanning tree problem enjoys more satisfactory algorithmic aspect and robustness, when faced with uncertainty in input data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces a target-based distributionally robust optimization framework for the minimum spanning tree (MST) problem on stochastic graphs where edge-weight distributions are unknown but partial statistical information (e.g., moments or support) is available. It formulates a target-based ambiguity set, derives a deterministic equivalent, and presents two exact algorithms—a Benders decomposition procedure and a modified Prim algorithm—whose correctness relies on the target-based structure preserving the matroid greedy property. The central claim is that the resulting model is computationally more tractable and less conservative than classical stochastic or robust MST formulations.

Significance. If the claimed exactness and termination properties hold, the work supplies a concrete algorithmic bridge between distributionally robust and combinatorial optimization by showing that a suitably chosen target-based ambiguity set admits polynomial-time solution methods while controlling conservatism. The explicit reduction to a deterministic equivalent that inherits the matroid structure, together with the two specialized algorithms, constitutes a technical strength that could be useful for network design under partial distributional information.

minor comments (3)
  1. Abstract: missing space after the comma in 'problems,The proposed'; 'continuingly' should be 'continuously'; the phrase 'more satisfactory algorithmic aspect' is imprecise and should be replaced by a concrete statement about complexity or empirical performance.
  2. Abstract: the sentence 'Compared with the NP-hard stochastic and robust spanning tree problems,…' asserts superiority without referencing the specific complexity results or numerical evidence that appear later in the manuscript; a forward pointer would improve readability.
  3. The abstract states that the proposed problem 'enjoys more satisfactory algorithmic aspect and robustness' but does not indicate whether the algorithms are polynomial-time or merely exact; a brief clarification of the complexity class would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive summary and positive evaluation of the significance of our target-based DRO MST framework. The recommendation for minor revision is appreciated. No major comments were listed in the report, so we have no specific points requiring rebuttal or revision at this stage. We remain available to address any additional feedback from the editor or further review.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a target-based DRO formulation for the MST using partial statistical information to define an ambiguity set, then derives Benders decomposition and a modified Prim algorithm whose correctness follows from the deterministic equivalent preserving the matroid greedy property. No step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The central tractability and robustness claims rest on explicit algorithmic reductions shown to terminate at the worst-case optimum under the stated ambiguity set, without load-bearing self-citations or ansatz smuggling. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no details on specific free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5729 in / 911 out tokens · 28184 ms · 2026-05-24T05:49:53.372348+00:00 · methodology

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Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    Minimal spanning tree subject to a side constraint

    Vijay Aggarwal, Yash P Aneja, and KPK Nair. Minimal spanning tree subject to a side constraint. Computers & Operations Research, 9 0 (4): 0 287--296, 1982

    work page 1982

  2. [2]

    A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs

    Javad Akbari Torkestani and Mohammad Reza Meybodi. A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs. The Journal of Supercomputing, 59 0 (2): 0 1035--1054, 2012

    work page 2012

  3. [3]

    State space partition algorithms for stochastic systems with applications to minimum spanning trees

    Christos Alexopoulos and Jay A Jacobson. State space partition algorithms for stochastic systems with applications to minimum spanning trees. Networks: An International Journal, 35 0 (2): 0 118--138, 2000

    work page 2000

  4. [4]

    On the complexity of the robust spanning tree problem with interval data

    Ionu t D Aron and Pascal Van Hentenryck. On the complexity of the robust spanning tree problem with interval data. Operations Research Letters, 32 0 (1): 0 36--40, 2004

    work page 2004

  5. [5]

    Robust discrete optimization and network flows

    Dimitris Bertsimas and Melvyn Sim. Robust discrete optimization and network flows. Mathematical programming, 98 0 (1): 0 49--71, 2003

    work page 2003

  6. [6]

    Theory and applications of robust optimization

    Dimitris Bertsimas, David B Brown, and Constantine Caramanis. Theory and applications of robust optimization. SIAM review, 53 0 (3): 0 464--501, 2011

    work page 2011

  7. [7]

    An ant-based algorithm for finding degree-constrained minimum spanning tree

    Thang N Bui and Catherine M Zrncic. An ant-based algorithm for finding degree-constrained minimum spanning tree. In Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 11--18, 2006

    work page 2006

  8. [8]

    Robust cara optimization

    Li Chen and Melvyn Sim. Robust cara optimization. Available at SSRN 3937474, 2021

    work page 2021

  9. [9]

    Goal-driven optimization

    Wenqing Chen and Melvyn Sim. Goal-driven optimization. Operations Research, 57 0 (2): 0 342--357, 2009 a

    work page 2009

  10. [10]

    Chen and M

    W.Q. Chen and M. Sim. Goal-driven optimization. Operations Research, 57 0 (2): 0 342--357, 2009 b

    work page 2009

  11. [11]

    A polynomial solvable minimum risk spanning tree problem with interval data

    Xujin Chen, Jie Hu, and Xiaodong Hu. A polynomial solvable minimum risk spanning tree problem with interval data. European Journal of Operational Research, 198 0 (1): 0 43--46, 2009

    work page 2009

  12. [12]

    Inventory routing problem under uncertainty

    Zheng Cui, Daniel Zhuoyu Long, Jin QI, and Lianmin Zhang. Inventory routing problem under uncertainty. Available at SSRN 3946807, 2021

    work page 2021

  13. [13]

    An evolutionary approach to solve minimum spanning tree problem with fuzzy parameters

    T Agostinho de Almeida, Akebo Yamakami, and M \'a rcia Tomie Takahashi. An evolutionary approach to solve minimum spanning tree problem with fuzzy parameters. In International Conference on Computational Intelligence for Modelling, Control and Automation and International Conference on Intelligent Agents, Web Technologies and Internet Commerce (CIMCA-IAWT...

    work page 2005

  14. [14]

    Transportation networks with random arc capacities

    P Doulliez and E Jamoulle. Transportation networks with random arc capacities. Revue fran c aise d'automatique, informatique, recherche op \'e rationnelle. Recherche op \'e rationnelle , 6 0 (V3): 0 45--59, 1972

    work page 1972

  15. [15]

    A randomly weighted minimum arborescence with a random cost constraint

    Alan M Frieze and Tomasz Tkocz. A randomly weighted minimum arborescence with a random cost constraint. Mathematics of Operations Research, 2021

    work page 2021

  16. [16]

    Raghavan

    Ioannis Gamvros, Bruce Golden, and S. Raghavan. The multilevel capacitated minimum spanning tree problem. INFORMS Journal on Computing, 18 0 (3): 0 348--365, 2006. doi:10.1287/ijoc.1040.0123. URL https://doi.org/10.1287/ijoc.1040.0123

    work page doi:10.1287/ijoc.1040.0123 2006

  17. [17]

    Modeling hop-constrained and diameter-constrained minimum spanning tree problems as steiner tree problems over layered graphs

    Luis Gouveia, Luidi Simonetti, and Eduardo Uchoa. Modeling hop-constrained and diameter-constrained minimum spanning tree problems as steiner tree problems over layered graphs. Mathematical Programming, 128 0 (1): 0 123--148, 2011

    work page 2011

  18. [18]

    Neighbourhood searches for the bounded diameter minimum spanning tree problem embedded in a vns, ea, and aco

    Martin Gruber, Jano van Hemert, and G \"u nther R Raidl. Neighbourhood searches for the bounded diameter minimum spanning tree problem embedded in a vns, ea, and aco. In Proceedings of the 8th annual conference on Genetic and Evolutionary Computation, pages 1187--1194, 2006

    work page 2006

  19. [19]

    Hall, D.Z

    N.G. Hall, D.Z. Long, J. Qi, and M. Sim. Managing underperformance risk in project portfolio selection. Operations Research, 63 0 (3): 0 660--675, 2015

    work page 2015

  20. [20]

    A model and algorithm for minimum spanning tree problems in uncertain networks

    Fangguo He and Huan Qi. A model and algorithm for minimum spanning tree problems in uncertain networks. In 2008 3rd International Conference on Innovative Computing Information and Control, pages 493--493. IEEE, 2008

    work page 2008

  21. [21]

    Efficient algorithms for the inverse spanning-tree problem

    Dorit S Hochbaum. Efficient algorithms for the inverse spanning-tree problem. Operations Research, 51 0 (5): 0 785--797, 2003

    work page 2003

  22. [22]

    Bounding distributions for the weight of a minimum spanning tree in stochastic networks

    Kevin R Hutson and Douglas R Shier. Bounding distributions for the weight of a minimum spanning tree in stochastic networks. Operations research, 53 0 (5): 0 879--886, 2005

    work page 2005

  23. [23]

    Minimum spanning trees in networks with varying edge weights

    Kevin R Hutson and Douglas R Shier. Minimum spanning trees in networks with varying edge weights. Annals of Operations Research, 146 0 (1): 0 3--18, 2006

    work page 2006

  24. [24]

    Confidence regional method of stochastic spanning tree problem

    H Ishii and T Matsutomi. Confidence regional method of stochastic spanning tree problem. Mathematical and computer modelling, 22 0 (10-12): 0 77--82, 1995

    work page 1995

  25. [25]

    Stochastic bottleneck spanning tree problem

    Hiroaki Ishii and Toshio Nishida. Stochastic bottleneck spanning tree problem. Networks, 13 0 (3): 0 443--449, 1983

    work page 1983

  26. [26]

    Stochastic spanning tree problem

    Hiroaki Ishii, Sh \=o go Shiode, Toshio Nishida, and Yoshikazu Namasuya. Stochastic spanning tree problem. Discrete Applied Mathematics, 3 0 (4): 0 263--273, 1981

    work page 1981

  27. [27]

    Routing optimization under uncertainty

    Patrick Jaillet, Jin Qi, and Melvyn Sim. Routing optimization under uncertainty. Operations research, 64 0 (1): 0 186--200, 2016

    work page 2016

  28. [28]

    Approximations for the random minimal spanning tree with application to network provisioning

    Anjani Jain and John W Mamer. Approximations for the random minimal spanning tree with application to network provisioning. Operations Research, 36 0 (4): 0 575--584, 1988

    work page 1988

  29. [29]

    Joseph B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7 0 (1): 0 48--50, 1956. ISSN 00029939, 10886826. URL http://www.jstor.org/stable/2033241

    work page arXiv 1956

  30. [30]

    Maximizing expected utility for stochastic combinatorial optimization problems

    Jian Li and Amol Deshpande. Maximizing expected utility for stochastic combinatorial optimization problems. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, pages 797--806. IEEE, 2011

    work page 2011

  31. [31]

    Distributionally robust discrete optimization with entropic value-at-risk

    Daniel Zhuoyu Long and Jin Qi. Distributionally robust discrete optimization with entropic value-at-risk. Operations Research Letters, 42 0 (8): 0 532--538, 2014

    work page 2014

  32. [32]

    Optimal trees

    Thomas L Magnanti and Laurence A Wolsey. Optimal trees. Handbooks in operations research and management science, 7: 0 503--615, 1995

    work page 1995

  33. [33]

    Survey of capital budgeting: Theory and practice

    James CT Mao. Survey of capital budgeting: Theory and practice. Journal of finance, pages 349--360, 1970

    work page 1970

  34. [34]

    Interval elimination method for stochastic spanning tree problem

    Ismail Bin Mohd. Interval elimination method for stochastic spanning tree problem. Applied Mathematics and Computation, 66 0 (2-3): 0 325--341, 1994

    work page 1994

  35. [35]

    A benders decomposition approach for the robust spanning tree problem with interval data

    Roberto Montemanni. A benders decomposition approach for the robust spanning tree problem with interval data. European Journal of Operational Research, 174 0 (3): 0 1479--1490, 2006

    work page 2006

  36. [36]

    A branch and bound algorithm for the robust spanning tree problem with interval data

    Roberto Montemanni and Luca Maria Gambardella. A branch and bound algorithm for the robust spanning tree problem with interval data. European Journal of Operational Research, 161 0 (3): 0 771--779, 2005

    work page 2005

  37. [37]

    Design of capacitated minimum spanning tree with uncertain cost and demand parameters

    Temel \"O ncan. Design of capacitated minimum spanning tree with uncertain cost and demand parameters. Information Sciences, 177 0 (20): 0 4354--4367, 2007

    work page 2007

  38. [38]

    A tabu search heuristic for the generalized minimum spanning tree problem

    Temel \"O ncan, Jean-Francois Cordeau, and Gilbert Laporte. A tabu search heuristic for the generalized minimum spanning tree problem. European Journal of Operational Research, 191 0 (2): 0 306--319, 2008

    work page 2008

  39. [39]

    An iterative algorithm for delay-constrained minimum-cost multicasting

    Mehrdad Parsa, Qing Zhu, and JJ Garcia-Luna-Aceves. An iterative algorithm for delay-constrained minimum-cost multicasting. IEEE/ACM transactions on networking, 6 0 (4): 0 461--474, 1998

    work page 1998

  40. [40]

    Budgeted prize-collecting traveling salesman and minimum spanning tree problems

    Alice Paul, Daniel Freund, Aaron Ferber, David B Shmoys, and David P Williamson. Budgeted prize-collecting traveling salesman and minimum spanning tree problems. Mathematics of Operations Research, 45 0 (2): 0 576--590, 2020

    work page 2020

  41. [41]

    Shortest connection networks and some generalizations

    Robert Clay Prim. Shortest connection networks and some generalizations. The Bell System Technical Journal, 36 0 (6): 0 1389--1401, 1957

    work page 1957

  42. [42]

    Preferences for travel time under risk and ambiguity: Implications in path selection and network equilibrium

    Jin Qi, Melvyn Sim, Defeng Sun, and Xiaoming Yuan. Preferences for travel time under risk and ambiguity: Implications in path selection and network equilibrium. Transportation Research Part B: Methodological, 94: 0 264--284, 2016

    work page 2016

  43. [43]

    Distributionally robust optimization: A review

    Hamed Rahimian and Sanjay Mehrotra. Distributionally robust optimization: A review. arXiv preprint arXiv:1908.05659, 2019

    work page arXiv 1908

  44. [44]

    The robust minimum spanning tree problem: compact and convex uncertainty

    Martha Salazar-Neumann. The robust minimum spanning tree problem: compact and convex uncertainty. Operations research letters, 35 0 (1): 0 17--22, 2007

    work page 2007

  45. [45]

    Chance-constrained programming models and approximations for general stochastic bottleneck spanning tree problems

    Siqian Shen, Murat Kurt, and Jue Wang. Chance-constrained programming models and approximations for general stochastic bottleneck spanning tree problems. INFORMS Journal on Computing, 27 0 (2): 0 301--316, 2015

    work page 2015

  46. [46]

    Solving inverse spanning tree problems through network flow techniques

    PT Sokkalingam, Ravindra K Ahuja, and James B Orlin. Solving inverse spanning tree problems through network flow techniques. Operations Research, 47 0 (2): 0 291--298, 1999

    work page 1999

  47. [47]

    A multiobjective branch-and-bound framework: Application to the biobjective spanning tree problem

    Francis Sourd and Olivier Spanjaard. A multiobjective branch-and-bound framework: Application to the biobjective spanning tree problem. INFORMS Journal on Computing, 20 0 (3): 0 472--484, 2008

    work page 2008

  48. [48]

    Learning automata-based algorithms for solving stochastic minimum spanning tree problem

    Javad Akbari Torkestani and Mohammad Reza Meybodi. Learning automata-based algorithms for solving stochastic minimum spanning tree problem. Applied Soft Computing, 11 0 (6): 0 4064--4077, 2011

    work page 2011

  49. [49]

    Integer programming formulations for minimum spanning tree interdiction

    Ningji Wei, Jose L Walteros, and Foad Mahdavi Pajouh. Integer programming formulations for minimum spanning tree interdiction. INFORMS Journal on Computing, 33 0 (4): 0 1461--1480, 2021

    work page 2021

  50. [50]

    The robust spanning tree problem with interval data

    Hande Yaman, Oya Ekin Kara s an, and Mustafa C P nar. The robust spanning tree problem with interval data. Operations research letters, 29 0 (1): 0 31--40, 2001

    work page 2001