arxiv: 2605.06319 · v1 · submitted 2026-05-07 · 💻 cs.DS · cs.NI

Recognition: unknown

Designing Capacitated Subnetworks for Shortest Path Routing

Markus Chimani, Max Ilsen

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:48 UTC · model grok-4.3

classification 💻 cs.DS cs.NI

keywords energy-efficient networkingshortest path routingnetwork designinteger linear programmingcolumn generationcapacitated subnetworkstraffic engineeringgreen networking

0 comments

The pith

A new integer linear program exactly solves the combined problem of deactivating network links and routing demands by shortest paths.

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

The goal is to improve energy efficiency in computer networks by selecting a smaller active subnetwork during low-traffic periods while still routing all demands via shortest paths without violating capacities. Earlier methods either ignored capacities, assumed routes stay fixed after deactivation, or solved design and routing separately using approximations. This paper develops an integer linear program that simultaneously chooses which links to keep and accounts for how those choices alter the shortest paths that will be used. The formulation is solved exactly with a tailored column generation procedure plus provably valid strengthening constraints. Experiments on small realistic instances show that the simplest heuristic—first pick any routing, then deactivate every unused link—produces solutions very close to the computed optimum, while a traffic-oblivious method works best when multiple demand matrices must be handled.

Core claim

We introduce a novel integer linear program that jointly optimizes the selection of a capacitated subnetwork and the routing of demands under shortest-path routing, explicitly modeling the fact that deactivating a link can change the shortest paths of other demands. The model is solved via a specialized column-generation algorithm augmented by additional strengthening constraints that remain valid in the capacitated setting. Using this exact solver we obtain true optimal solutions for small practical networks and, for the first time, quantify the gap to several common simplifying heuristics; the simplest of these—compute a routing first, then turn off all superfluous connections—turns out to

What carries the argument

The integer linear program that models dynamically changing shortest paths induced by link deactivation, solved by column generation.

If this is right

True optimal solutions become available for small practical instances.
The simple routing-first-then-deactivate heuristic is near-optimal on the tested instances.
Among compared methods, the traffic-oblivious TOCA approach performs best when multiple demand matrices must be accommodated.
The absolute quality of earlier simplifying algorithms can now be measured directly.

Where Pith is reading between the lines

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

The surprisingly good performance of the simplest heuristic suggests that shortest-path routings are often robust to moderate link removal.
The column-generation framework could be adapted to obtain exact solutions for medium-sized networks by adding more aggressive bounding or decomposition.
Similar dynamic-path modeling might be useful for other routing protocols whose paths depend on the active topology.

Load-bearing premise

The ILP formulation together with its column-generation procedure correctly encodes every possible change to shortest paths that can result from deactivating links, and the added strengthening constraints remain valid once capacities are present.

What would settle it

For any small network whose optimal subnetwork and corresponding shortest-path routing can be enumerated exhaustively by hand, the ILP must return exactly the same objective value and the same set of active links.

read the original abstract

In pursuit of higher energy efficiency in computer networks, one subfield of green traffic engineering aims at reducing the size of a network during times of low traffic, while still guaranteeing the ability to route all occurring demands. In this setting, we have to simultaneously solve a network design problem (choosing connections to deactivate) and a routing problem (routing paths in the active subnetwork, adhering to some routing protocol). Interestingly, there seems to be no available method to tackle the problem as a whole for the simplest (and still most commonly used) routing paradigm: shortest path routing. State-of-the-art methods either do not consider capacities, or assume that the routing paths should not change when deactivating network connections, or separate the problem into its two constituents, first solving the network design problem (using some estimators in lieu of the precise routing protocol) and only then the actual routing problem. In this paper, we present an algorithm to tackle the full combined problem exactly via a novel integer linear program, modeling dynamically changing shortest paths. To solve it, we need to devise a special-purpose column generation method. To speed up the solution process, we further propose additional provably strengthening constraints. Now having the means to yield true optimal solutions for (small) practical instances, we can for the first time give an in-depth experimental evaluation that includes the absolute quality intrinsic to the above simplifying algorithms. It turns out that the arguably simplest method--first computing a routing, fixing it, and turning off all superfluous connections--yields solutions surprisingly close to the true optimum in practice. When considering multiple different traffic demands, a recent traffic-oblivious approach (TOCA) performs best, while being comparatively straightforward to implement.

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.

Desk Editor's Note private letter to a colleague

The paper gives the first exact ILP plus column generation for joint capacitated network design and dynamic shortest-path routing, and shows that the simplest prune-after-route heuristic is often near-optimal on small instances.

read the letter

The core advance is a new integer linear program that chooses which links to keep while forcing the routing to use shortest paths in the induced active subnetwork and respecting capacities. They solve it with a tailored column-generation scheme and add some provably valid strengthening cuts. This is genuinely new; earlier work either dropped capacities, froze the routing, or solved the two pieces separately with proxies for the protocol. Having an exact method lets them run the first direct comparison on small practical instances, and the main empirical takeaway is useful: computing a routing first and then turning off unused links gets solutions surprisingly close to the true optimum. The traffic-oblivious TOCA approach also comes out ahead when demands vary. That kind of benchmark is worth having for deployment decisions in green traffic engineering. The modeling claim is the soft spot. The abstract asserts that the ILP correctly tracks changing shortest paths after arbitrary deactivations, but the description stays high-level and supplies no derivation or proof that the reduced-cost or potential constraints stay valid once capacities are added. If the interaction between capacity cuts and shortest-path conditions has a gap, the reported optima could include infeasible or non-shortest-path solutions. Experiments are limited to small cases with no error bars or scaling data, so the “close in practice” statement is suggestive rather than quantified. The paper is aimed at network-optimization researchers who care about exact methods for energy-aware routing. A reader who wants to see how far simple heuristics are from optimality will find concrete numbers here. It is worth sending to peer review because the problem is practical, the technical step is new, and the central modeling question is checkable once the full formulation is on the table. Ask the referees to verify the shortest-path enforcement and run a few larger instances if possible.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce a novel integer linear program (ILP) with a custom column-generation procedure that exactly solves the combined capacitated network-design and shortest-path routing problem, where link deactivations are chosen simultaneously with routings that remain shortest paths in the induced active subnetwork. Strengthening cuts are added, and computational experiments on small instances are used to benchmark heuristics, concluding that the simple 'fix routing then deactivate' approach is surprisingly close to optimal while a traffic-oblivious method (TOCA) performs best under multiple demand matrices.

Significance. If the ILP formulation is shown to be correct, the work would be significant: it supplies the first exact benchmark for a practically relevant green-networking problem that prior literature had to decompose or approximate, and the experimental finding that a trivial heuristic is near-optimal would immediately inform engineering practice.

major comments (2)

[ILP formulation and column-generation procedure] The central claim of exact optimality rests on the novel ILP correctly enforcing that every selected path remains a shortest path in the residual capacitated graph after arbitrary link deactivations. The abstract states that the model 'captures dynamically changing shortest paths' but supplies neither the explicit reduced-cost or potential-based constraints nor a proof that these constraints remain valid once capacities are introduced; this modeling step is load-bearing for every optimality guarantee reported later.
[ILP formulation and column-generation procedure] No derivation or small-instance verification is referenced that would confirm the interaction between the capacity constraints and the shortest-path conditions (e.g., that a deactivated link cannot create a cheaper path that violates the chosen routing). Without such a check, reported 'optimal' solutions may be infeasible under true shortest-path semantics.

minor comments (2)

[Abstract] The abstract asserts that 'an in-depth experimental evaluation' is provided, yet no tables, instance sizes, runtimes, or optimality gaps appear in the summary; the reader cannot assess the 'surprisingly close' claim without the actual data.
[Strengthening constraints] The strengthening constraints are described as 'provably' valid, but the abstract does not indicate whether the proof is included in the main text or relegated to an appendix.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and insightful review. The comments correctly identify that the correctness of the ILP is central to the paper's claims, and we appreciate the opportunity to clarify and strengthen the presentation of the formulation. We address each major comment below.

read point-by-point responses

Referee: [ILP formulation and column-generation procedure] The central claim of exact optimality rests on the novel ILP correctly enforcing that every selected path remains a shortest path in the residual capacitated graph after arbitrary link deactivations. The abstract states that the model 'captures dynamically changing shortest paths' but supplies neither the explicit reduced-cost or potential-based constraints nor a proof that these constraints remain valid once capacities are introduced; this modeling step is load-bearing for every optimality guarantee reported later.

Authors: The manuscript body (Section 3) presents the full ILP, which employs a potential-based formulation to enforce the shortest-path property. Node potentials are introduced such that, for every active link, the reduced cost of each routed path is zero, while deactivated links (capacity fixed at zero) are excluded from the residual graph used for shortest-path verification. The column-generation procedure generates only paths satisfying these conditions with respect to the current set of active links and capacities. We agree that the abstract is concise and does not reproduce the constraints or the validity argument. In the revision we will expand the abstract's modeling description and add an explicit proof (in an appendix) showing that the potential constraints continue to guarantee shortest-path semantics after arbitrary deactivations. revision: yes
Referee: [ILP formulation and column-generation procedure] No derivation or small-instance verification is referenced that would confirm the interaction between the capacity constraints and the shortest-path conditions (e.g., that a deactivated link cannot create a cheaper path that violates the chosen routing). Without such a check, reported 'optimal' solutions may be infeasible under true shortest-path semantics.

Authors: We acknowledge that an explicit small-instance verification would improve readability and confidence in the interaction between capacities and shortest-path constraints. In the revised manuscript we will insert a dedicated subsection containing a small hand-solved example network. For this instance we will (i) state the ILP solution, (ii) manually enumerate all possible paths in the induced active subnetwork, and (iii) confirm that the chosen routes are shortest paths under the computed potentials and that no cheaper path appears after any deactivation. This verification will directly address the concern that a deactivated link could create an unintended cheaper route. revision: yes

Circularity Check

0 steps flagged

No circularity: direct ILP modeling of capacitated shortest-path subnetwork design

full rationale

The paper's core contribution is an explicit integer linear program that encodes network design decisions, dynamic shortest-path routing validity after link deactivations, and capacity constraints using standard graph-theoretic primitives (reduced costs, potentials, and flow conservation). Column generation is introduced as a solution technique for the resulting master problem, and strengthening cuts are derived and proven valid from the same external definitions. No equation or claim reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation; correctness is independently checkable against any shortest-path algorithm and capacity oracle. Experimental comparisons evaluate the ILP against heuristic baselines rather than deriving results from the heuristics themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard integer-linear-programming theory and column-generation techniques for path-based routing; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the modeling framework itself.

axioms (1)

standard math Standard assumptions that an ILP solver returns a globally optimal solution when the model is correctly formulated and that column generation converges to the optimal LP relaxation.
Invoked when claiming the algorithm yields true optimal solutions for small instances.

pith-pipeline@v0.9.0 · 5606 in / 1298 out tokens · 35324 ms · 2026-05-08T05:48:16.413211+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

107 extracted references · 30 canonical work pages

[1]

Kodialam, and T

Randeep Bhatia, Fang Hao, Murali S. Kodialam, and T. V. Lakshman. Optimized network traffic engineering using segment routing. In Proc. INFOCOM 2015 , pages 657--665. IEEE , 2015. https://doi.org/10.1109/INFOCOM.2015.7218434 doi:10.1109/INFOCOM.2015.7218434

work page doi:10.1109/infocom.2015.7218434 2015
[5]

Directed capacity-preserving subgraphs: hardness and exact polynomial algorithms

Markus Chimani and Max Ilsen. Directed capacity-preserving subgraphs: hardness and exact polynomial algorithms. Acta Informatica , 62(10), 2025. https://doi.org/10.1007/s00236-024-00475-7 doi:10.1007/s00236-024-00475-7

work page doi:10.1007/s00236-024-00475-7 2025
[8]

The design of survivable directed networks

Geir Dahl. The design of survivable directed networks. Telecommun. Syst. , 2(1):349--377, 1993. https://doi.org/10.1007/BF02109865 doi:10.1007/BF02109865

work page doi:10.1007/bf02109865 1993
[9]

Directed steiner problems with connectivity constraints

Geir Dahl. Directed steiner problems with connectivity constraints. Discret. Appl. Math. , 47(2):109--128, 1993. https://doi.org/10.1016/0166-218X(93)90086-4 doi:10.1016/0166-218X(93)90086-4

work page doi:10.1016/0166-218x(93)90086-4 1993
[12]

Segment routing architecture

Clarence Filsfils, Stefano Previdi, Les Ginsberg, Bruno Decraene, Stephane Litkowski, and Rob Shakir. Segment routing architecture. RFC , 8402:1--32, 2018. https://doi.org/10.17487/RFC8402 doi:10.17487/RFC8402

work page doi:10.17487/rfc8402 2018
[15]

A variable fixing heuristic with local branching for the fixed charge uncapacitated network design problem with user-optimal flow

Pedro Henrique Gonz \' a lez, Luidi Simonetti, Philippe Michelon, Carlos Alberto de Jesus Martinhon, and Edcarllos Santos. A variable fixing heuristic with local branching for the fixed charge uncapacitated network design problem with user-optimal flow. Comput. Oper. Res. , 76:134--146, 2016. https://doi.org/10.1016/J.COR.2016.06.016 doi:10.1016/J.COR.2016.06.016

work page doi:10.1016/j.cor.2016.06.016 2016
[16]

Solving segment routing problems with hybrid constraint programming techniques

Renaud Hartert, Pierre Schaus, Stefano Vissicchio, and Olivier Bonaventure. Solving segment routing problems with hybrid constraint programming techniques. In Proc. CP 2015 , volume 9255 of LNCS , pages 592--608. Springer, 2015. https://doi.org/10.1007/978-3-319-23219-5_41 doi:10.1007/978-3-319-23219-5_41

work page doi:10.1007/978-3-319-23219-5_41 2015
[18]

Network utilization: The flow view

Avinatan Hassidim, Danny Raz, Michal Segalov, and Ariel Shaqed. Network utilization: The flow view. In Proc.\ INFOCOM , pages 1429--1437. IEEE , 2013. https://doi.org/10.1109/INFCOM.2013.6566937 doi:10.1109/INFCOM.2013.6566937

work page doi:10.1109/infcom.2013.6566937 2013
[26]

Predictive traffic engineering with 2-segment routing considering requirements of a carrier IP network

Timmy Sch \" u ller, Nils Aschenbruck, Markus Chimani, Martin Horneffer, and Stefan Schnitter. Predictive traffic engineering with 2-segment routing considering requirements of a carrier IP network. In Proc.\ LCN , pages 667--675, 2017

2017
[27]

Construction of minimum-weight spanners

Mikkel Sigurd and Martin Zachariasen. Construction of minimum-weight spanners. In Proc.\ ESA 2004 , Lecture Notes in Computer Science, pages 797--808. Springer, 2004. https://doi.org/10.1007/978-3-540-30140-0_70 doi:10.1007/978-3-540-30140-0_70

work page doi:10.1007/978-3-540-30140-0_70 2004
[29]

Journal of Mathematical Analysis and Applications , volume =

Richard James Duffin , title =. Journal of Mathematical Analysis and Applications , volume =. 1965 , issn =

1965
[30]

Tarjan and Eugene L

Jacobo Valdes and Robert E. Tarjan and Eugene L. Lawler , title =. 1982 , doi =

1982
[31]

Traffic engineering using segment routing and considering requirements of a carrier

Timmy Sch. Traffic engineering using segment routing and considering requirements of a carrier. Proc. 2017 , doi =

2017
[32]

Traffic Engineering Using Segment Routing and Considering Requirements of a Carrier

Timmy Sch. Traffic Engineering Using Segment Routing and Considering Requirements of a Carrier. 2018 , doi =

2018
[33]

Garey and David S

Michael R. Garey and David S. Johnson , title =. 1979 , isbn =

1979
[34]

1974 , doi =

Sartaj Sahni , title =. 1974 , doi =

1974
[35]

Tarjan , title =

Robert E. Tarjan , title =. 1972 , doi =

1972
[36]

Adrian Vetta , title =. Proc. 2001 , url =

2001
[37]

Piotr Berman and Bhaskar DasGupta and Marek Karpinski , title =. Proc. 2009 , doi =

2009
[38]

Young , title =

Samir Khuller and Balaji Raghavachari and Neal E. Young , title =. 1995 , doi =

1995
[39]

Liang Zhao and Hiroshi Nagamochi and Toshihide Ibaraki , title =. Inf. Process. Lett. , volume =. 2003 , doi =

2003
[40]

The minimum spanning strong subdigraph problem is fixed parameter tractable , journal =

J. The minimum spanning strong subdigraph problem is fixed parameter tractable , journal =. 2008 , doi =

2008
[41]

Aho and Michael R

Alfred V. Aho and Michael R. Garey and Jeffrey D. Ullman , title =. 1972 , doi =

1972
[42]

Richey and R

Michael B. Richey and R. Gary Parker and Ronald L. Rardin , title =. Networks , volume =. 1985 , doi =

1985
[43]

Young , title =

Samir Khuller and Balaji Raghavachari and Neal E. Young , title =. Inf. Process. Lett. , volume =. 1994 , doi =

1994
[44]

Irit Dinur and David Steurer , title =. Proc. 2014 , doi =

2014
[45]

Theory Comput

Dana Moshkovitz , title =. Theory Comput. , volume =. 2015 , doi =

2015
[46]

Karp , title =

Richard M. Karp , title =. Proc. 1972 , doi =

1972
[47]

Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs , journal =

Andr. Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs , journal =. 2015 , doi =

2015
[48]

Spielman and Shang

Daniel A. Spielman and Shang. Spectral Sparsification of Graphs , journal =. 2011 , doi =

2011
[49]

2020 , howpublished=

Cisco , title =. 2020 , howpublished=

2020
[50]

Capacity Trends and Limits of Optical Communication Networks , journal =

Ren. Capacity Trends and Limits of Optical Communication Networks , journal =. 2012 , doi =

2012
[51]

The Lockdown Effect: Implications of the

Anja Feldmann and Oliver Gasser and Franziska Lichtblau and Enric Pujol and Ingmar Poese and Christoph Dietzel and Daniel Wagner and Matthias Wichtlhuber and Juan Tapiador and Narseo Vallina. The Lockdown Effect: Implications of the. 2020 , doi =

2020
[52]

Wong , title =

Richard T. Wong , title =. Networks , volume =. 2021 , doi =

2021
[53]

Mingui Zhang and Cheng Yi and Bin Liu and Beichuan Zhang , title =. Proc. 2010 , doi =

2010
[54]

Luca Chiaraviglio and Marco Mellia and Fabio Neri , title =. Proc. 2009 , doi =

2009
[55]

On Sparse Spanners of Weighted Graphs , journal =

Ingo Alth. On Sparse Spanners of Weighted Graphs , journal =. 1993 , doi =

1993
[56]

Piotr Berman and Arnab Bhattacharyya and Konstantin Makarychev and Sofya Raskhodnikova and Grigory Yaroslavtsev , title =. Inf. Comput. , volume =. 2013 , doi =

2013
[57]

Vertex Sparsifiers: New Results from Old Techniques , journal =

Matthias Englert and Anupam Gupta and Robert Krauthgamer and Harald R. Vertex Sparsifiers: New Results from Old Techniques , journal =. 2014 , doi =

2014
[58]

2019 , doi =

Michael Elkin and Ofer Neiman , title =. 2019 , doi =

2019
[59]

Random Struct

Surender Baswana and Sandeep Sen , title =. Random Struct. Algorithms , volume =. 2007 , doi =

2007
[60]

Kobourov and Richard Spence , title =

Abu Reyan Ahmed and Greg Bodwin and Faryad Darabi Sahneh and Keaton Hamm and Mohammad Javad Latifi Jebelli and Stephen G. Kobourov and Richard Spence , title =. Comput. Sci. Rev. , volume =. 2020 , doi =

2020
[61]

CoRR , volume =

Reid Andersen and Uriel Feige , title =. CoRR , volume =. 2009 , eprinttype =. 0907.3631 , timestamp =

work page arXiv 2009
[62]

Minimizing Congestion in General Networks , booktitle =

Harald R. Minimizing Congestion in General Networks , booktitle =. 2002 , doi =

2002
[63]

Optimal hierarchical decompositions for congestion minimization in networks , booktitle =

Harald R. Optimal hierarchical decompositions for congestion minimization in networks , booktitle =. 2008 , doi =

2008
[64]

Kleinberg and Harald R

Mohammad Taghi Hajiaghayi and Robert D. Kleinberg and Harald R. Oblivious routing on node-capacitated and directed graphs , journal =. 2007 , doi =

2007
[65]

Amir Abboud and Robert Krauthgamer and Ohad Trabelsi , title =. Proc. 2022 , doi =

2022
[66]

Torben Hagerup and Jyrki Katajainen and Naomi Nishimura and Prabhakar Ragde , title =. J. Comput. Syst. Sci. , volume =. 1998 , doi =

1998
[67]

Ankur Moitra , title =. Proc. 2009 , doi =

2009
[68]

Frank Thomson Leighton and Ankur Moitra , title =. Proc. 2010 , doi =

2010
[69]

Alexandr Andoni and Anupam Gupta and Robert Krauthgamer , title =. Proc. 2014 , doi =

2014
[70]

Approximating

Andr. Approximating. Proc. 1996 , doi =

1996
[71]

Cohen and Jonathan A

Michael B. Cohen and Jonathan A. Kelner and John Peebles and Richard Peng and Anup B. Rao and Aaron Sidford and Adrian Vladu , title =. Proc. 2017 , doi =

2017
[72]

Cohen and Jonathan A

Michael B. Cohen and Jonathan A. Kelner and Rasmus Kyng and John Peebles and Richard Peng and Anup B. Rao and Aaron Sidford , title =. Proc. 2018 , doi =

2018
[73]

Ruoxu Cen and Yu Cheng and Debmalya Panigrahi and Kevin Sun , title =. Proc. 2021 , doi =

2021
[74]

Ying Zhang and Zhiqiang Zhao and Zhuo Feng , title =. Proc. 2019 , doi =

2019
[75]

Kamal Jain , title =. Comb. , volume =. 2001 , doi =

2001
[76]

Geir Dahl , title =. Discret. Appl. Math. , volume =. 1993 , doi =

1993
[77]

Telecommun

Geir Dahl , title =. Telecommun. Syst. , volume =. 1993 , doi =

1993
[78]

Design of Survivable Networks:

Herv. Design of Survivable Networks:. Networks , volume =. 2005 , doi =

2005
[79]

2023 , eprint=

Markus Chimani and Max Ilsen , title=. 2023 , eprint=

2023
[80]

2025 , doi =

Markus Chimani and Max Ilsen , title =. 2025 , doi =

2025
[81]

Vazirani , title =

Vijay V. Vazirani , title =. 2001 , url =

2001
[82]

Williamson and David B

David P. Williamson and David B. Shmoys , title =. 2011 , url =

2011
[83]

Daniel Otten and Max Ilsen and Markus Chimani and Nils Aschenbruck , title =. Proc. 2023 , doi =

2023
[84]

2018 , doi =

Clarence Filsfils and Stefano Previdi and Les Ginsberg and Bruno Decraene and Stephane Litkowski and Rob Shakir , title =. 2018 , doi =

2018
[85]

Renaud Hartert and Pierre Schaus and Stefano Vissicchio and Olivier Bonaventure , title =. Proc. 2015 , doi =

2015
[86]

Kodialam and T

Randeep Bhatia and Fang Hao and Murali S. Kodialam and T. V. Lakshman , title =. Proc. 2015 , doi =

2015
[87]

Markus Chimani and Max Ilsen , title =. Proc. 2023 , doi =

2023
[88]

Algorithms for a network design problem with crossing supermodular demands , journal =

Vardges Melkonian and. Algorithms for a network design problem with crossing supermodular demands , journal =. 2004 , doi =

2004
[89]

Ahuja and Thomas L

Ravindra K. Ahuja and Thomas L. Magnanti and James B. Orlin , title =. 1993 , isbn =

1993
[90]

Khodakaram Salimifard and Sara Bigharaz , title =. Oper. Res. , volume =. 2022 , doi =

2022
[91]

Telecommunications Network Planning , year=

Gendron, Bernard and Crainic, Teodor Gabriel and Frangioni, Antonio , title=. Telecommunications Network Planning , year=
[92]

2014 , doi =

Bernard Gendron and Mathieu Larose , title =. 2014 , doi =

2014
[93]

Transportation Research Part B: Methodological , volume =

A multi-commodity flow network design problem , author =. Transportation Research Part B: Methodological , volume =. 1981 , issn =

1981
[94]

Sibel Salman and Joseph Cheriyan and R

F. Sibel Salman and Joseph Cheriyan and R. Ravi and S. Subramanian , title =. Proc. 1997 , url =

1997
[95]

Spyridon Antonakopoulos , title =. Proc. 2010 , doi =

2010
[96]

Salavatipour , title =

Chandra Chekuri and Mohammad Taghi Hajiaghayi and Guy Kortsarz and Mohammad R. Salavatipour , title =. 2010 , doi =

2010
[97]

New economical virtual private networks , journal =

Walid Ben. New economical virtual private networks , journal =. 2003 , doi =

2003
[98]

Optimization and Engineering , volume =

Routing of uncertain traffic demands , author =. Optimization and Engineering , volume =. 2005 , publisher =

2005

Showing first 80 references.