Robust Dynamic Edge Service Placement Under Spatio-Temporal Correlated Demand Uncertainty

Duong Thuy Anh Nguyen; Duong Tung Nguyen; Jiaming Cheng

REVIEW 3 major objections 6 minor 30 references

Robust Dynamic Edge Service Placement Under Spatio-Temporal Correlated Demand Uncertainty

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that a two-stage robust edge service placement problem with integer recourse variables can be solved exactly by an iterative decomposition algorithm with finite convergence, and that dynamic spatio-temporal uncertainty…

desk verdict Genuinely new model combining dynamic placement with an autoregressive uncertainty set, but the finite exact convergence claim for ROD is not proven because the worst-case MILP-recourse demand need not be an extreme point of D2. read the letter →

arxiv 2412.15608 v1 pith:FRNCNWA7 submitted 2024-12-20 math.OC cs.SYeess.SY

Jiaming Cheng , Duong Thuy Anh Nguyen , Duong Tung Nguyen This is my paper · ORCID

classification math.OCcs.SYeess.SY MSC 90C1190C1590C4790C90

keywords edgeserviceplacementadjustablerobustoptimizationintegerrecoursedynamicuncertaintysetspatio-temporalcorrelationworkloadallocationcolumn-and-constraintgenerationfiniteconvergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that edge service providers can reserve computing resources for an entire planning horizon while keeping the right to move services and buy or sell capacity after demand is observed, and that this flexibility is worth the added computational difficulty. The authors formulate the problem as a two-stage multi-period adjustable robust optimization with binary service-placement variables in the second stage, producing a tri-level min-max-min program with integer recourse. They prove that their iterative decomposition algorithm, ROD, converges to an exact optimal solution in a finite number of iterations. If the claim holds, service providers can use spatio-temporal demand correlations to shrink the uncertainty set and lower operating costs without sacrificing worst-case service guarantees.

What carries the argument

The two load-bearing objects are the dynamic uncertainty set $\mathcal{D}_2$ and the ROD decomposition. $\mathcal{D}_2$ models the demand deviation at each access point as a multivariate auto-regressive process of order $L$ with a spatial coupling term, so the set of possible future demands shrinks as historical deviations are revealed; this is what makes the robust solution less conservative. ROD splits the tri-level problem into an outer column-and-constraint generation loop over the extreme points of $\mathcal{D}_2$ and an inner loop that handles the binary service-placement and download variables. The inner loop's master problem is reformulated into a single-level MILP using strong duality for the continuous recourse and McCormick envelopes for the bilinear products between dual variables and demand deviations. The convergence argument rests on $\mathcal{D}_2$ being a compact convex polytope with finitely many extreme points: any repeated extreme point forces the upper and lower bounds to coincide.

What would settle it

Solve a one-period, one-EN, two-AP instance of the inner problem with $\Gamma=1$, so the uncertainty set is a line segment between two extreme demands, and place the service-placement cost so that placing or not placing the service is optimal exactly at an interior demand value; if the worst-case value occurs at that interior point rather than at an endpoint, ROD's repeated-extreme-point argument no longer applies to that instance.

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Extended reading notes

Core claim

The central claim is that a two-stage robust edge service placement problem with integer recourse variables can be solved exactly, not just approximately, by decomposing it into an outer loop over extreme demand scenarios and an inner loop that iteratively enumerates the binary placement decisions that matter. Given first-stage resource reservation $s$, the inner loop alternates between a mixed-integer subproblem that fixes the worst-case demand and a master problem that finds the next worst-case demand by strong duality and McCormick linearization; when the inner loop finishes it returns a true worst-case demand and cost for that $s$. The outer loop then adds this demand as a cut to the reservation master problem and repeats. The paper proves this terminates finitely, bounded by the number of extreme points of the dynamic uncertainty set, and that at termination the reservation decision is optimal for the original tri-level problem. The paper further claims that using a dynamic uncertainty set built from an auto-regressive model of demand, rather than a static polyhedral set, reduces conservatism and total cost in numerical experiments with real demand traces.

Load-bearing premise

The worst-case demand for the second-stage problem with binary service-placement decisions is assumed to occur at an extreme point of the dynamic uncertainty set, even though the second-stage value function is not convex in demand.

Editorial extensions

If this is right

Service providers can plan day-ahead resource reservations that are provably optimal under the modeled uncertainty, not just heuristic.
Dynamic service placement, where services are downloaded from nearby edge nodes after demand is known, becomes computationally tractable despite binary recourse variables.
Using spatio-temporal correlations systematically reduces the uncertainty set and lowers total cost relative to static uncertainty sets at the same robustness budget.
The algorithm's runtime depends on network size and uncertainty budget, and the paper reports that a looser optimality gap for large networks cuts runtime substantially.
The model extends to multiple resource types and additional placement constraints, as the authors note.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication the authors do not pursue is that the same outer/inner decomposition applies to other two-stage robust problems with binary recourse, such as facility location or network function placement, whenever the uncertainty set is a polytope with integer budget.
The extreme-point assumption before eq. (26) is the step we would test first: if a worst-case demand can occur inside a face of $\mathcal{D}_2$ when the binary placement decision changes there, the finite-convergence proof would need an additional argument beyond repeated extreme points.
One could calibrate $\mathcal{D}_2$ online from streaming demand and re-run ROD at each re-optimization horizon; the convergence guarantee would still hold per instance, and the value of updating the auto-regressive coefficients could be measured empirically.
A direct comparison with distributionally robust or stochastic benchmarks would quantify how much of the cost saving comes from the correlation structure versus from the integer-recourse dynamic placement itself; the paper separates these only partially through the D-ARO-SUS benchmark.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this is a genuinely new model wrapped around an exactness claim that does not actually hold as proven. The model contribution is real; the algorithm proof is the soft spot.

What's actually new: dynamic service placement with binary recourse inside a two-stage robust optimization with an autoregressive spatio-temporal uncertainty set. I don't know of prior work combining those two. The cost model (reservation, buy-more/sell-back, placement, download, delay) is sensible, and the time series estimation from Shanghai Telecom traces is a solid data-driven touch. The numerical comparisons against static uncertainty and static placement show consistent cost reductions, so the empirical case is at least suggestive.

Where it gets shaky: the proof of finite exact convergence (Prop. IV.2) depends on the worst-case demand being an extreme point of D2. That's true when the second-stage value function is convex in λ, which is the continuous-recourse case. Here the second stage has binary variables, so the value function is the minimum of finitely many LP value functions and is generally nonconvex. A global maximum of a nonconvex function over a polytope can lie strictly inside a face, so the enumeration over extreme points in (18) can miss it. The citation to [4] does not cover integer recourse. This is not a cosmetic gap; it is the load-bearing part of the claimed exactness. Also, the McCormick relaxation in the inner master (the set Mδ,g,σ) is written incorrectly for δ = σg, and the text treats g as binary at one point while D2 defines it as continuous in [-1,1]. That needs cleanup.

One more thing: the paper does not appear to be circular. AR parameters come from separate historical data, baselines are compared fairly, and no parameter is tuned to produce the savings. The runtime analysis is honest.

Overall: this is a paper for the robust-optimization and edge-computing community. The model will attract citations if the proof is repaired. Right now, my recommendation is to send it to peer review with a clear signal that the exactness theorem must either be fixed or downgraded to 'practical convergence.' If the authors can prove the extreme-point property under a restricted class (or handle interior optima with a valid-inequality strategy), the paper would be solid. As is, the abstract overstates.

Referee Report

3 major / 6 minor

Summary. The paper proposes a two-stage multi-period robust optimization model for edge service placement and resource reservation under demand uncertainty. The first stage decides resource reservations before demand is known, while the second stage chooses dynamic service placement, service download, resource adjustment, and workload allocation after observing demand. The uncertainty is modeled by a dynamic set D2 that incorporates spatio-temporal correlation through a multivariate autoregressive structure. Because the second-stage problem contains binary recourse variables, the paper develops an iterative decomposition algorithm called ROD, which alternates between an outer master problem and an inner max-min subproblem, and claims finite convergence to an exact optimal solution of the tri-level problem P1. Numerical experiments compare the proposed D-ARO-DUS model against static uncertainty-set and static placement benchmarks.

Significance. The modeling direction is relevant and timely: dynamic service placement with integer recourse and a data-driven spatio-temporal uncertainty set addresses a practically important gap in edge service placement. The numerical study is extensive, uses real traffic traces, reports runtimes, and includes sensitivity analyses. However, the paper's central theoretical contribution, namely the claim that ROD converges finitely to an exact optimum, is not established. The proof relies on an extreme-point optimality property that is false in general for mixed-integer recourse, and the McCormick linearization used in the inner master problem is mis-specified. If the exactness claim cannot be repaired, the algorithm remains a heuristic, and the paper's main advertised advantage over existing decomposition methods is unsupported.

major comments (3)

[Section IV-C, equations after (26) and (27i)] The assertion that 'the worst-case scenarios can be achieved when g is either -1 or 1 [4]' is not valid for the mixed-integer recourse problem considered here. For fixed first-stage decisions, the inner value function Q(ŝ,λ) = min_{q,z,x,y} cost is the minimum of finitely many convex LP value functions and is therefore generally nonconvex in λ. Maximizing a nonconvex function over a polytope need not attain its maximum at an extreme point. Monotonicity in λ does not rescue the argument: a nondecreasing concave function such as Q(λ)=min{λ1,λ2}, which can arise as a binary-choice recourse value function, attains its maximum at the interior point (1/2,1/2) on the face conv{(1,0),(0,1)}. The citation to [4] concerns continuous recourse, not integer recourse. Consequently, the equivalence between P1 and the finite enumeration in (18) is not established, and the finite-convergence proof of Proposition IV.2, which assumes that each iteration generates a new extreme point of D2, does not cover the true worst case. This is a load-bearing gap for the paper's main exactness claim.
[Section IV-E, proof of Proposition IV.2] The displayed set M(δ,g,σ) is claimed to be the convex hull of the bilinear set {(δ,g,σ): δ=σg, g∈[-1,1], σ∈[0,M]}, but the inequalities given do not describe that convex hull. For example, with g=1 and σ=1, the constraints δ ≥ -σ - M(1-g) and δ ≤ -σ + M(1-g) force δ = -1, whereas the true relation δ=σg gives δ = 1. Similarly, for g=-1 the constraints do not force δ=-σ. The standard McCormick convex hull for this set includes the constraints δ ≥ M g + σ - M and δ ≤ M g - σ + M, among others; the paper's formulation is not equivalent. Since Inner-MP in (27) relies on this linearization to produce an MILP reformulation of Q(ŝ), the inner master problem is not a valid exact reformulation, and the upper bounds produced by Step 2 of Algorithm 2 are not trustworthy. This error directly affects the numerical results produced by the ROD algorithm.
[Section IV-E, proof of Proposition IV.2] The proof of Proposition IV.2 states that 'D2 is a finite set with K elements' and bounds the number of iterations by K, the number of extreme points of D2. This is a category error: D2 is a convex polytope with infinitely many points; it has only finitely many extreme points. More importantly, the proof assumes without support that the λ produced by the inner loop is always an extreme point of D2. If the true worst-case demand is interior, the algorithm may add cuts for non-extreme points that do not appear in the enumerated set D*, and the finite-repetition argument cannot be applied. Thus, even if the extreme-point property held for the continuous-relaxation case, the proof as written would still not establish finite exact convergence for the integer-recourse problem.

minor comments (6)

[Section V-B, Eq. (33b)] Equation (33b) appears to have two typos: the term should be Σ_{s=1}^L A_s^i λtilde_i^{t-s}, not Σ_{s=1}^L A_s^i λtilde_i^t, and the B_i g_i^t term that appears in (16b) is missing here. Please align the two definitions.
[Section IV-C, Eq. (23d)] In the Inner-SP constraints, the demand constraint should sum workload over edge nodes j, i.e., x_{i,0}^t + Σ_j x_{i,j}^t ≥ λ_i^{t,*}; the displayed expression sums over i, which is inconsistent with (10).
[Section III-A, paragraph after Fig. 2] The sentence 'the network delay between AP i and the cloud is denoted bywe di,0' contains a typo ('bywe').
[Section II and Introduction] Several citations use 'el at.' instead of 'et al.', for example in the sentences citing [8], [9], [11], and [18]; the reference list itself should also be checked for consistent formatting.
[Section IV-B, Eq. (19c)] The first-stage constraints are indexed by l in (19c), although s is a single first-stage decision common to all cuts. This makes the formulation ambiguous; either the index should be removed or the notation should be clarified so that s is not replicated per cut.
[Section V-D, Figs. 11(a)-(b)] The comparison between D-ARO-DUS and D-ARO-SUS in the α-sweep is not fully explained: it is unclear how the manually adjusted deviation α relates to the parameters of the dynamic uncertainty set, and whether the comparison holds the amount of information used by both models equal. A short explanation would improve reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the model, algorithm, and benchmarks are self-contained, and the notable weakness (extreme-point optimality for integer recourse) is an unsupported proof step rather than a circular dependency.

full rationale

The paper's central derivation chain is not circular. The DUS parameters (A_i^s, B_i) are estimated from Shanghai Telecom traces via AR/regression, not from the robust model's outputs, so the reported cost comparison with SUS is not a fitted-input-called-prediction. The ROD algorithm is tested against independent baselines (D-ARO-SUS, S-ARO from [3]) and no parameter is tuned to force the headline savings. The one load-bearing step that warrants scrutiny is the assertion before Eq. (26) that "the worst-case scenarios can be achieved when g is either -1 or 1 [4]" and the resulting equivalence of P1 to the finite extreme-point problem (18). For the second-stage MILP recourse (binary z and q), the value function is generally nonconvex in lambda, so the extreme-point property does not follow from convexity/compactness of D2 and is not established for integer recourse; the citation to [4] (a same-author prior work without binary recourse) does not fill this gap. This is a genuine correctness/proof gap in the finite-exact-convergence argument (Prop. IV.2), but it is not circular: the paper does not define its conclusion into existence or fit a parameter to the target result. Thus the circularity score is low.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central proof relies on the extreme-point property and on the exactness of the McCormick linearization; the remaining items are data-fitting and economic assumptions. No new entities are postulated.

free parameters (4)

Uncertainty budget Γ1 = 5 (default)
User-specified robustness parameter; the paper varies it in sensitivity analysis to control conservatism.
AR order L
Time lag order chosen for the autoregressive model in eq. (16b); the selected value used in experiments is not reported.
AR coefficients A_i^s and B_i
Estimated from the Shanghai Telecom dataset using multivariate time-series tools; fitted values and standard errors are not reported.
Forecast coefficients φ1...φ5
Estimated by linear regression for the seasonal demand forecast in eq. (34); fitted values are not reported.

assumptions (5)

ad hoc to paper Worst-case demand for the second-stage MILP occurs at an extreme point of D2, with g_i^t at its bounds.
Stated before eq. (26) citing [4]. This is load-bearing for finite convergence but is not generally true for nonconvex recourse value functions.
ad hoc to paper The McCormick inequalities in the M_{δ,g,σ} set give the exact convex hull of δ = σ g.
The inequalities as written do not match the standard McCormick envelope; this can make the inner master a mis-specified model.
domain assumption Demand deviations follow the AR process in eq. (16b) with scalar B_i, and this captures spatio-temporal correlation.
The recursion lacks cross-AP terms; the budget constraint couples APs only by count, and the covariance-based Cholesky derivation is not reflected in the equation.
standard math Strong duality holds for the inner LP after fixing integer variables.
Used to derive the dual in eq. (26); valid for linear programs under standard regularity.
domain assumption Prices satisfy a <= p <= e to prevent arbitrage.
Assumed in Section III-A and used to make the resource adjustment model economically meaningful.

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Cite this review

Pith. "Pith review of Robust Dynamic Edge Service Placement Under Spatio-Temporal Correlated Demand Uncertainty." pith.science (2026). https://pith.science/paper/FRNCNWA7

@misc{pith2026241215608,
  author       = {Pith},
  title        = {Pith review of: Robust Dynamic Edge Service Placement Under Spatio-Temporal Correlated Demand Uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRNCNWA7}},
  note         = {Machine review of arXiv:2412.15608}
}

read the original abstract

Edge computing allows Service Providers (SPs) to enhance user experience by placing their services closer to the network edge. Determining the optimal provisioning of edge resources to meet the varying and uncertain demand cost-effectively is a critical task for SPs. This paper introduces a novel two-stage multi-period robust model for edge service placement and workload allocation, aiming to minimize the SP's operating costs while ensuring service quality. The salient feature of this model lies in its ability to enable SPs to utilize dynamic service placement and leverage spatio-temporal correlation in demand uncertainties to mitigate the inherent conservatism of robust solutions. In our model, resource reservation is optimized in the initial stage, preemptively, before the actual demand is disclosed, whereas dynamic service placement and workload allocation are determined in the subsequent stage, following the revelation of uncertainties. To address the challenges posed by integer recourse variables in the second stage of the resulting tri-level adjustable robust optimization problem, we propose a novel iterative, decomposition-based approach, ensuring finite convergence to an exact optimal solution. Extensive numerical results are provided to demonstrate the efficacy of the proposed model and approach.

Figures

Figures reproduced from arXiv: 2412.15608 by the authors.

**Figure 1.** Resource trading model Our proposed model aims to assist the SP in optimizing resource reservation under uncertainty. Besides resource trading, the SP must optimize service placement, assuming the service is consistently available in the cloud. Let the binary parameter z 0 j signify the initial service placement status at EN j. We [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** System model introduce the binary variable z t j , which equals one when the SP places the service onto EN j at time t. This allows the SP to adapt its decisions based on actual demand realization. We also define the binary variable q t m,j to indicate whether the service is downloaded from the source EN m to the destination EN j at time t. The binary variable q t 0,j indicates if the service is downloaded from the … view at source ↗

**Figure 3.** Uncertainty comparison The error terms in the model, represented by the vector [ϵ t 1 , . . . , ϵt I ] T, are independent across different time slots and follow a normal distribution with mean 0 and covariance matrix Σ. Similar to SUS, Γ1 controls the size of uncertainty and influences the robustness of the optimal solution, while the parameter g t i governs the maximum forecast error relative to the forecast demand… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Next, we prove that, given first-stage variable [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 4.** Figure 4: Flow diagram of the ROD algorithm Proof. This proposition can be shown through contradiction, where the repetition of any λ ∗ implies UBinner = LBinner. Let λ ∗ represent the worst-case uncertainty at iteration r for InnerMP, and let the optimal decisions obtained by …

**Figure 5.** Figure 5: Time series analysis C. Sensitivity Analysis This section assesses the impact of key system parameters on the optimal solution. These parameters include the uncertainty factor Γ, demand deviation α, and delay penalty ρ. To understand the influence of varying cost para…

**Figure 6.** Figure 6: Total cost: varying ρ and Γ 2) Impacts of cost parameters: Fig.7(a) - 7(c) demonstrate the impacts of resource reservation cost (p) and resource adjustment cost (e). Initially, the total cost and payment rise and then reach a plateau after a certain point. This trend o…

**Figure 7.** Figure 7: The impact of cost parameters 3) Convergence and running time analysis of ROD: In our simulations, ϵ1 and ϵ2 denote the optimality gaps for the outer and inner loop problems, respectively, both set at 0.1%. Firstly, we examine the convergence properties of our proposed…

**Figure 8.** Figure 8: Convergence property For our runtime analysis, we calculate the average computational time across 100 different problem instances for each problem size. Table III shows the time taken by our proposed model for various problem sizes and uncertain budget Γ. The increase…

**Figure 9.** Figure 9: The impact of the cost parameters Unlike the static model, our dynamic model allows the SP to postpone immediate decisions on service download locations. This increased flexibility empowers the SP to exploit favorable conditions, such as lower h costs, and choose to do…

**Figure 10.** Figure 10: The impacts of cost parameter Figs. 10(c) - 10(d) illustrate that the S-ARO model incurs no cost for resource adjustment, as the SP is required to reserve all necessary resources prior to the operational stage. Notably, even when the on-spot price e is similar to the …

**Figure 11.** Figure 11: The impacts of uncertain parameters VI. CONCLUSIONS This paper introduced a new two-stage multi-period dynamic edge service placement framework designed to optimize edge service provisioning and allocation decisions. The framework operates in two stages: the first sta…

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User allocation-aware edge cloud placement in mobile edge computing,

Y . Guo, S. Wang, A. Zhou, J. Xu, J. Yuan, and C.-H. Hsu, “User allocation-aware edge cloud placement in mobile edge computing,” Soft- ware: Practice and Experience , vol. 50, no. 5, pp. 489–502, 2020

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G. C. Reinsel, Elements of multivariate time series analysis . Springer Science & Business Media, 1997. 1 VII. A PPENDIX A. Static ARO (S-ARO) We consider static adjustable robust optimization (S-ARO) over a time horizon, which is the proposed model from previous work in [3]. ...

1997

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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J. Cheng, D. T. Nguyen, and V . K. Bhargava, “Resilient edge service placement under demand and node failure uncertainties,” IEEE Trans. Netw. Serv. Manag., pp. 1–1, 2023

work page 2023

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Edge computing: Vision and challenges,

W. Shi, J. Cao, Q. Zhang, Y . Li, and L. Xu, “Edge computing: Vision and challenges,” IEEE Internet Things J. , vol. 3, no. 5, pp. 637–646, 2016

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Ben-Tal, L

A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust optimization . Princeton university press, 2009

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Two-stage robust edge service placement and sizing under demand uncertainty,

D. T. Nguyen, H. T. Nguyen, N. Trieu, and V . K. Bhargava, “Two-stage robust edge service placement and sizing under demand uncertainty,” IEEE Internet Things J. , vol. 9, no. 2, pp. 1560–1574, 2022

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A short-term traffic flow prediction method based on spatial–temporal correlation using edge computing,

X. Yu, L. Sun, Y . Yan, and G. Liu, “A short-term traffic flow prediction method based on spatial–temporal correlation using edge computing,” Computers & Electrical Engineering , vol. 93, p. 107219, 2021

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Cloudlstm: A recurrent neural model for spatiotemporal point-cloud stream forecasting,

C. Zhang, M. Fiore, I. Murray, and P. Patras, “Cloudlstm: A recurrent neural model for spatiotemporal point-cloud stream forecasting,” in Proceedings of the AAAI Conference on Artificial Intelligence , vol. 35, no. 12, 2021, pp. 10 851–10 858

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Solving two-stage robust optimization problems using a column-and-constraint generation method,

B. Zeng and L. Zhao, “Solving two-stage robust optimization problems using a column-and-constraint generation method,” Operations Research L., vol. 41, no. 5, pp. 457–461, 2013

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A scalable algorithm for the placement of service function chains,

M. Mechtri, C. Ghribi, and D. Zeghlache, “A scalable algorithm for the placement of service function chains,” IEEE Trans. Netw. Serv. Manag. , vol. 13, no. 3, pp. 533–546, 2016

work page 2016