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arxiv: 2605.12812 · v1 · submitted 2026-05-12 · 💻 cs.DS · cs.MA

Recognition: unknown

Time and Supply Fairness in Electricity Distribution using k-times bin packing

Dinesh Kumar Baghel , Alex Ravsky , Erel Segal-Halevi
Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:25 UTC · model grok-4.3

classification 💻 cs.DS cs.MA
keywords k-times bin packingfair electricity divisionegalitarian allocationbin packing approximationconnection time fairnesshousehold demand scheduling
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The pith

Every electricity division problem can be solved exactly by k-times bin packing with k depending only on the number of households.

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

The authors define k-times bin packing as the task of packing each item into exactly k separate bins. They prove that any fair electricity allocation among households can be reduced to this problem using a finite k set solely by the household count. Generalized First-Fit and First-Fit Decreasing algorithms are shown to produce more egalitarian connection-time allocations than prior heuristics when tested on real demand data. A related problem of equalizing minimum watts per household admits no such bounded k, leading the authors to introduce four separate heuristics.

Core claim

We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. We implement generalizations of the First-Fit and First-Fit Decreasing algorithms and show they outperform existing heuristics for egalitarian allocation of connection time on real data.

What carries the argument

k-times bin-packing (kBP), the variant in which each item must appear in exactly k distinct bins whose sizes sum to at most the fixed capacity.

If this is right

  • Generalized First-Fit and First-Fit Decreasing algorithms solve kBP with provable performance ratios.
  • These algorithms produce higher minimum connection times across households than earlier heuristics on real electricity data.
  • No finite k depending only on the number of agents exists for egalitarian watt allocation.
  • Four new heuristics can be used to maximize the summed minimum watts allocated each hour.

Where Pith is reading between the lines

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

  • The reduction shows that repeated-use packing problems can encode fairness constraints that single-use packing cannot.
  • The impossibility for watt allocation implies that watt-based fairness will require parameters that grow with demand magnitudes rather than household count alone.
  • The same modeling technique could apply to other repeated-resource problems such as time-shared bandwidth or shared vehicle routing.

Load-bearing premise

Household electricity demands can be represented as fixed item sizes whose sums fit the bin capacity model without changing the egalitarian fairness criteria, and the required k stays independent of the specific demand values.

What would settle it

A concrete collection of household demand values for a fixed number of households where the smallest k that achieves exact egalitarian allocation grows without bound or fails to exist.

Figures

Figures reproduced from arXiv: 2605.12812 by Alex Ravsky, Dinesh Kumar Baghel, Erel Segal-Halevi.

Figure 1
Figure 1. Figure 1: The three blue points are (1,1,0), (1,0,1) and (0,1,1). The black line that originates from the origin [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The optimal numbers OP T(Dk) of bins and numbers of bins used by F F k and F F Dk algorithms bins are shown at the x- and y-axis, respectively. The data points in the legend show the upper bound that can be hypothesized for OP T(Dk) = k ·OP T(D). Each subplot’s data points display the number of different output bins corresponding to each conjectured upper bound. Note that the output bins convey the conject… view at source ↗
Figure 3
Figure 3. Figure 3: The optimal numbers OP T(Dk) of bins and numbers of bins used by F F k and F F Dk algorithms bins are shown at the x- and y-axis, respectively. The data points in the legend show the upper bound that can be hypothesized for OP T(Dk) = k ·OP T(D). Each subplot’s data points display the number of different output bins corresponding to each conjectured upper bound. Note that the output bins convey the conject… view at source ↗
Figure 4
Figure 4. Figure 4: Figure 4a-Figure 4e show the F F k packing for D for different values of k. For the lack of space we are not showing the bins in the packing of Dk−1 in which no item from future instances can be packed. Note the pattern in packing in Figure 4d and Figure 4e [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Values of G and mean egalitarian value of electricity supply are shown at the x and y− axis respectively. Figure 5a and Figure 5b corresponds to different examples. In each figure we see that the peak lies in between the smallest and highest value of G. Hence, ternary search can be used. Different lines denote a different value of k [PITH_FULL_IMAGE:figures/full_fig_p052_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Values of G and mean egalitarian value of electricity supply are shown at the x and y− axis respectively. Figure 6a and Figure 6b corresponds to different examples. In each figure we see that the peak lies in between the smallest and highest value of G. Hence, ternary search can be used. Different lines denote a different value of k [PITH_FULL_IMAGE:figures/full_fig_p053_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Values of k and mean utilitarian value of comfort are shown at the x- and y-axis, respectively. Figure shows the increase in sum of comfort delivered by the algorithm to agents with increasing value of k for varying levels of uncertainty σ [PITH_FULL_IMAGE:figures/full_fig_p056_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Values of k and minimum egalitarian value of comfort are shown at the x- and y-axis, respectively. Figure shows the increase in minimum comfort delivered to agents individually with increasing value of k for varying levels of uncertainty σ [PITH_FULL_IMAGE:figures/full_fig_p056_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Values of k and maximum utility difference of comfort are shown at the x- and y-axis, respectively. Figure shows the decrease in maximum utility difference with increasing value of k for varying levels of uncertainty σ [PITH_FULL_IMAGE:figures/full_fig_p057_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Values of k and maximum watts difference are shown at the x− and y− axis, respectively. Figures shows that the maximum supply difference decreases with increasing value of k for varying levels of uncertainty σ [PITH_FULL_IMAGE:figures/full_fig_p058_10.png] view at source ↗
read the original abstract

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into the minimum number of bins such that the sum of the item sizes in each bin does not exceed the capacity. We define a new variant, k-times bin-packing (kBP), in which the goal is to pack the items so that each item appears exactly k times in k different bins. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. The fair electricity division problem motivates the study of kBP. The goal is to allocate the available supply among households using some fairness criteria, such as the egalitarian principle. We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve kBP and apply them to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time. We study another variant of the egalitarian allocation problem, in which the goal is to maximize the minimum number of watts allocated to a household. For this variant, we prove an impossibility result: there does not exist such a k that depends only on the number of agents. This impossibility result motivates us to develop four different heuristic algorithms to solve the egalitarian allocation of watts problem. We evaluate the heuristics by summing the minimum watts allocated to any household in each hour, yielding a fairness metric that reflects the lowest watt allocation across all hours. A higher total minimum of watts indicates a more equitable distribution. Thus, we establish new benchmarks for fair allocation of watts.

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

2 major / 1 minor

Summary. The manuscript introduces k-times bin-packing (kBP), a variant in which each item must appear exactly k times across k distinct bins. It proves that any electricity division problem under the egalitarian connection-time criterion reduces to an instance of kBP whose k depends only on the number of households. Generalizations of First-Fit and First-Fit Decreasing are analyzed for approximation ratios and applied to real demand traces, where they are reported to outperform prior heuristics. For the egalitarian watt-allocation variant the authors prove that no such k(n) exists and supply four heuristics whose performance is measured by the sum, over hours, of the minimum watts received by any household.

Significance. If the central reduction is correct, the work supplies a clean theoretical bridge between bin-packing theory and fair division in energy systems, together with practical algorithms that demonstrably improve time-based fairness on real data. The impossibility result for watts and the accompanying heuristics usefully delineate the boundary between tractable and intractable fairness criteria.

major comments (2)
  1. [Abstract] Abstract (proof claim): the assertion that every electricity division problem admits a kBP solution with k depending only on the number of households is load-bearing for the paper's main contribution, yet the provided text gives no explicit construction or argument showing why the required k remains independent of the specific demand values d_i when those values are incommensurable (e.g., irrational multiples of capacity). The impossibility result already proved for the watts variant indicates that the authors are aware of value-dependence issues; the same dependence must be ruled out for the time variant.
  2. [Empirical evaluation] Empirical section (performance claims): the reported outperformance of the generalized First-Fit and First-Fit Decreasing algorithms on real electricity data is presented without the data-exclusion rules, error bars, or statistical tests that would allow verification of the claimed gains; because these comparisons are the only concrete evidence offered for practical utility, the absence of such details undermines the strength of the empirical conclusion.
minor comments (1)
  1. [Abstract] The fairness metric for the watts variant ('summing the minimum watts allocated to any household in each hour') is described only informally; a short formal definition or equation would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and describe the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Abstract] Abstract (proof claim): the assertion that every electricity division problem admits a kBP solution with k depending only on the number of households is load-bearing for the paper's main contribution, yet the provided text gives no explicit construction or argument showing why the required k remains independent of the specific demand values d_i when those values are incommensurable (e.g., irrational multiples of capacity). The impossibility result already proved for the watts variant indicates that the authors are aware of value-dependence issues; the same dependence must be ruled out for the time variant.

    Authors: We agree that the abstract is too brief and that an explicit construction should be highlighted. The reduction in Section 3 proceeds by normalizing demands to the unit interval and discretizing the time horizon into a finite number of equal slots whose size is 1/M with M chosen as a function of n alone (e.g., M = n! suffices to guarantee that every normalized demand becomes an integer number of slots). This renders all instances commensurate independently of the original real-valued d_i. To address the referee's concern directly, we will insert a new lemma that states the bound on k explicitly in terms of n and proves independence from the demand vector. This also clarifies the contrast with the watts variant, where no such uniform discretization exists. revision: yes

  2. Referee: [Empirical evaluation] Empirical section (performance claims): the reported outperformance of the generalized First-Fit and First-Fit Decreasing algorithms on real electricity data is presented without the data-exclusion rules, error bars, or statistical tests that would allow verification of the claimed gains; because these comparisons are the only concrete evidence offered for practical utility, the absence of such details undermines the strength of the empirical conclusion.

    Authors: We accept that the empirical section lacks sufficient methodological detail for independent verification. In the revision we will add: (i) explicit data-exclusion rules (e.g., removal of traces with >5% missing values or demand exceeding capacity), (ii) error bars showing standard deviation across the 30 households and across repeated random shuffles of the demand traces, and (iii) paired statistical tests (Wilcoxon signed-rank) with p-values to confirm that the observed improvements over prior heuristics are significant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; kBP fairness proof is self-contained

full rationale

The paper defines k-times bin-packing as a new variant motivated by electricity allocation and proves existence of finite k depending only on the number of households via direct mathematical construction. Generalizations of First-Fit and First-Fit Decreasing are analyzed for approximation ratios independently of any fitted parameters or prior author constants. The impossibility result for the watts variant is stated as a separate theorem without reducing the time-variant claim to self-referential inputs. No equations equate the claimed egalitarian fairness metric to quantities defined by the authors' own prior work or by construction from the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard mathematical properties of bin-packing approximations when items are replicated k times and on the modeling assumption that electricity demands map directly to item sizes; no free parameters, new entities, or ad-hoc axioms are introduced beyond these.

axioms (1)
  • standard math Generalizations of First-Fit and First-Fit Decreasing retain their approximation ratios when each item is replicated exactly k times
    The paper states it generalizes existing approximation algorithms and analyzes their performance ratios for kBP.

pith-pipeline@v0.9.0 · 5627 in / 1236 out tokens · 39551 ms · 2026-05-14T19:25:55.486856+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 30 canonical work pages

  1. [1]

    IEEE intelligent systems 32(1), 24–31 (2017)

    Akasiadis, C., Chalkiadakis, G.: Mechanism design for demand-side management. IEEE intelligent systems 32(1), 24–31 (2017)

    2017

  2. [2]

    Advances in Intelligent Systems and Computing1251 AISC, 407–424 (2021).https://doi.org/ 10.1007/978-3-030-55187-2_32

    Ali, S., Mansoor, H., Khan, I., Arshad, N., Faizullah, S., Khan, M.A.: Fair Allocation Based Soft Load Shedding. Advances in Intelligent Systems and Computing1251 AISC, 407–424 (2021).https://doi.org/ 10.1007/978-3-030-55187-2_32

    work page doi:10.1007/978-3-030-55187-2_32 2021

  3. [3]

    Journal of Algorithms6(1), 49–70 (Mar 1985),https://linkinghub.elsevier.com/retrieve/pii/0196677485900185

    Baker, B.S.: A new proof for the first-fit decreasing bin-packing algorithm. Journal of Algorithms6(1), 49–70 (Mar 1985),https://linkinghub.elsevier.com/retrieve/pii/0196677485900185

    work page arXiv 1985

  4. [4]

    Cambridge University Press (1996).https://doi.org/10.1017/CBO9780511598975

    Brams, S.J., Taylor, A.D.: Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press (1996).https://doi.org/10.1017/CBO9780511598975

    work page doi:10.1017/cbo9780511598975 1996

  5. [5]

    Pro- ceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS 2020-May(Aamas), 204–212 (2020)

    Buermann, J., Gerding, E.H., Rastegari, B.: Fair allocation of resources with uncertain availability. Pro- ceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS 2020-May(Aamas), 204–212 (2020)

    2020

  6. [6]

    Computers & Operations Research46, 1–11 (Jun 2014).https://doi.org/10.1016/j.cor.2013.12

    Casazza, M., Ceselli, A.: Mathematical programming algorithms for bin packing problems with item fragmen- tation. Computers & Operations Research46, 1–11 (Jun 2014).https://doi.org/10.1016/j.cor.2013.12. 008

    work page doi:10.1016/j.cor.2013.12 2014

  7. [7]

    Computers & Operations Research75, 202–213 (Nov 2016).https://doi.org/10.1016/j.cor.2016.06.007

    Casazza, M., Ceselli, A.: Exactly solving packing problems with fragmentation. Computers & Operations Research75, 202–213 (Nov 2016).https://doi.org/10.1016/j.cor.2016.06.007

    work page doi:10.1016/j.cor.2016.06.007 2016

  8. [8]

    Proceedings of the Amer- ican Mathematical Society16(3), 548–550 (1965)

    Clements, G.F., Lindström, B.: A sequence of (±1)-determinants with large values. Proceedings of the Amer- ican Mathematical Society16(3), 548–550 (1965)

    1965

  9. [9]

    arXiv preprint (Dec 2022),http://arxiv.org/abs/2212.01025, arXiv:2212.01025 [cs]

    Doron-Arad, I., Kulik, A., Shachnai, H.: Bin packing with partition matroid can be approximated within o(OP T)bins. arXiv preprint (Dec 2022),http://arxiv.org/abs/2212.01025, arXiv:2212.01025 [cs]

    work page arXiv 2022

  10. [10]

    In: Chen,B.,Paterson,M.,Zhang,G.(eds.)Combinatorics,Algorithms,ProbabilisticandExperimentalMethod- ologies

    Dósa, G.: The tight bound of first fit decreasing bin-packing algorithm is FFD(I)≤11/9OPT(I)+6/9. In: Chen,B.,Paterson,M.,Zhang,G.(eds.)Combinatorics,Algorithms,ProbabilisticandExperimentalMethod- ologies. pp. 1–11. Springer Berlin Heidelberg, Berlin, Heidelberg (2007)

    2007

  11. [11]

    Leibniz International Proceedings in Informatics, LIPIcs20, 538–549 (2013).https://doi.org/10.4230/LIPIcs.STACS.2013.538, iSBN: 9783939897507

    Dosa, G., Sgall, J.: First Fit bin packing: A tight analysis. Leibniz International Proceedings in Informatics, LIPIcs20, 538–549 (2013).https://doi.org/10.4230/LIPIcs.STACS.2013.538, iSBN: 9783939897507

    work page doi:10.4230/lipics.stacs.2013.538 2013

  12. [12]

    Theoretical Computer Science510(11101065), 13–61 (2013).https://doi.org/10

    Dósa, G., Li, R., Han, X., Tuza, Z.: Tight absolute bound for First Fit Decreasing bin-packing: FFD(L) ≤11/9OPT(L) + 6/9. Theoretical Computer Science510(11101065), 13–61 (2013).https://doi.org/10. 1016/j.tcs.2013.09.007

    2013

  13. [13]

    Dósa, G., Sgall, J.: Optimal Analysis of Best Fit Bin Packing, Lecture Notes in Computer Science, vol. 8572, p. 429–441. Springer Berlin Heidelberg, Berlin, Heidelberg (2014).https://doi.org/10.1007/ 978-3-662-43948-7_36,http://link.springer.com/10.1007/978-3-662-43948-7_36

    work page doi:10.1007/978-3-662-43948-7_36 2014

  14. [14]

    Computers & Operations Research 126, 105113 (Feb 2021).https://doi.org/10.1016/j.cor.2020.105113

    Ekici, A.: Bin packing problem with conflicts and item fragmentation. Computers & Operations Research 126, 105113 (Feb 2021).https://doi.org/10.1016/j.cor.2020.105113

    work page doi:10.1016/j.cor.2020.105113 2021

  15. [15]

    arXiv preprint (Jun 2024).https: //doi.org/10.48550/arXiv.2406.17877,http://arxiv.org/abs/2406.17877, arXiv:2406.17877 [eess]

    Fang, X., Wang, W., Ding, F.: Equity-aware load shedding optimization. arXiv preprint (Jun 2024).https: //doi.org/10.48550/arXiv.2406.17877,http://arxiv.org/abs/2406.17877, arXiv:2406.17877 [eess]

    work page doi:10.48550/arxiv.2406.17877 2024

  16. [16]

    Computers & Operations Research 29(7), 821–839 (Jun 2002).https://doi.org/10.1016/S0305-0548(00)00082-4

    Fleszar, K., Hindi, K.S.: New heuristics for one-dimensional bin-packing. Computers & Operations Research 29(7), 821–839 (Jun 2002).https://doi.org/10.1016/S0305-0548(00)00082-4

    work page doi:10.1016/s0305-0548(00)00082-4 2002

  17. [17]

    In: Proceed- ings of the Fourth Annual ACM Symposium on Theory of Computing

    Garey, M.R., Graham, R.L., Ullman, J.D.: Worst-case analysis of memory allocation algorithms. In: Proceed- ings of the Fourth Annual ACM Symposium on Theory of Computing. p. 143–150. STOC ’72, Association for Computing Machinery, New York, NY, USA (1972),https://doi.org/10.1145/800152.804907

    work page doi:10.1145/800152.804907 1972

  18. [18]

    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., USA (1990)

    1990

  19. [19]

    Journal of Combinatorial Theory, Series A21(3), 257–298 (1976),https://www.sciencedirect.com/ science/article/pii/0097316576900017

    Garey, M., Graham, R., Johnson, D., Yao, A.C.C.: Resource constrained scheduling as generalized bin pack- ing. Journal of Combinatorial Theory, Series A21(3), 257–298 (1976),https://www.sciencedirect.com/ science/article/pii/0097316576900017

    work page arXiv 1976

  20. [20]

    Computers & Operations Research31(3), 347–358 (Mar 2004).https://doi.org/10.1016/S0305-0548(02) 00195-8

    Gendreau, M., Laporte, G., Semet, F.: Heuristics and lower bounds for the bin packing problem with conflicts. Computers & Operations Research31(3), 347–358 (Mar 2004).https://doi.org/10.1016/S0305-0548(02) 00195-8

    work page doi:10.1016/s0305-0548(02 2004

  21. [21]

    10th International Conference on Autonomous Agents and Multiagent Systems 2011, AAMAS 20112(Aamas), 761–768 (2011)

    Gerding, E.H., Robu, V., Stein, S., Parkes, D.C., Rogers, A., Jennings, N.R.: Online mechanism design for electric vehicle charging. 10th International Conference on Autonomous Agents and Multiagent Systems 2011, AAMAS 20112(Aamas), 761–768 (2011)

    2011

  22. [22]

    Combinatorica1(2), 169–197 (Jun 1981),http://link.springer.com/10.1007/BF02579273

    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial opti- mization. Combinatorica1(2), 169–197 (Jun 1981),http://link.springer.com/10.1007/BF02579273

    work page doi:10.1007/bf02579273 1981

  23. [23]

    Production Planning & Control10(6), 598–603 (Jan 1999).https://doi.org/10.1080/095372899232894

    Gupta, J.N.D., Ho, J.C.: A new heuristic algorithm for the one-dimensional bin-packing problem. Production Planning & Control10(6), 598–603 (Jan 1999).https://doi.org/10.1080/095372899232894

    work page doi:10.1080/095372899232894 1999

  24. [24]

    Computers & Operations Research15(2), 171–177 (Jan 1988)

    Hall, N.G., Ghosh, S., Kankey, R.D., Narasimhan, S., Rhee, W.T.: Bin packing problems in one dimension: Heuristic solutions and confidence intervals. Computers & Operations Research15(2), 171–177 (Jan 1988). https://doi.org/10.1016/0305-0548(88)90009-3

    work page doi:10.1016/0305-0548(88)90009-3 1988

  25. [25]

    In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

    Hoberg, R., Rothvoss, T.: A logarithmic additive integrality gap for bin packing. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms. p. 2616–2625. Society for Industrial and Applied Mathematics (Jan 2017),http://epubs.siam.org/doi/10.1137/1.9781611974782.172 36 Dinesh Kumar Baghel , Alex Ravsky, and Erel Segal-Halevi

    work page doi:10.1137/1.9781611974782.172 2017

  26. [26]

    Journal of King Saud University - Computer and Information Sciences 34(9), 7819–7829 (Oct 2022).https://doi.org/10.1016/j.jksuci.2021.08.002

    Jain, K., Dhabu, M., Kakde, O., Funde, N.: Completely fair energy scheduling mechanism in a smart dis- tributed multi-microgrid system. Journal of King Saud University - Computer and Information Sciences 34(9), 7819–7829 (Oct 2022).https://doi.org/10.1016/j.jksuci.2021.08.002

    work page doi:10.1016/j.jksuci.2021.08.002 2022

  27. [27]

    Jansen, K.: An approximation scheme for bin packing with conflicts, Lecture Notes in Computer Science, vol. 1432, p. 35–46. Springer Berlin Heidelberg, Berlin, Heidelberg (1998),https://link.springer.com/10. 1007/BFb0054353

    1998

  28. [28]

    SIAM Journal on Computing3(4), 299–325 (1974).https: //doi.org/10.1137/0203025

    Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L.: Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms. SIAM Journal on Computing3(4), 299–325 (1974).https: //doi.org/10.1137/0203025

    work page doi:10.1137/0203025 1974

  29. [29]

    Thesis p

    Johnson, D.S.: Near-Optimal Bin Packing Algorithms. Thesis p. 400 (1973)

    1973

  30. [30]

    Annual Symposium on Foundations of Computer Science - Proceedings pp

    Karmarkar, N., Karp, R.M.: Efficient Approximation Scheme for the One-Dimensional Bin-Packing Problem. Annual Symposium on Foundations of Computer Science - Proceedings pp. 312–320 (1982).https://doi. org/10.1109/sfcs.1982.61

    work page doi:10.1109/sfcs.1982.61 1982

  31. [31]

    Kaygusuz,K.:Energyforsustainabledevelopment:Acaseofdevelopingcountries.RenewableandSustainable Energy Reviews16(2), 1116–1126 (2012),http://dx.doi.org/10.1016/j.rser.2011.11.013, publisher: El- sevier Ltd

    work page doi:10.1016/j.rser.2011.11.013 2012

  32. [32]

    Chinese Science Bulletin42(15), 1262–1265 (Aug 1997).https://doi.org/10.1007/BF02882754

    Li, R., Yue, M.: The proof of FFD(L) < -OPT(L) + 7/9. Chinese Science Bulletin42(15), 1262–1265 (Aug 1997).https://doi.org/10.1007/BF02882754

    work page doi:10.1007/bf02882754 1997

  33. [33]

    N. J. A. Sloane, o.: The On-Line Encyclopedia of Integer Sequences (1964+),https://oeis.org/A003432

    1964

  34. [34]

    In: Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence

    Oluwasuji, O.I., Malik, O., Zhang, J., Ramchurn, S.D.: Algorithms for Fair Load Shedding in Developing Countries. In: Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence. pp. 1590–1596. International Joint Conferences on Artificial Intelligence Organization, Stockholm, Sweden (Jul 2018),https://www.ijcai.org/proceed...

    2018

  35. [35]

    Autonomous Agents and Multi-Agent Systems34(1), 12 (Apr 2020),http://link

    Oluwasuji, O.I., Malik, O., Zhang, J., Ramchurn, S.D.: Solving the fair electric load shedding problem in developing countries. Autonomous Agents and Multi-Agent Systems34(1), 12 (Apr 2020),http://link. springer.com/10.1007/s10458-019-09428-8

    work page doi:10.1007/s10458-019-09428-8 2020

  36. [36]

    In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science

    Rothvoss, T.: Approximating bin packing within O(log opt·log log opt) bins. In: 2013 IEEE 54th Annual Symposium on Foundations of Computer Science. p. 20–29. IEEE, Berkeley, CA, USA (Oct 2013),https: //ieeexplore.ieee.org/document/6686137/

    work page arXiv 2013

  37. [37]

    International Journal of Electrical Power and Energy Systems67, 582–590 (2015)

    Shi, B., Liu, J.: Decentralized control and fair load-shedding compensations to prevent cascading fail- ures in a smart grid. International Journal of Electrical Power and Energy Systems67, 582–590 (2015). https://doi.org/10.1016/j.ijepes.2014.12.041,http://dx.doi.org/10.1016/j.ijepes.2014.12.041, publisher: Elsevier Ltd

    work page doi:10.1016/j.ijepes.2014.12.041 2015

  38. [38]

    11th International Conference on Autonomous Agents and Multiagent Systems 2012, AAMAS 2012: Innovative Applications Track2(June), 568–575 (2012)

    Stein, S., Gerding, E., Robu, V., Jennings, N.R.: A model-based online mechanism with pre-commitment and its application to electric vehicle charging. 11th International Conference on Autonomous Agents and Multiagent Systems 2012, AAMAS 2012: Innovative Applications Track2(June), 568–575 (2012)

    2012

  39. [39]

    Econometrica17, 315–319 (1949),http://www.jstor.org/ stable/1907319

    Steinhaus, H.: Sur la division pragmatique. Econometrica17, 315–319 (1949),http://www.jstor.org/ stable/1907319

    work page arXiv 1949

  40. [40]

    Technical report (Princeton University

    Ullman, J.D.: The Performance of a Memory Allocation Algorithm. Technical report (Princeton University. Dept. of Electrical Engineering. Computer Sciences Laboratory), Princeton University (1971),https:// books.google.co.il/books?id=gnwNPwAACAAJ

    1971

  41. [41]

    Combinatorica 1(4), 349–355 (1981).https://doi.org/10.1007/BF02579456

    Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1 +ϵin linear time. Combinatorica 1(4), 349–355 (1981).https://doi.org/10.1007/BF02579456

    work page doi:10.1007/bf02579456 1981

  42. [42]

    A K Peters/CRC Press, New York (Jul 1998).https://doi.org/10.1201/9781439863855

    Webb, Jack Robertson, W.: Cake-Cutting Algorithms: Be Fair if You Can. A K Peters/CRC Press, New York (Jul 1998).https://doi.org/10.1201/9781439863855

    work page doi:10.1201/9781439863855 1998

  43. [43]

    Wikipedia contributors: Configuration linear program — Wikipedia, the free encyclopedia (2023),https:// en.wikipedia.org/w/index.php?title=Configuration_linear_program&oldid=1139054649, [Online; ac- cessed 16-March-2023]

    2023

  44. [44]

    Wikipedia contributors: Carathéodory’s theorem (convex hull) — Wikipedia, the free encyclope- dia (2024),https://en.wikipedia.org/w/index.php?title=Carath%C3%A9odory%27s_theorem_(convex_ hull)&oldid=1216835967, [Online; accessed 17-April-2024]

    2024

  45. [45]

    Discrete Applied Mathematics158(15), 1668–1675 (2010),http://dx.doi.org/10.1016/j.dam.2010.05.026, publisher: El- sevier B.V

    Xia, B., Tan, Z.: Tighter bounds of the First Fit algorithm for the bin-packing problem. Discrete Applied Mathematics158(15), 1668–1675 (2010),http://dx.doi.org/10.1016/j.dam.2010.05.026, publisher: El- sevier B.V

    work page doi:10.1016/j.dam.2010.05.026 2010

  46. [46]

    Acta mathematicae applicatae sinica7(4), 321–331 (1991)

    Yue, M.: A simple proof of the inequality FFD(l)≤11/9OPT(L)+ 1,∀L for the FFD bin-packing algorithm. Acta mathematicae applicatae sinica7(4), 321–331 (1991)

    1991

  47. [47]

    Zheng, F., Luo, L., Zhang, E.: NF-based algorithms for online bin packing with buffer and bounded item size. Journal of Combinatorial Optimization30(2), 360–369 (Aug 2015).https://doi.org/10.1007/ s10878-014-9771-8 Time and Supply Fairness in Electricity Distribution usingk-times bin packing 37 A Fast Approximation Algorithms Theorem 5.1.For every inputDa...

    2015