On bin packing with clustering and bin packing with delays

Leah Epstein

REVIEW 1 major objections 5 minor 27 references

On bin packing with clustering and bin packing with delays

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bin packing bounds tighten: 1.93667 for clusters, 3.1551 for delays.

desk verdict Real improvement on the price of clustering with a clean analytic proof, and a solid delay-algorithm result — but the delay theorem is proved only for unbounded delay functions, which the abstract doesn't state. read the letter →

arxiv 1908.06727 v1 pith:WT4JISDW submitted 2019-08-19 cs.DS cs.DMmath.COmath.OC

Leah Epstein This is my paper · ORCID

classification cs.DScs.DMmath.COmath.OC MSC 68W2768W25

keywords binpackingpriceofclusteringonlinewithdelayscompetitiveratioweightingfunctionsFirst-FitDecreasingphase-basedalgorithm

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 narrows the worst-case gap for two bin packing variants. In bin packing with clustering, items carry colors and a clustered solution may only pack same-color bins; the paper proves that, when every color class needs at least three bins, the price of clustering—the worst-case ratio of the best clustered solution to the best unconstrained solution—lies between 1.93558 and 1.93667, improving the previous interval of [1.93344, 1.951]. The upper bound comes from a weight function that charges each bin of the global solution at most 581/300 and shows each cluster's greedy packing uses no more bins than its total weight. In online bin packing with delays, where bins may be opened at chosen times and waiting items accrue delay costs, the paper gives a phase-based algorithm with competitive ratio below 3.1551, improving the previous 3.951. A reader should care because the paper replaces a computer-assisted proof with a checkable analytic one and shows the phase-based waiting strategy can be tuned much closer to the known lower bound of 2.

What carries the argument

For the clustering bound, the carrying object is a two-part weight function w(x) = \frac{21}{13}x + bonus(x), where bonus(x) is 997/3900 for x>1/2, 64/975 for 1/3<x≤1/2, 18/325 for 1/4<x≤1/3, 2/195 for 1/6<x≤1/4, and 0 below 1/6. It does two jobs at once: every bin of a global optimum weighs at most 581/300, and every cluster's FFD output has at least as much weight as it has bins; the second inequality is what forces the cluster-optimal cost to be paid for in weight. For the online problem, the mechanism is the phase threshold ρ: the algorithm waits until the total delay of all unpacked items reaches ρ≈0.4640251938, packs that phase by FFD, and repeats; the analysis uses a known FFD weight function v with FFD(J)≤V(J)+1 for every set J and V(J')≤π∞ for every set J' of total size at most 1, turning the phase count into a linear expression in the optimal bins B and delay D.

What would settle it

For clustering, extend the paper's lower-bound construction with the next item families (clusters of 13 items just above 1/7, 85 just above 1/43, and 3613 just above 1/1807, with larger M) and compute the exact clustered/global ratio: any value above 581/300≈1.93667 refutes Theorem 2.11. For delays, simulate the phase algorithm with ρ=0.4640251938 on an adversarial input with, say, delay functions d(t)=$t^{2}$ and compare against the optimal offline cost: a ratio above 3.1550554008 refutes Theorem 3.1.

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Extended reading notes

Core claim

The paper's central discovery is that both previously known bounds were loose. For price of clustering with cluster optimal cost at least 3, it constructs an explicit family of instances whose clustered-to-global ratio approaches 1.9355858244424, and proves an upper bound of 581/300 ≈ 1.93667 by assigning weights to item sizes: w(x) = (21/13)x plus bonuses for sizes above 1/6, 1/4, 1/3, and 1/2. The proof shows every bin in a globally optimal solution has total weight at most 581/300, while every cluster's First-Fit-Decreasing packing has total weight at least its bin count, using the fact that each cluster needs at least three bins. For online bin packing with delays, the paper generalizes the phase-based algorithm to an arbitrary waiting parameter ρ and proves that with ρ ≈ 0.4640251938 its competitive ratio is at most 3.1550554008, by combining a known weight function for FFD with a per-bin weight bound π∞ ≈ 1.691030207 and the accounting D ≥ ℓ2·ρ for phases in which the optimal solution has no associated bin.

Load-bearing premise

For the clustering bound, the load-bearing premise is that every cluster's optimal packing uses at least three bins; for the delay bound, it is that every delay function is continuous, non-decreasing, unbounded, and starts at zero, so every phase eventually reaches the threshold ρ.

Editorial extensions

If this is right

For inputs where every cluster's optimal packing uses at least three bins, monochromatic-only packing costs at most about 1.93667 times the globally optimal packing, so the loss from forbidding color mixing is under a factor of 1.937.
The same weight-function approach gives an upper bound of about 1.88049612 for the case where every cluster needs at least four bins, and the paper provides lower-bound formulas for all k≥4 that tend near 1.691 as k grows.
The online bin-packing-with-delays algorithm is polynomial time and needs no optimal solutions: it only waits until total delay reaches ρ and packs by FFD, so it can be implemented whenever delay values are computable.
The delay competitive ratio of 3.1550554008 narrows the gap to the unconditional lower bound of 2 inherited from TCP acknowledgment.
The clustering upper-bound proof is fully analytic, so it can be checked and adapted without running the earlier computer-assisted linear program.

Reading between the lines

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

The tiny gap between 1.93558 and 1.93667 suggests the true k=3 price of clustering is very close to the paper's lower-bound construction; extending that construction with further reciprocal-type clusters is a natural way to try to close the gap.
The same phase-based waiting scheme could likely be improved further by replacing the per-bin weight bound π∞≈1.691030207 with sharper harmonic-type weights from online bin packing, which would push the delay competitive ratio below 3.1551.
For bounded delay functions the paper only sketches what to do in the final phase; a hybrid rule that packs remaining items when the input ends, with an additive term in the competitive ratio, is a testable extension.
Because the price of clustering for very large clusters coincides with a known batched bin packing ratio, improvements to either problem's bounds can transfer to the other.

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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 main thing to know: this paper is worth taking seriously. The price-of-clustering upper bound for k=3 drops from 1.951 to 581/300 ≈ 1.93667 using an explicit weight function and case analysis instead of the earlier computer-assisted proof, and the lower bound rises to ≈1.93558. I went through the weight-function argument and the FFD-based lemmas; they are coherent and the case splits are organized. The derivation does not assume the target bound. That part is solid.

The second contribution, the online bin-packing-with-delays competitive ratio, is also a genuine improvement: a phase-based algorithm with parameter rho ≈ 0.464025 gives ratio < 3.155055, down from 3.951. The analysis using the [17] weight function and the two-phase-type accounting is compact, and I don't see a hole in it — for unbounded delay functions.

The soft spot is exactly what the stress-test flags: the theorem is stated and proven only under the assumption that delay functions are continuous, non-decreasing, unbounded, and satisfy d(0)=0, which the paper does say at the start of Section 3. But the abstract and introduction do not carry that qualifier. If delays are bounded, the last phase may end with total delay less than rho, so the inequality D ≥ l2·rho used in the proof does not hold, and the claimed competitive ratio is not established. The paper's offhand note that one could pack remaining items once the input stops is not an analysis. This is a real scope mismatch, not a nitpick. The fix could be small — restrict the theorem statement and abstract, or prove the bounded case — but as written the central claim for delays is narrower than it appears.

A smaller issue: Section 2.3 gives a general k≥4 lower bound formula, Proposition 2.12, with only a sketch of the construction. That is probably fine as a forward-looking note, but it should be marked as a sketch or given a real proof.

Overall, the clustering result deserves to be published, and the delay result deserves refereeing with the bounded-delay qualifier addressed. I would send this to review and, as a referee, would ask for the bounded-delay case to be handled or the claims to be scoped honestly. The paper is within-field, but the improvements are substantial and the proof methods are cleaner than what came before.

Referee Report

1 major / 5 minor

Summary. This paper studies two bin packing variants. For bin packing with clustering, it improves the known bounds on the price of clustering for the case where each cluster's optimal cost is at least 3, proving a lower bound of 1.93558 and an upper bound of 581/300 ≈ 1.93667, and it gives a simplified analytical proof of the earlier 1.95 upper bound. For bin packing with delays, it analyzes a phase-based online algorithm with a tunable parameter ρ and proves, for continuous, non-decreasing, unbounded delay functions satisfying d(0)=0, a competitive ratio of at most 3.1550554008, improving the previous 3.951.

Significance. These are meaningful quantitative improvements. The clustering gap is narrowed to about 0.0011, and the delay competitive ratio is reduced by about 0.8. The clustering upper bound uses an elementary weight-function argument rather than computer-assisted LP, which is a methodological plus. The delay analysis is compact and builds on a clean weight-function framework. The main caveat is that the delay theorem is proved under an unbounded-delay assumption not stated in the abstract; this limits the scope of the headline claim as written.

major comments (1)

[Section 3, Theorem 3.1 and Abstract] The abstract and introduction claim a competitive ratio below 3.1551 for 'online bin packing with delays' without qualification, but the proof of Theorem 3.1 relies on the delay functions being unbounded (Section 3, first paragraph). The algorithm's phases are defined only when the total delay reaches ρ, and the inequality D ≥ ℓ2·ρ for second-type phases uses that each such phase has accumulated exactly ρ delay. For bounded delay functions the final phase may end with delay δ<ρ, and the paper's parenthetical remark ('Alternatively, it is possible to use bounded delay functions...') does not provide a competitive-ratio analysis. As stated, the theorem does not cover the general problem as introduced, where delay functions are merely monotonically non-decreasing. Please either prove the bound for bounded delay functions (e.g., by arguing that the last incomplete phase is necessarily of the first type and hence the existing inequalities still hold) or qualify the abstract, introduction, and theorem statement to the unbounded case.

minor comments (5)

[Section 2.2, Lemma 2.10] In the case θ ∈ (1/7,1/6], the sentence 'If there is at least one item of size above 1/4, or at least four items with positive bonuses, we are done' is too terse; a single item just above 1/4 contributes only 40/3900 in bonus, so the reader must supply the structural argument that such an item forces an inner bin of load at least 6/7 (making the first paragraph of the lemma applicable). Please expand this step.
[Section 3, types of phases] The assertion that in second-type phases OPT pays at least the same delay as the algorithm is correct only under the association of each OPT bin with its maximum-index item, and the argument that a second-type phase's items cannot be packed before the phase ends is implicit. A one-sentence explanation would help.
[Section 2.1, first paragraph] The text 'divisible by 5000! ·9M' should read 5000!·9^M; as written '9M' is ambiguous.
[Section 2.3, Claim 2.13] The last displayed equation ends with '= λ/δ' but should be '= λ/Δ' (with Δ=77805).
[Section 2.1, global optimality] The statement 'It is obvious that there is no global solution whose cost is below N' could be made explicit by noting that the total size of all items is N - O(N^2 μ), which exceeds N-1 for suitable small μ, forcing at least N bins.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the price-of-clustering bounds are proved by explicit weight-function inequalities and an explicit lower-bound construction, and the online delay ratio follows from an externally cited harmonic weight-function theorem plus a parameter optimized in the final inequality.

full rationale

The paper's central derivations are self-contained against external benchmarks and do not reduce to their own inputs. For the price of clustering, the upper bound (Theorem 2.11) is proved by defining a concrete weight function w(x), proving W <= (581/300)OPT for every optimal bin, and proving Wi >= Ai for every FFD output on each cluster; the conclusion sum OPT_i <= sum Ai <= sum Wi <= (581/300)OPT follows by algebra, not by assuming the target ratio. The lower bound (Theorem 2.1) is an explicit item/cluster construction whose cluster costs and global cost are computed directly, so the claimed lower bound is not assumed. For bin packing with delays, the main external input is the harmonic weight function from Epstein and Levin [17], cited as Lemmas 2 and 5: it states two general properties for arbitrary item sets, FFD(J) <= V(J)+1 and V(J') <= pi_infinity for any bin-content set J' of total size at most 1. This is a parameter-free, externally published theorem whose assumptions do not include the competitive ratio being proved, so citing it is legitimate independent support even though one of the current authors is also an author of [17]. The new contribution is the phase parameter rho, chosen by minimizing max{1+1/a, 1+a+pi_infinity} in the final inequality, and the phase-classification argument giving (1+rho)ell <= (1+rho)B + (1+1/rho)D. None of these steps fits the patterns of self-definition, fitted-input-called-prediction, or ansatz-smuggled-via-citation. The bounded-delay caveat in Section 3, where the text notes that with bounded delay functions the final phase may not reach rho and the algorithm should pack remaining items once the input stops, is a scope limitation rather than a circular step: it narrows the proven theorem to unbounded delay functions but does not make the unbounded-delay proof depend on its own conclusion. Overall, the derivations are honest proofs, and no load-bearing step reduces by construction to its inputs.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claims rest on standard bin packing theorems, namely the FFD structural properties proved in the paper and the harmonic weight function from [17], plus explicit problem assumptions such as k>=3 clusters and unbounded delay functions. No new entities are postulated. The only ad hoc numbers are the proof-certificate constants in the weight functions and the optimization parameter rho, which are transparently derived rather than fitted to data.

free parameters (3)

rho (phase threshold) = approximately 0.4640251938
Parameter of the online bin packing with delays algorithm. Chosen to balance the two terms in the competitive ratio bound, (1+1/rho) and (1+rho+pi_infinity). It is an explicit design choice, not fitted to data.
Clustering weight function constants = 21/13 base; bonuses 997/3900, 256/3900, 216/3900, 40/3900
Introduced ad hoc to satisfy the inequalities of Claim 2.9 and Lemma 2.10. They serve as a proof certificate and are not empirical fits.
mu (perturbation in lower bound) = arbitrarily small positive value, tending to 0
Used in the lower bound construction to separate item sizes by a tiny margin. The price-of-clustering lower bound is obtained in the limit mu goes to 0, N goes to infinity, M goes to infinity.

assumptions (3)

standard math There exists a weight function v such that FFD(J) <= V(J) + 1 and V(J') <= pi_infinity for any set J' of total size at most 1, with pi_infinity = sum 1/c_i < 1.691030207.
Invoked in Theorem 3.1 (Section 3) to bound the number of bins used by FFD in each phase. This is a published theorem from Epstein and Levin [17], not proved in this paper.
domain assumption Delay functions are continuous, monotonically non-decreasing, unbounded, and satisfy d(0)=0.
Stated at the start of Section 3. The proof that every phase generates delay exactly rho, and the inequality D >= l2 times rho, depends on the total delay eventually reaching rho, which requires unboundedness.
domain assumption Every cluster has an optimal bin packing cost of at least 3 (parameter k=3).
This is the problem definition inherited from Azar et al. [3]. The price of clustering bounds proved here apply to this case; for k=2 the price is 2 and for k=1 it is unbounded.

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Cite this review

Pith. "Pith review of On bin packing with clustering and bin packing with delays." pith.science (2026). https://pith.science/paper/WT4JISDW

@misc{pith2026190806727,
  author       = {Pith},
  title        = {Pith review of: On bin packing with clustering and bin packing with delays},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WT4JISDW}},
  note         = {Machine review of arXiv:1908.06727}
}

read the original abstract

We continue the study of two recently introduced bin packing type problems, called bin packing with clustering, and online bin packing with delays. A bin packing input consists of items of sizes not larger than 1, and the goal is to partition or pack them into bins, where the total size of items of every valid bin cannot exceed 1. In bin packing with clustering, items also have colors associated with them. A globally optimal solution can combine items of different colors in bins, while a clustered solution can only pack monochromatic bins. The goal is to compare a globally optimal solution to an optimal clustered solution, under certain constraints on the coloring provided with the input. We show close bounds on the worst-case ratio between these two costs, called "the price of clustering", improving and simplifying previous results. Specifically, we show that the price of clustering does not exceed 1.93667, improving over the previous upper bound of 1.951, and that it is at least 1.93558, improving over the previous lower bound of 1.93344. In online bin packing with delays, items are presented over time. Items may wait to be packed, and an algorithm can create a new bin at any time, packing a subset of already existing unpacked items into it, under the condition that the bin is valid. A created bin cannot be used again in the future, and all items have to be packed into bins eventually. The objective is to minimize the number of used bins plus the sum of waiting costs of all items, called delays. We build on previous work and modify a simple phase-based algorithm. We combine the modification with a careful analysis to improve the previously known competitive ratio from 3.951 to below 3.1551.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [17]

Epstein and A

L. Epstein and A. Levin. On bin packing with conﬂicts. SIAM Journal on Optimization , 19(3):1270–1298, 2008

work page 2008

[2] [1]

Ahlroth, A

L. Ahlroth, A. Schumacher, and P. Orponen. Online bin pac king with delay and holding costs. Operations Research Letters, 41(1):1–6, 2013

work page 2013

[3] [2]

Albers and H

S. Albers and H. Bals. Dynamic TCP acknowledgment: Penal izing long delays. SIAM Journal on Discrete Mathatics , 19(4):938–951, 2005

work page 2005

[4] [3]

Y. Azar, Y. Emek, R. van Stee, and D. Vainstein. The price o f clustering in bin-packing with applications to bin-packing with delays. In The 31st ACM on Symposium on Parallelism in Algorithms and Architectures, (SPAA2019) , pages 1–10, 2019

work page 2019

[5] [4]

Y. Azar, A. Ganesh, R. Ge, and D. Panigrahi. Online servic e with delay. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC2017) , pages 551–563, 2017

work page 2017

[6] [5]

B. S. Baker and E. G. Coﬀman, Jr. A tight asymptotic bound fo r next-ﬁt-decreasing bin- packing. SIAM J. on Algebraic and Discrete Methods , 2(2):147–152, 1981

work page 1981

[7] [6]

Balogh, J

J. Balogh, J. B´ ek´ esi, G. D´ osa, L. Epstein, and A. Levin. A new and improved algorithm for online bin packing. In Proc. of the 26th European Symposium on Algorithms (ESA2018 ), pages 5:1–5:14, 2018

work page 2018

[8] [7]

A new lower bound for classic online bin packing

J. Balogh, J. B´ ek´ esi, G. D´ osa, L. Epstein, and A. Levin. A new lower bound for classic online bin packing. CoRR, abs/1807.05554, 2018

work page Pith review arXiv 2018