Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

Andreas Karrenbauer; Angikar Ghosal

REVIEW 1 major objections 6 minor 13 references

Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

T0 review · 1 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The 3×3 Domino matrix shows the fractional binary rank is not multiplicative under the Kronecker product, and its asymptotic rank lies in [2, 2.373].

desk verdict Rigorous counterexample to multiplicativity of fractional binary rank, with a fragile computational upper bound that needs verification. read the letter →

arxiv 2502.06730 v1 pith:PDPSDKLM submitted 2025-02-10 cs.DS

Angikar Ghosal , Andreas Karrenbauer This is my paper · ORCID

classification cs.DS MSC 05C7015A2390C0590C27

keywords fractionalbinaryrankbicliquepartitionKroneckerproductasymptoticcolumngenerationDominographcoverlinearprogramming

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

The paper tries to settle whether the fractional binary rank—the least total weight of bicliques needed to cover a matrix's 1-entries exactly—is multiplicative under Kronecker products. It answers no: the Domino matrix $D$ satisfies $\mathrm{bp}_f(D\otimes D)=6$, strictly below $\mathrm{bp}_f(D)^2=6.25$, so the asymptotic fractional binary rank of the Domino is strictly smaller than its fractional binary rank. The paper also proves that the fractional biclique cover number lower-bounds the asymptotic fractional binary rank, and it engineers a column-generation algorithm that computes $\mathrm{bp}_f(D^{\otimes k})$ for $k$ up to 5, giving the interval $[2, 2.372713]$ for the Domino. If correct, this establishes the fractional binary rank as another rank function—alongside nonnegative and boolean rank—whose asymptotic value can be strictly less than its finite value.

What carries the argument

The central objects are the fractional biclique partition number $\mathrm{bp}_f(A)$ (equivalently the fractional binary rank), the Domino matrix $D$, and its Kronecker powers $D^{\otimes k}$. The argument is carried by two mechanisms: (1) an explicit 12-biclique fractional decomposition of $D\otimes D$ with uniform weight $1/2$, accompanied by a dual certificate proving optimality at value 6, which shows the rank function is not multiplicative; and (2) a projection lemma that maps bicliques of $A\otimes A'$ to bicliques of $A$ by intersecting with blocks of the form $a_{ij}A'$, normalizing by $\mathrm{bp}_f(A')$, to convert an optimal fractional partition of the product into a fractional cover of $A$. The computational upper bound rests on a column-generation scheme whose pricing problem is a maximum-weight submatrix ILP posed per maximal biclique, with maximal bicliques computed by the MBEA algorithm, initialized inductively from Kronecker products of optimal biclique sets, and pruned by counting slack iterations.

What would settle it

Independently solve the fractional biclique partition LP for $D^{\otimes 5}$ with a different solver or an exact rational LP algorithm: if the optimum is below 75.201302, the reported optimality claim is wrong, and if it is above, the upper bound is false. For the non-multiplicativity result, check the twelve 0-1 matrices and the dual certificate in Section 3 by hand to confirm $\mathrm{bp}_f(D\otimes D)=6$.

Watch

Extended reading notes

Core claim

The central discovery is a counterexample to multiplicativity of the fractional binary rank: for the Domino adjacency matrix $D$, $\mathrm{bp}_f(D\otimes D) = 6 < 6.25 = \mathrm{bp}_f(D)^2$. The 12-biclique fractional decomposition with weight $1/2$ on each biclique, together with an explicit dual feasible solution, certifies this value. Consequently, by Fekete's lemma the asymptotic fractional binary rank $\mathrm{bp}^\infty_f(D)$ is well defined and strictly less than $\mathrm{bp}_f(D)$. The paper additionally proves that $\mathrm{bp}_f(A\otimes A') \ge \max\{ \mathrm{bc}_f(A)\cdot\mathrm{bp}_f(A'),\ \mathrm{bp}_f(A)\cdot\mathrm{bc}_f(A') \}$, which implies that the fractional biclique cover number $\mathrm{bc}_f(A)$ is a lower bound on the asymptotic fractional binary rank; for the Domino this yields the lower bound $2$. Finally, the engineered column-generation algorithm computes $\mathrm{bp}_f(D^{\otimes 5}) = 75.201302$, giving $\mathrm{bp}^\infty_f(D) \le 2.372713$.

Load-bearing premise

The claimed upper bound 2.372713 depends on the correctness of the engineered column-generation solver with Gurobi 10.0.0 on the $k=5$ instance, whose reported value 75.201302 is not accompanied by a solver certificate or versioned code, so it cannot be independently verified from the paper alone.

Editorial extensions

If this is right

The Domino graph is a concrete witness that the fractional binary rank is not multiplicative, placing it alongside nonnegative and boolean rank as rank functions whose asymptotic value can be strictly smaller than the finite value.
The interval $2 \le \mathrm{bp}^\infty_f(D) \le 2.372713$ is now the best-known bound, and solving $k=6$ would tighten the upper endpoint since Fekete's lemma identifies the infimum over all $k$.
The proven chain $i(A) \le \mathrm{bc}_f(A) \le \mathrm{bp}^\infty_f(A) \le \mathrm{bp}_f(A) \le \mathrm{bp}(A)$ gives a new general lower bound for the asymptotic fractional binary rank in terms of the fractional biclique cover number, which is at least as strong as the fooling-set bound.
The column-generation machinery with inductive initialization and slack-based pruning handles problems with double-exponentially many bicliques, so the same pipeline can be retargeted at other small graphs and their Kronecker powers.

Reading between the lines

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

If further Kronecker powers keep pushing the $k$-th root toward 2, the asymptotic fractional binary rank of the Domino would equal its fractional biclique cover number, which would match the value predicted by the fractional boolean rank—an equality the paper does not itself claim.
The projection lemma gives a general template for lower-bounding fractional binary rank of products: project an optimal fractional partition of $A\otimes A'$ onto one factor and compare with a fractional cover; applying this to graphs other than the Domino could reveal whether non-multiplicativity is rare or common.
Since the fractional relaxation already fails multiplicativity at the second power, the integer binary rank question likely requires a genuinely different approach; a fractional counterexample does not transfer to the integer case, but it rules out simple fractional-based proofs of multiplicativity.
The suggested rational value $2059/149$ for $\mathrm{bp}_f(D^{\otimes 3})$ invites a combinatorial construction that would make the upper-bound sequence exactly computable for small $k$ without rerunning the optimizer.

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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

Read this one for the counterexample. The authors give an explicit 12-biclique decomposition of D⊗D, each with weight 1/2, showing bpf(D⊗D) ≤ 6 < 6.25 = bpf(D)^2. That rigorously kills multiplicativity of fractional binary rank under Kronecker products, and it is the real contribution. The fractional boolean (cover) rank is multiplicative, so this distinction is meaningful.

The rest is a clean lower-bound lemma plus an engineered column-generation pipeline. Lemma 3 proves bcf(A) ≤ bp∞_f(A) via a projection argument, and it is at least as strong as the isolating-set bound. The proof is correct as far as I can see, aside from one typo: the sentence after the first half says 'bcf(A⊗A′)≤bpf(A)·bcf(A′)' which is not the bound you need; swapping roles gives bpf(A⊗A') ≥ bpf(A) bcf(A'), and since bpf is invariant under swapping factors, that works.

The computational part is clever and fragile. The column-generation scheme—inductive initialization from Kronecker products, pricing via QUBOs over maximal bicliques, pruning by a slack counter—is sensible and the reported running times are plausible. But the upper bound in Theorem 2, bpf(D^⊗5)=75.201302, comes from a Gurobi run with no solver certificate, no log files, and no pinned repository revision. Pricing uses heuristic thresholds (ε=1e-6, pruning after 3 slacks), and floating-point tolerances could mean the final solution is not exactly feasible. As written, an independent group cannot audit that number. If it is off, the interval [2,2.372713] is not established. However, the rigorous decomposition for k=2 already gives bp∞_f(D) ≤ sqrt(6) ≈ 2.449, so the qualitative conclusion—asymptotic fractional binary rank strictly below 2.5—does not depend on the computational result.

There is also a small verifiability gap in Section 3: the paper claims a dual certificate for bpf(D⊗D) ≥ 6 but does not display it. This is minor and easily fixed; the disproof only needs the upper bound.

Bottom line: the core result is new, correct, and worth a serious referee. Citations are appropriate, and the distinction from Watts' multiplicativity result is clearly drawn. The authors should be asked to supply the missing dual certificate and to make the computational run reproducible with certificates, logs, and a commit hash. This is for people working on combinatorial matrix ranks, communication complexity, or exponential-size LPs.

Referee Report

1 major / 6 minor

Summary. The paper studies the fractional binary rank (equivalently, the fractional biclique partition number bp_f) of Kronecker powers of binary matrices, motivated by the open question whether binary rank is multiplicative under Kronecker products. The central mathematical contribution is an explicit fractional biclique decomposition of D⊗D, where D is the 3×3 Domino matrix, using twelve bicliques each with weight 1/2, which yields bp_f(D⊗D) ≤ 6 < 6.25 = bp_f(D)^2 and thereby disproves multiplicativity of the fractional binary rank. The authors prove Lemma 3, a lower bound bp_f(A⊗A') ≥ max{bcf(A) bp_f(A'), bp_f(A) bcf(A')}, which implies bcf(A) ≤ bp∞_f(A) and gives the lower bound 2 for the Domino. On the computational side, the paper engineers a column-generation method with inductive initialization, ILP-based pricing over maximal bicliques, and pruning, and reports bp_f(D^{⊗k}) values for k=1,...,5, leading to the interval bp∞_f(D) ∈ [2, 2.372713]. The paper also discusses the relation between biclique cover/partition numbers and the isolating-set bound.

Significance. The explicit disproof of multiplicativity is a clean and checkable result: it settles a natural conjecture for the fractional relaxation of binary rank in the negative, and it is the kind of counterexample that can be verified by direct inspection. Lemma 3 is a useful strengthening of the standard isolating-set lower bound and is proved with a projection argument that, once the details are spelled out, is sound. The computational engineering is thoughtful and the reported upper bounds are plausible, but the exactness of the k=5 optimum is not independently auditable from the manuscript. Even without the computed upper bound, the elementary bound bp∞_f(D) ≤ sqrt(6) ≈ 2.449 obtained from the explicit decomposition already implies the qualitative phenomenon that the asymptotic fractional binary rank can be strictly below the fractional binary rank. The paper therefore contains a solid core result plus a potentially interesting but less fully certified computational upper bound.

major comments (1)

[5.3–5.4] The exactness of the reported values in the table of Section 5.4, in particular bp_f(D^{⊗5}) = 75.201302, is not verifiable from the manuscript. The column-generation loop stops when no column in the current pool has reduced cost above 1+ε, but the paper does not report the final maximum pricing objective α, the implied dual lower bound, or any certificate that the final primal solution satisfies Mx=1. Pruning in Section 5.3 can delete columns permanently, and although pricing over all maximal bicliques should in principle regenerate useful columns, the completeness of the search and the final duality gap are asserted rather than documented. Please provide the final dual bound for each k, a machine-checkable certificate (e.g., the list of bicliques and weights for k=5), and the repository commit hash; alternatively, state explicitly that the table gives upper bounds obtained from feasible fractional partitions and adjust the wording of Theorem 2 and the abstract.

minor comments (6)

[Section 3] The sentence 'this decomposition is also optimal because there is a dual feasible solution certifying bp_f(D⊗D) ≥ 6' is not supported by an explicit certificate in the text; no dual vector is shown, in contrast to the certificate Y* provided for D. While the upper bound 6 alone is enough for the non-multiplicativity claim, the table in Section 5.4 reports 6.0 as an exact value, so the claimed optimality of the 12-biclique decomposition should be backed by the dual vector or a file reference.
[Section 4] Please expand the sentence 'The intersection ... yields a feasible fractional biclique partition of G_A′' in the proof of Lemma 3 to explain that restricting each intersecting biclique to the block gives a biclique in the block and that the edge-cover weights sum to one for every edge in the block.
[Section 5.2] The dual variables in Eq. (5) are unrestricted in sign because the primal problem has equality constraints; please state this explicitly when introducing the dual.
[Section 5.3] The sentence beginning 'To ensure that the number of bicliques does not blow up too fast when we It is possible...' is grammatically broken and should be rewritten.
[Section 5.4] The speculation that bp_f(D^{⊗3}) could equal 2059/149 is unsupported; either verify this identity or remove the sentence.
[Section 5.4] Please include the repository commit hash and the exact Gurobi parameter settings (e.g., MIP gap for the pricing ILPs, barrier crossover tolerances) to make the computational claims reproducible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the non-multiplicativity result is an explicit decomposition with a dual certificate, and the upper bound is a feasible-solution computation rather than a fitted input.

full rationale

The paper's central claim, that the fractional binary rank is not multiplicative, is established in Section 3 by an explicit 12-biclique weighted decomposition of D⊗D giving bpf(D⊗D) ≤ 6 < 6.25 = bpf(D)^2, together with a stated dual feasible solution certifying optimality from below. This is a self-contained mathematical proof, not a fitted parameter renamed as a prediction. The lower bound in Theorem 2 is proved in Section 4 via Lemma 3, whose projection argument is given in full and only borrows a proof strategy from reference [8]; it does not assume the target result. The upper bound bpf(D^{⊗5}) = 75.201302 is obtained by running column generation on the defining LP (3); any feasible restricted-master solution is a feasible fractional biclique partition for the full problem, so its objective value is a genuine upper bound, not a value derived from the theorem it is used to prove. No load-bearing self-citations appear: the reference list contains no works by the present authors, and the cited uniqueness/multiplicativity facts (e.g., fractional boolean rank multiplicativity [12]) are used only for contrast, not to justify the paper's own conclusions. The computational endpoint is not shipped with certificates or a pinned commit hash, which is a reproducibility limitation and a verification-gap concern, but it is not circularity: an upper bound from a feasible LP solution does not reduce by construction to the quantity it bounds. Even if the k=5 run were disregarded, the rigorous sqrt(6) ≈ 2.449 bound from the Section 3 decomposition already suffices to prove strict asymptotic descent below 2.5. Overall, the derivation chain is independent of its own outputs.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The mathematical lower bound and the non-multiplicativity counterexample rely only on standard LP duality, Fekete's lemma, and elementary combinatorial projection arguments. The computational upper bound additionally depends on the correctness of the MBEA algorithm and the Gurobi solver. There are no fitted constants in the derivation; the two hand-chosen engineering parameters (pricing tolerance and pruning threshold) affect runtime and memory but not the validity of the final bounds.

free parameters (2)

Pricing tolerance epsilon = 1e-6
Numerical threshold used in the pricing problem to decide if a new column has negative reduced cost (Section 5.2); a hand-chosen engineering constant that does not affect the mathematical claims.
Pruning slack threshold = 3
A biclique column is pruned if its dual slack is below 1 for more than 3 consecutive column generation iterations (Section 5.3); chosen by hand to balance memory and convergence, not part of the theory.

assumptions (6)

standard math Fekete's lemma: for sub-multiplicative sequences, the limit of the k-th root equals the infimum.
Used in Section 1 and Section 3 to define the asymptotic fractional binary rank and to justify that any feasible fractional partition of D^k yields an upper bound.
standard math Strong LP duality and existence of optimal solutions for the primal-dual pair in (3) and (5).
Used in Section 4 to certify optimality of the Domino decomposition and in the column generation algorithm (master problem and pricing).
domain assumption The restriction of a biclique in the Kronecker product to a block a_ij A' is a biclique in A'.
Core of the projection argument in Lemma 3 (Section 4); the assumption is stated in the proof and is checkable.
domain assumption Known result from [12]: the fractional boolean rank (fractional biclique cover number) is multiplicative and i(A) ≤ bcf(A).
Used in Section 1 and Section 4 to establish the lower bound chain; these are external results taken as given.
domain assumption Correctness of MBEA algorithm [13] for enumerating maximal bicliques.
Used in Section 5.2 to generate the set of maximal bicliques for the pricing problem; the paper does not prove this algorithm.
domain assumption Correctness of the LP/IP solver (Gurobi 10.0.0) for the master and pricing problems.
The computational results in Section 5.4 depend on the solver returning exact optima within tolerance; no certificates are shipped.

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Cite this review

Pith. "Pith review of Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices." pith.science (2026). https://pith.science/paper/PDPSDKLM

@misc{pith2026250206730,
  author       = {Pith},
  title        = {Pith review of: Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PDPSDKLM}},
  note         = {Machine review of arXiv:2502.06730}
}

abstract

We investigate structural properties of the binary rank of Kronecker powers of binary matrices, equivalently, the biclique partition numbers of the corresponding bipartite graphs. To this end, we engineer a Column Generation approach to solve linear optimization problems for the fractional biclique partition number of bipartite graphs, specifically examining the Domino graph and its Kronecker powers. We address the challenges posed by the double exponential growth of the number of bicliques in increasing Kronecker powers. We discuss various strategies to generate suitable initial sets of bicliques, including an inductive method for increasing Kronecker powers. We show how to manage the number of active bicliques to improve running time and to stay within memory limits. Our computational results reveal that the fractional binary rank is not multiplicative with respect to the Kronecker product. Hence, there are binary matrices, and bipartite graphs, respectively, such as the Domino, where the asymptotic fractional binary rank is strictly smaller than the fractional binary rank. While we used our algorithm to reduce the upper bound, we formally prove that the fractional biclique cover number is a lower bound, which is at least as good as the widely used isolating (or fooling set) bound. For the Domino, we obtain that the asymptotic fractional binary rank lies in the interval $[2,2.373]$. Since our computational resources are not sufficient to further reduce the upper bound, we encourage further exploration using more substantial computing resources or further mathematical engineering techniques to narrow the gap and advance our understanding of biclique partitions, particularly, to settle the open question whether binary rank and biclique partition number are multiplicative with respect to the Kronecker product.

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Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

[1]

Beasley, Hartmut Klauck, Troy Lee, and Dirk Oliver Theis

LeRoy B. Beasley, Hartmut Klauck, Troy Lee, and Dirk Oliver Theis. Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082) . Dagstuhl Reports , 3(2):127--143, 2013. URL: https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.2.127, https://doi.org/10.4230/DagRep.3.2.127 doi:10.423...

work page doi:10.4230/dagrep.3.2.127 2013
[2]

On the asymptotic nonnegative rank of matrices and its applications in information theory

Yeow Meng Chee, Quoc-Tung Le, and Hoang Ta. On the asymptotic nonnegative rank of matrices and its applications in information theory. In 2024 IEEE International Symposium on Information Theory (ISIT) , pages 1955--1960, 2024. https://doi.org/10.1109/ISIT57864.2024.10619320 doi:10.1109/ISIT57864.2024.10619320

arXiv 2024
[3]

Submatrix with largest sum, 2021

Yet Another Math Programming Consultant. Submatrix with largest sum, 2021. URL: https://yetanothermathprogrammingconsultant.blogspot.com/2021/01/submatrix-with-largest-sum.html

work page 2021
[4]

Maximal-sum submatrix search using a hybrid contraint programming/linear programming approach

Guillaume Derval and Pierre Schaus. Maximal-sum submatrix search using a hybrid contraint programming/linear programming approach. European Journal of Operational Research , 297(3):853--865, 2022

work page 2022
[5]

\"U ber die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten

Michael Fekete. \"U ber die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten. Mathematische Zeitschrift , 17(1):228--249, 1923

work page 1923
[6]

Upper bounds on the boolean rank of kronecker products

Ishay Haviv and Michal Parnas. Upper bounds on the boolean rank of kronecker products. Procedia Computer Science , 195:172–180, 2021. Proceedings of the XI Latin and American Algorithms, Graphs and Optimization Symposium. URL: https://www.sciencedirect.com/science/article/pii/S1877050921021621, https://doi.org/10.1016/j.procs.2021.11.023 doi:10.1016/j.pro...

work page doi:10.1016/j.procs.2021.11.023 2021
[7]

On the Binary and Boolean Rank of Regular Matrices

Ishay Haviv and Michal Parnas. On the binary and boolean rank of regular matrices, 2023. URL: https://arxiv.org/abs/2203.13073, https://arxiv.org/abs/2203.13073 arXiv:2203.13073

work page Pith review arXiv 2023
[8]

On biclique cover and partition problems

Tetiana Lavynska. On biclique cover and partition problems. Bachelor's thesis, University of Magdeburg, Faculty of Computer Science, March 2018

work page 2018

Show all 13 references

[9]

u bbecke. Column generation. In M. L\

M. L\" u bbecke. Column generation. In M. L\" u bbecke, editor, Wiley Encyclopedia of Operations Research and Management Science . John Wiley & Sons, Chichester, Jan 2011. https://doi.org/10.1002/9780470400531.eorms0158 doi:10.1002/9780470400531.eorms0158

2011 doi
[10]

Nonnegative rank vs

Thomas Watson. Nonnegative rank vs. binary rank. Chicago Journal of Theoretical Computer Science , 2016(2), February 2016

2016
[11]

Boolean rank of kronecker products

Valerie L Watts. Boolean rank of kronecker products. Linear Algebra and its Applications , 336(1-3):261--264, 2001

2001
[12]

Fractional biclique covers and partitions of graphs

Valerie L Watts. Fractional biclique covers and partitions of graphs. the electronic journal of combinatorics , pages R74--R74, 2006

2006
[13]

Phillips, Gary L

Yun Zhang, Charles A. Phillips, Gary L. Rogers, Erich J. Baker, Elissa J. Chesler, and Michael A. Langston. On finding bicliques in bipartite graphs: a novel algorithm and its application to the integration of diverse biological data types. BMC Bioinformatics , 15(110), 2014. ...

2014 doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Beasley, Hartmut Klauck, Troy Lee, and Dirk Oliver Theis

LeRoy B. Beasley, Hartmut Klauck, Troy Lee, and Dirk Oliver Theis. Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082) . Dagstuhl Reports , 3(2):127--143, 2013. URL: https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.2.127, https://doi.org/10.4230/DagRep.3.2.127 doi:10.423...

work page doi:10.4230/dagrep.3.2.127 2013

[2] [2]

On the asymptotic nonnegative rank of matrices and its applications in information theory

Yeow Meng Chee, Quoc-Tung Le, and Hoang Ta. On the asymptotic nonnegative rank of matrices and its applications in information theory. In 2024 IEEE International Symposium on Information Theory (ISIT) , pages 1955--1960, 2024. https://doi.org/10.1109/ISIT57864.2024.10619320 doi:10.1109/ISIT57864.2024.10619320

arXiv 2024

[3] [3]

Submatrix with largest sum, 2021

Yet Another Math Programming Consultant. Submatrix with largest sum, 2021. URL: https://yetanothermathprogrammingconsultant.blogspot.com/2021/01/submatrix-with-largest-sum.html

work page 2021

[4] [4]

Maximal-sum submatrix search using a hybrid contraint programming/linear programming approach

Guillaume Derval and Pierre Schaus. Maximal-sum submatrix search using a hybrid contraint programming/linear programming approach. European Journal of Operational Research , 297(3):853--865, 2022

work page 2022

[5] [5]

\"U ber die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten

Michael Fekete. \"U ber die verteilung der wurzeln bei gewissen algebraischen gleichungen mit ganzzahligen koeffizienten. Mathematische Zeitschrift , 17(1):228--249, 1923

work page 1923

[6] [6]

Upper bounds on the boolean rank of kronecker products

Ishay Haviv and Michal Parnas. Upper bounds on the boolean rank of kronecker products. Procedia Computer Science , 195:172–180, 2021. Proceedings of the XI Latin and American Algorithms, Graphs and Optimization Symposium. URL: https://www.sciencedirect.com/science/article/pii/S1877050921021621, https://doi.org/10.1016/j.procs.2021.11.023 doi:10.1016/j.pro...

work page doi:10.1016/j.procs.2021.11.023 2021

[7] [7]

On the Binary and Boolean Rank of Regular Matrices

Ishay Haviv and Michal Parnas. On the binary and boolean rank of regular matrices, 2023. URL: https://arxiv.org/abs/2203.13073, https://arxiv.org/abs/2203.13073 arXiv:2203.13073

work page Pith review arXiv 2023

[8] [8]

On biclique cover and partition problems

Tetiana Lavynska. On biclique cover and partition problems. Bachelor's thesis, University of Magdeburg, Faculty of Computer Science, March 2018

work page 2018