2 Search Results for "Ta, Hoang"

Document

DOI: 10.4230/LIPIcs.SEA.2025.18

Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

Authors: Angikar Ghosal and Andreas Karrenbauer

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

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.

Cite as

Angikar Ghosal and Andreas Karrenbauer. Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 18:1-18:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

Copy BibTex To Clipboard

@InProceedings{ghosal_et_al:LIPIcs.SEA.2025.18,
  author =	{Ghosal, Angikar and Karrenbauer, Andreas},
  title =	{{Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{18:1--18:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.18},
  URN =		{urn:nbn:de:0030-drops-232568},
  doi =		{10.4230/LIPIcs.SEA.2025.18},
  annote =	{Keywords: Asymptotic Binary Rank, Algorithm Engineering, Combinatorics of Bipartite Graphs, Linear Programming}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2022.48

Larger Corner-Free Sets from Combinatorial Degenerations

Authors: Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

There is a large and important collection of Ramsey-type combinatorial problems, closely related to central problems in complexity theory, that can be formulated in terms of the asymptotic growth of the size of the maximum independent sets in powers of a fixed small hypergraph, also called the Shannon capacity. An important instance of this is the corner problem studied in the context of multiparty communication complexity in the Number On the Forehead (NOF) model. Versions of this problem and the NOF connection have seen much interest (and progress) in recent works of Linial, Pitassi and Shraibman (ITCS 2019) and Linial and Shraibman (CCC 2021). We introduce and study a general algebraic method for lower bounding the Shannon capacity of directed hypergraphs via combinatorial degenerations, a combinatorial kind of "approximation" of subgraphs that originates from the study of matrix multiplication in algebraic complexity theory (and which play an important role there) but which we use in a novel way. Using the combinatorial degeneration method, we make progress on the corner problem by explicitly constructing a corner-free subset in F₂ⁿ × F₂ⁿ of size Ω(3.39ⁿ/poly(n)), which improves the previous lower bound Ω(2.82ⁿ) of Linial, Pitassi and Shraibman (ITCS 2019) and which gets us closer to the best upper bound 4^{n - o(n)}. Our new construction of corner-free sets implies an improved NOF protocol for the Eval problem. In the Eval problem over a group G, three players need to determine whether their inputs x₁, x₂, x₃ ∈ G sum to zero. We find that the NOF communication complexity of the Eval problem over F₂ⁿ is at most 0.24n + 𝒪(log n), which improves the previous upper bound 0.5n + 𝒪(log n).

Cite as

Matthias Christandl, Omar Fawzi, Hoang Ta, and Jeroen Zuiddam. Larger Corner-Free Sets from Combinatorial Degenerations. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 48:1-48:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{christandl_et_al:LIPIcs.ITCS.2022.48,
  author =	{Christandl, Matthias and Fawzi, Omar and Ta, Hoang and Zuiddam, Jeroen},
  title =	{{Larger Corner-Free Sets from Combinatorial Degenerations}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{48:1--48:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.48},
  URN =		{urn:nbn:de:0030-drops-156441},
  doi =		{10.4230/LIPIcs.ITCS.2022.48},
  annote =	{Keywords: Corner-free sets, communication complexity, number on the forehead, combinatorial degeneration, hypergraphs, Shannon capacity, eval problem}
}

Refine by Type
2 Document/PDF
1 Document/HTML

Refine by Publication Year
1 2025
1 2022

Refine by Author
1 Christandl, Matthias
1 Fawzi, Omar
1 Ghosal, Angikar
1 Karrenbauer, Andreas
1 Ta, Hoang
Show More...

Refine by Series/Journal
2 LIPIcs

Refine by Classification
1 Mathematics of computing → Discrete mathematics
1 Theory of computation → Algebraic complexity theory
1 Theory of computation → Communication complexity
1 Theory of computation → Discrete optimization

Refine by Keyword
1 Algorithm Engineering
1 Asymptotic Binary Rank
1 Combinatorics of Bipartite Graphs
1 Corner-free sets
1 Linear Programming
Show More...

2 Search Results for "Ta, Hoang"

Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices

Abstract

Cite as

Larger Corner-Free Sets from Combinatorial Degenerations

Abstract

Cite as

Thanks for your feedback!

Could not send message