14 Search Results for "Ivanyos, Gábor"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.47
Computing the Skyscraper Invariant

Authors: Marc Fersztand and Jan Jendrysiak

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
We develop the first algorithms for computing the Skyscraper Invariant [FJNT24]. This is a filtration of the classical rank invariant for multiparameter persistence modules defined by the Harder-Narasimhan filtrations along every central charge supported at a single parameter value. Cheng’s algorithm [Cheng24] can be used to compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension, but in practice, the large dimension of persistence modules makes this direct approach infeasible. We show that by exploiting the additivity of the HN filtration and the special central charges, one can get away with a brute-force approach. For d-parameter modules, this produces an FPT ε-approximate algorithm with runtime dominated by 𝒪(1/ε^d ⋅ T_dec), where T_dec is the time for decomposition, which we compute with aida [DJK25]. We show that the wall-and-chamber structure of the module can be computed via lower envelopes of degree d - 1 polynomials. This allows for an exact computation of the Skyscraper Invariant roughly in 𝒪(n^d ⋅ T_dec) time for n the size of the presentation and enables a fast hybrid algorithm. For 2-parameter modules, we have implemented not only our algorithms but also, for the first time, Cheng’s algorithm. We compare all algorithms and, as a proof of concept for data analysis, compute a filtered version of the Multiparameter Landscape for biomedical data.
Cite as

Marc Fersztand and Jan Jendrysiak. Computing the Skyscraper Invariant. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 47:1-47:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{fersztand_et_al:LIPIcs.SoCG.2026.47,
  author =	{Fersztand, Marc and Jendrysiak, Jan},
  title =	{{Computing the Skyscraper Invariant}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{47:1--47:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.47},
  URN =		{urn:nbn:de:0030-drops-258535},
  doi =		{10.4230/LIPIcs.SoCG.2026.47},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence, Persistence, Harder-Narasimhan Filtration, Skyscraper Invariant}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.32
Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem

Authors: Jin-Yi Cai and Ben Young

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Valiant’s Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets ℱ and 𝒢 of tensors (a.k.a. constraint functions or signatures) are related by a holographic transformation, then ℱ and 𝒢 are Holant-indistinguishable, i.e., every tensor network using tensors from ℱ, respectively from 𝒢, contracts to the same value. Xia (ICALP 2010) conjectured the converse of the Holant theorem, but a counterexample was found based on vanishing signatures, those which are Holant-indistinguishable from 0. We prove two near-converses of the Holant theorem using techniques from invariant theory. (I) Holant-indistinguishable ℱ and 𝒢 always admit two sequences of holographic transformations mapping them arbitrarily close to each other, i.e., their GL_q-orbit closures intersect. (II) We show that vanishing signatures are the only true obstacle to a converse of the Holant theorem. As corollaries of the two theorems we obtain the first characterization of homomorphism-indistinguishability over graphs of bounded degree, a long standing open problem, and show that two graphs with invertible adjacency matrices are isomorphic if and only if they are homomorphism-indistinguishable over graphs with maximum degree at most three. We also show that Holant-indistinguishability is complete for a complexity class TOCI introduced by Lysikov and Walter [Vladimir Lysikov and Michael Walter, 2024], and hence hard for graph isomorphism.
Cite as

Jin-Yi Cai and Ben Young. Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2026.32,
  author =	{Cai, Jin-Yi and Young, Ben},
  title =	{{Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{32:1--32:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.32},
  URN =		{urn:nbn:de:0030-drops-253198},
  doi =		{10.4230/LIPIcs.ITCS.2026.32},
  annote =	{Keywords: Holant, Orbit Closure Intersection, Homomorphism Indistinguishability, Tensor Network}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.110
Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing

Authors: Jaikumar Radhakrishnan, Chaitanya Reddy, and Rakesh Venkat

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We consider the communication complexity of the graph connectivity problem, where the edges of an n-vertex undirected graph G are distributed between two parties Alice and Bob, who are then required to communicate to determine if G is connected. We show that in any randomized protocol with two-rounds of communication, Alice and Bob must exchange Ω(nlog n) bits; such a lower bound for one-round protocols was shown by Sun and Woodruff (APPROX/RANDOM 2015). A one-round deterministic protocol, where Alice sends O(n log n) bits and Bob determines the answer, was observed by Hajnal, Maass and Turan (STOC 1988); they also showed a matching lower bound of Ω(n log n) bits for deterministic protocols with unbounded rounds of communication. For randomized protocols, a reduction from the set disjointness problem due to Babai, Frankl and Simon (FOCS 1986) implies a randomized lower bound of Ω(n) even with unbounded rounds of communication. Whether this lower bound can be improved to Ω(n log n) has been an outstanding open question, whose algorithmic implications were recently emphasized by Apers, Efron, Gawrychowski, Lee, Mukopadhyay and Nanongkai (FOCS 2022). Our lower bound for randomized two-round protocols is based on a reduction from a restricted version of the two-player pointer chasing problem originally studied by Papadimitriou and Sipser (JCSS 1984). Using this reduction, we show an ω(n) lower bounds on graph connectivity for any constant number of rounds by extending deterministic lower bounds shown by Ponzio, Radhakrishnan and Venkatesh (JCSS 2001) to the randomized setting.
Cite as

Jaikumar Radhakrishnan, Chaitanya Reddy, and Rakesh Venkat. Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 110:1-110:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{radhakrishnan_et_al:LIPIcs.ITCS.2026.110,
  author =	{Radhakrishnan, Jaikumar and Reddy, Chaitanya and Venkat, Rakesh},
  title =	{{Optimal Two-Round Communication Lower Bound for Graph Connectivity via Pointer Chasing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{110:1--110:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.110},
  URN =		{urn:nbn:de:0030-drops-253974},
  doi =		{10.4230/LIPIcs.ITCS.2026.110},
  annote =	{Keywords: Communication complexity}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.7
An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem

Authors: Marco Aldi, Sevag Gharibian, and Dorian Rudolph

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash equilibria, subclasses of TFNP remain arguably few and far between. In this work, we define two new subclasses of TFNP borne of the study of complex polynomial systems: Multi-homogeneous Systems (MHS) and Sparse Fundamental Theorem of Algebra (SFTA). The first of these is based on Bézout’s theorem from algebraic geometry, marking the first TFNP subclass based on an algebraic geometric principle. At the heart of our study is the computational problem known as Quantum SAT (QSAT) with a System of Distinct Representatives (SDR), first studied by [Laumann, Läuchli, Moessner, Scardicchio, and Sondhi 2010]. Among other results, we show that QSAT with SDR is MHS-complete, thus giving not only the first link between quantum complexity theory and TFNP, but also the first TFNP problem whose classical variant (SAT with SDR) is easy but whose quantum variant is hard. We also show how to embed the roots of a sparse, high-degree, univariate polynomial into QSAT with SDR, obtaining that SFTA is contained in a zero-error version of MHS. We conjecture this construction also works in the low-error setting, which would imply SFTA ⊆ MHS.
Cite as

Marco Aldi, Sevag Gharibian, and Dorian Rudolph. An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aldi_et_al:LIPIcs.ITCS.2026.7,
  author =	{Aldi, Marco and Gharibian, Sevag and Rudolph, Dorian},
  title =	{{An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{7:1--7:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.7},
  URN =		{urn:nbn:de:0030-drops-252946},
  doi =		{10.4230/LIPIcs.ITCS.2026.7},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), Quantum Satisfiability Problem (QSAT), total function NP (TFNP)}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.154
Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations

Authors: Ruiwen Dong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We show that Submonoid Membership is decidable in n-dimensional lamplighter groups (ℤ/pℤ) ≀ ℤⁿ for any prime p and integer n. More generally, we show decidability of Submonoid Membership in semidirect products of the form 𝒴 ⋊ ℤⁿ, where 𝒴 is any finitely presented module over the Laurent polynomial ring 𝔽_p[X₁^{±}, …, X_n^{±}]. Combined with a result of Shafrir (2024), this gives the first example of a group G and a finite index subgroup G̃ ≤ G, such that Submonoid Membership is decidable in G̃ but undecidable in G. To obtain our decidability result, we reduce Submonoid Membership in 𝒴 ⋊ ℤⁿ to solving S-unit equations over 𝔽_p[X₁^{±}, …, X_n^{±}]-modules. We show that the solution set of such equations is effectively p-automatic, extending a result of Adamczewski and Bell (2012). As an intermediate result, we also obtain that the solution set of the Knapsack Problem in 𝒴 ⋊ ℤⁿ is effectively p-automatic.
Cite as

Ruiwen Dong. Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 154:1-154:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dong:LIPIcs.ICALP.2025.154,
  author =	{Dong, Ruiwen},
  title =	{{Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{154:1--154:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.154},
  URN =		{urn:nbn:de:0030-drops-235316},
  doi =		{10.4230/LIPIcs.ICALP.2025.154},
  annote =	{Keywords: Submonoid Membership, lamplighter groups, S-unit equations, p-automatic sets, Knapsack in groups}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.99
Algorithmic Aspects of Semistability of Quiver Representations

Authors: Yuni Iwamasa, Taihei Oki, and Tasuku Soma

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and σ-semistability, finding the maximizers of King’s criterion, and computing the Harder-Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King’s criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the σ-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.
Cite as

Yuni Iwamasa, Taihei Oki, and Tasuku Soma. Algorithmic Aspects of Semistability of Quiver Representations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 99:1-99:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{iwamasa_et_al:LIPIcs.ICALP.2025.99,
  author =	{Iwamasa, Yuni and Oki, Taihei and Soma, Tasuku},
  title =	{{Algorithmic Aspects of Semistability of Quiver Representations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{99:1--99:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.99},
  URN =		{urn:nbn:de:0030-drops-234762},
  doi =		{10.4230/LIPIcs.ICALP.2025.99},
  annote =	{Keywords: quivers, \sigma-semistability, King’s criterion, operator scaling, submodular flow}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.41
Decomposing Multiparameter Persistence Modules

Authors: Tamal K. Dey, Jan Jendrysiak, and Michael Kerber

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.
Cite as

Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.41,
  author =	{Dey, Tamal K. and Jendrysiak, Jan and Kerber, Michael},
  title =	{{Decomposing Multiparameter Persistence Modules}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.41},
  URN =		{urn:nbn:de:0030-drops-231939},
  doi =		{10.4230/LIPIcs.SoCG.2025.41},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence Modules, Persistence, Decomposition}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.87
Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras

Authors: Gábor Ivanyos, Tushant Mittal, and Youming Qiao

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

Abstract
One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is, matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lovász (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).
Cite as

Gábor Ivanyos, Tushant Mittal, and Youming Qiao. Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 87:1-87:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ivanyos_et_al:LIPIcs.ITCS.2022.87,
  author =	{Ivanyos, G\'{a}bor and Mittal, Tushant and Qiao, Youming},
  title =	{{Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{87:1--87:21},
  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.2022.87},
  URN =		{urn:nbn:de:0030-drops-156837},
  doi =		{10.4230/LIPIcs.ITCS.2022.87},
  annote =	{Keywords: derandomization, polynomial identity testing, symbolic determinant, non-commutative rank, Lie algebras}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.8
From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge Between Graphs and Alternating Matrix Spaces

Authors: Xiaohui Bei, Shiteng Chen, Ji Guan, Youming Qiao, and Xiaoming Sun

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
In the 1970’s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration. (Dedicated to the memory of Ker-I Ko)
Cite as

Xiaohui Bei, Shiteng Chen, Ji Guan, Youming Qiao, and Xiaoming Sun. From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge Between Graphs and Alternating Matrix Spaces. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 8:1-8:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bei_et_al:LIPIcs.ITCS.2020.8,
  author =	{Bei, Xiaohui and Chen, Shiteng and Guan, Ji and Qiao, Youming and Sun, Xiaoming},
  title =	{{From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge Between Graphs and Alternating Matrix Spaces}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{8:1--8:48},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.8},
  URN =		{urn:nbn:de:0030-drops-116932},
  doi =		{10.4230/LIPIcs.ITCS.2020.8},
  annote =	{Keywords: independent set, vertex coloring, graphs, matrix spaces, isotropic subspace}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.66
On Learning Linear Functions from Subset and Its Applications in Quantum Computing

Authors: Gábor Ivanyos, Anupam Prakash, and Miklos Santha

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
Let F_{q} be the finite field of size q and let l: F_{q}^{n} -> F_{q} be a linear function. We introduce the Learning From Subset problem LFS(q,n,d) of learning l, given samples u in F_{q}^{n} from a special distribution depending on l: the probability of sampling u is a function of l(u) and is non zero for at most d values of l(u). We provide a randomized algorithm for LFS(q,n,d) with sample complexity (n+d)^{O(d)} and running time polynomial in log q and (n+d)^{O(d)}. Our algorithm generalizes and improves upon previous results [Friedl et al., 2014; Gábor Ivanyos, 2008] that had provided algorithms for LFS(q,n,q-1) with running time (n+q)^{O(q)}. We further present applications of our result to the Hidden Multiple Shift problem HMS(q,n,r) in quantum computation where the goal is to determine the hidden shift s given oracle access to r shifted copies of an injective function f: Z_{q}^{n} -> {0, 1}^{l}, that is we can make queries of the form f_{s}(x,h) = f(x-hs) where h can assume r possible values. We reduce HMS(q,n,r) to LFS(q,n, q-r+1) to obtain a polynomial time algorithm for HMS(q,n,r) when q=n^{O(1)} is prime and q-r=O(1). The best known algorithms [Andrew M. Childs and Wim van Dam, 2007; Friedl et al., 2014] for HMS(q,n,r) with these parameters require exponential time.
Cite as

Gábor Ivanyos, Anupam Prakash, and Miklos Santha. On Learning Linear Functions from Subset and Its Applications in Quantum Computing. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ivanyos_et_al:LIPIcs.ESA.2018.66,
  author =	{Ivanyos, G\'{a}bor and Prakash, Anupam and Santha, Miklos},
  title =	{{On Learning Linear Functions from Subset and Its Applications in Quantum Computing}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.66},
  URN =		{urn:nbn:de:0030-drops-95299},
  doi =		{10.4230/LIPIcs.ESA.2018.66},
  annote =	{Keywords: Learning from subset, hidden shift problem, quantum algorithms, linearization}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.55
Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time

Authors: Gábor Ivanyos, Youming Qiao, and K Venkata Subrahmanyam

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
Let {\mathcal B} be a linear space of matrices over a field {\mathbb spanned by n\times n matrices B_1, \dots, B_m. The non-commutative rank of {\mathcal B}$ is the minimum r\in {\mathbb N} such that there exists U\leq {\mathbb F}^n satisfying \dim(U)-\dim( {\mathcal B} (U))\geq n-r, where {\mathcal B}(U):={\mathrm span}(\cup_{i\in[m]} B_i(U)). Computing the non-commutative rank generalizes some well-known problems including the bipartite graph maximum matching problem and the linear matroid intersection problem. In this paper we give a deterministic polynomial-time algorithm to compute the non-commutative rank over any field {\mathbb F}. Prior to our work, such an algorithm was only known over the rational number field {\mathbb Q}, a result due to Garg et al, [GGOW]. Our algorithm is constructive and produces a witness certifying the non-commutative rank, a feature that is missing in the algorithm from [GGOW]. Our result is built on techniques which we developed in a previous paper [IQS1], with a new reduction procedure that helps to keep the blow-up parameter small. There are two ways to realize this reduction. The first involves constructivizing a key result of Derksen and Makam [DM2] which they developed in order to prove that the null cone of matrix semi-invariants is cut out by generators whose degree is polynomial in the size of the matrices involved. We also give a second, simpler method to achieve this. This gives another proof of the polynomial upper bound on the degree of the generators cutting out the null cone of matrix semi-invariants. Both the invariant-theoretic result and the algorithmic result rely crucially on the regularity lemma proved in [IQS1]. In this paper we improve on the constructive version of the regularity lemma from [IQS1] by removing a technical coprime condition that was assumed there.
Cite as

Gábor Ivanyos, Youming Qiao, and K Venkata Subrahmanyam. Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ivanyos_et_al:LIPIcs.ITCS.2017.55,
  author =	{Ivanyos, G\'{a}bor and Qiao, Youming and Subrahmanyam, K Venkata},
  title =	{{Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{55:1--55:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.55},
  URN =		{urn:nbn:de:0030-drops-81667},
  doi =		{10.4230/LIPIcs.ITCS.2017.55},
  annote =	{Keywords: invariant theory, non-commutative rank, null cone, symbolic determinant identity testing, semi-invariants of quivers}
}
Document
DOI: 10.4230/LIPIcs.CCC.2017.30
On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

Authors: Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang

Published in: LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

Abstract
The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.
Cite as

Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 30:1-30:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{belovs_et_al:LIPIcs.CCC.2017.30,
  author =	{Belovs, Aleksandrs and Ivanyos, G\'{a}bor and Qiao, Youming and Santha, Miklos and Yang, Siyi},
  title =	{{On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz}},
  booktitle =	{32nd Computational Complexity Conference (CCC 2017)},
  pages =	{30:1--30:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-040-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{79},
  editor =	{O'Donnell, Ryan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.30},
  URN =		{urn:nbn:de:0030-drops-75260},
  doi =		{10.4230/LIPIcs.CCC.2017.30},
  annote =	{Keywords: Chevalley-Warning Theorem, Combinatorail Nullstellensatz, Polynomial Parity Argument, arithmetic circuit}
}
Document
DOI: 10.4230/LIPIcs.STACS.2014.397
Generalized Wong sequences and their applications to Edmonds' problems

Authors: Gábor Ivanyos, Marek Karpinski, Youming Qiao, and Miklos Santha

Published in: LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

Abstract
We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the nxn matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, while symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one. Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independent of the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.
Cite as

Gábor Ivanyos, Marek Karpinski, Youming Qiao, and Miklos Santha. Generalized Wong sequences and their applications to Edmonds' problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 397-408, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ivanyos_et_al:LIPIcs.STACS.2014.397,
  author =	{Ivanyos, G\'{a}bor and Karpinski, Marek and Qiao, Youming and Santha, Miklos},
  title =	{{Generalized Wong sequences and their applications to Edmonds' problems}},
  booktitle =	{31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)},
  pages =	{397--408},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-65-1},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{25},
  editor =	{Mayr, Ernst W. and Portier, Natacha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.397},
  URN =		{urn:nbn:de:0030-drops-44741},
  doi =		{10.4230/LIPIcs.STACS.2014.397},
  annote =	{Keywords: symbolic determinantal identity testing, Edmonds' problem, maximum rank matrix completion, derandomization, Wong sequences}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2012.148
New bounds on the classical and quantum communication complexity of some graph properties

Authors: Gábor Ivanyos, Hartmut Klauck, Troy Lee, Miklos Santha, and Ronald de Wolf

Published in: LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

Abstract
We study the communication complexity of a number of graph properties where the edges of the graph G are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: 1. An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. 2. A deterministic upper bound of O(n^{3/2} log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. 3. A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for its quantum communication complexity. 4. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon [Babai et al 1986]. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if $G$ is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.
Cite as

Gábor Ivanyos, Hartmut Klauck, Troy Lee, Miklos Santha, and Ronald de Wolf. New bounds on the classical and quantum communication complexity of some graph properties. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 148-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ivanyos_et_al:LIPIcs.FSTTCS.2012.148,
  author =	{Ivanyos, G\'{a}bor and Klauck, Hartmut and Lee, Troy and Santha, Miklos and de Wolf, Ronald},
  title =	{{New bounds on the classical and quantum communication complexity of some graph properties}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)},
  pages =	{148--159},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-47-7},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{18},
  editor =	{D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.148},
  URN =		{urn:nbn:de:0030-drops-38523},
  doi =		{10.4230/LIPIcs.FSTTCS.2012.148},
  annote =	{Keywords: Graph properties, communication complexity, quantum communication}
}
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