12 Search Results for "Koiran, Pascal"

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
DOI: 10.4230/LIPIcs.MFCS.2025.19
Monotone Bounded-Depth Complexity of Homomorphism Polynomials

Authors: C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
For every fixed graph H, it is known that homomorphism counts from H and colorful H-subgraph counts can be determined in O(n^{t+1}) time on n-vertex input graphs G, where t is the treewidth of H. On the other hand, a running time of n^{o(t / log t)} would refute the exponential-time hypothesis. Komarath, Pandey, and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph polynomials for fixed graphs H. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the monotone circuit complexity of the homomorphism polynomial for H is Θ(n^{tw(H)+1}). In this paper, we characterize the power of monotone bounded-depth circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters tw_Δ(H), for fixed Δ ∈ ℕ, which capture the width of tree-decompositions for H when the underlying tree is required to have depth at most Δ. We prove that monotone circuits of product-depth Δ computing the homomorphism polynomial for H require size Θ(n^{tw_Δ(H^{†})+1}), where H^{†} is the graph obtained from H by removing all degree-1 vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.
Cite as

C.S. Bhargav, Shiteng Chen, Radu Curticapean, and Prateek Dwivedi. Monotone Bounded-Depth Complexity of Homomorphism Polynomials. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhargav_et_al:LIPIcs.MFCS.2025.19,
  author =	{Bhargav, C.S. and Chen, Shiteng and Curticapean, Radu and Dwivedi, Prateek},
  title =	{{Monotone Bounded-Depth Complexity of Homomorphism Polynomials}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.19},
  URN =		{urn:nbn:de:0030-drops-241269},
  doi =		{10.4230/LIPIcs.MFCS.2025.19},
  annote =	{Keywords: algebraic complexity, homomorphisms, monotone circuit complexity, bounded-depth circuits, treewidth, pathwidth}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.92
On Piecewise Affine Reachability with Bellman Operators

Authors: Anton Varonka and Kazuki Watanabe

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
A piecewise affine map is one of the simplest mathematical objects exhibiting complex dynamics. The reachability problem of piecewise affine maps is as follows: Given two vectors s, t ∈ ℚ^d and a piecewise affine map f: ℚ^d → ℚ^d, is there n ∈ ℕ such that fⁿ(s) = t? Koiran, Cosnard, and Garzon show that the reachability problem of piecewise affine maps is undecidable even in dimension 2. Most of the recent progress has been focused on decision procedures for one-dimensional piecewise affine maps, where the reachability problem has been shown to be decidable for some subclasses. However, the general undecidability discouraged research into positive results in arbitrary dimension. In this work, we investigate a rich subclass of piecewise affine maps arising as Bellman operators of Markov decision processes (MDPs). We consider the reachability problem restricted to this subclass and examine its decidability in arbitrary dimensions. We establish that the reachability problem for Bellman operators is decidable in any dimension under either of the following conditions: (i) the target vector t is not the fixed point of the operator f; or (ii) the initial and target vectors s and t are comparable with respect to the componentwise order. Furthermore, we show that the reachability problem for two-dimensional Bellman operators is decidable for arbitrary s, t ∈ ℚ^d, in contrast to the known undecidability of reachability for general piecewise affine maps.
Cite as

Anton Varonka and Kazuki Watanabe. On Piecewise Affine Reachability with Bellman Operators. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 92:1-92:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{varonka_et_al:LIPIcs.MFCS.2025.92,
  author =	{Varonka, Anton and Watanabe, Kazuki},
  title =	{{On Piecewise Affine Reachability with Bellman Operators}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{92:1--92:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.92},
  URN =		{urn:nbn:de:0030-drops-241998},
  doi =		{10.4230/LIPIcs.MFCS.2025.92},
  annote =	{Keywords: piecewise affine map, reachability, value iteration, Markov decision process, Bellman operator}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.29
A Universal Uniform Approximation Theorem for Neural Networks

Authors: Olivier Bournez, Johanne Cohen, and Adrian Wurm

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We show the existence of a fixed recurrent network capable of approximating any computable function with arbitrary precision, provided that an encoding of the function is given in the initial input. While uniform approximation over a compact domain is a well-known property of neural networks, we go further by proving that our network ensures effective uniform approximation - simultaneously ensuring: - Uniform approximation in the sup-norm sense, guaranteeing precision across the compact domain {[0,1]^d}; - Uniformity in the sense of computability theory (also referred to as effectivity or universality), meaning the same network works for all computable functions. Our result is obtained constructively, using original arguments. Moreover, our construction bridges computation theory with neural network approximation, providing new insights into the fundamental connections between circuit complexity and function representation. Furthermore, this connection extends beyond computability to complexity theory. The obtained network is efficient: if a function is computable or approximable in polynomial time in the Turing machine model, then the network requires only a polynomial number of recurrences or iterations to achieve the same level of approximation, and conversely. Moreover, the recurrent network can be assumed to be very narrow, strengthening the link our results and existing models of very deep learning, where uniform approximation properties have already been established.
Cite as

Olivier Bournez, Johanne Cohen, and Adrian Wurm. A Universal Uniform Approximation Theorem for Neural Networks. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bournez_et_al:LIPIcs.MFCS.2025.29,
  author =	{Bournez, Olivier and Cohen, Johanne and Wurm, Adrian},
  title =	{{A Universal Uniform Approximation Theorem for Neural Networks}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.29},
  URN =		{urn:nbn:de:0030-drops-241365},
  doi =		{10.4230/LIPIcs.MFCS.2025.29},
  annote =	{Keywords: Models of computation, Complexity theory, Formal neural networks}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.79
Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space

Authors: Eike Neumann

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We study the problem of deciding whether a point escapes a closed subset of ℝ^d under the iteration of a continuous map f : ℝ^d → ℝ^d in the bit-model of real computation. We give a sound partial decision method for this problem which is complete in the sense that its halting set contains the halting set of all sound partial decision methods for the problem. Equivalently, our decision method terminates on all problem instances whose answer is robust under all sufficiently small perturbations of the function. We further show that the halting set of our algorithm is dense in the set of all problem instances. While our algorithm applies to general continuous functions, we demonstrate that it also yields complete decision methods for much more rigid function families: affine linear systems and quadratic complex polynomials. In the latter case, completeness is subject to the density of hyperbolicity conjecture in complex dynamics. This in particular yields an alternative proof of Hertling’s (2004) conditional answer to a question raised by Penrose (1989) regarding the computability of the Mandelbrot set.
Cite as

Eike Neumann. Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{neumann:LIPIcs.MFCS.2025.79,
  author =	{Neumann, Eike},
  title =	{{Deciding Robust Instances of an Escape Problem for Dynamical Systems in Euclidean Space}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{79:1--79:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.79},
  URN =		{urn:nbn:de:0030-drops-241866},
  doi =		{10.4230/LIPIcs.MFCS.2025.79},
  annote =	{Keywords: Dynamical Systems, Computability in Analysis, Non-Linear Functions}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.72
One-Parametric Presburger Arithmetic Has Quantifier Elimination

Authors: Alessio Mansutti and Mikhail R. Starchak

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
We give a quantifier elimination procedure for one-parametric Presburger arithmetic, the extension of Presburger arithmetic with the function x ↦ t ⋅ x, where t is a fixed free variable ranging over the integers. This resolves an open problem proposed in [Bogart et al., Discrete Analysis, 2017]. As conjectured in [Goodrick, Arch. Math. Logic, 2018], quantifier elimination is obtained for the extended structure featuring all integer division functions x ↦ ⌊x/(f(t))⌋, one for each integer polynomial f. Our algorithm works by iteratively eliminating blocks of existential quantifiers. The elimination of a block builds on two sub-procedures, both running in non-deterministic polynomial time. The first one is an adaptation of a recently developed and efficient quantifier elimination procedure for Presburger arithmetic, modified to handle formulae with coefficients over the ring ℤ[t] of univariate polynomials. The second is reminiscent of the so-called "base t division method" used by Bogart et al. As a result, we deduce that the satisfiability problem for the existential fragment of one-parametric Presburger arithmetic (which encompasses a broad class of non-linear integer programs) is in NP, and that the smallest solution to a satisfiable formula in this fragment is of polynomial bit size.
Cite as

Alessio Mansutti and Mikhail R. Starchak. One-Parametric Presburger Arithmetic Has Quantifier Elimination. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mansutti_et_al:LIPIcs.MFCS.2025.72,
  author =	{Mansutti, Alessio and Starchak, Mikhail R.},
  title =	{{One-Parametric Presburger Arithmetic Has Quantifier Elimination}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{72:1--72:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.72},
  URN =		{urn:nbn:de:0030-drops-241794},
  doi =		{10.4230/LIPIcs.MFCS.2025.72},
  annote =	{Keywords: decision procedures, quantifier elimination, non-linear integer arithmetic}
}
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
DOI: 10.4230/LIPIcs.FSTTCS.2022.24
Black Box Absolute Reconstruction for Sums of Powers of Linear Forms

Authors: Pascal Koiran and Subhayan Saha

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial f ∈ K[x_1 , . . . , x_n] (where K ⊆ ℂ) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are: - For d = 3, we improve by a factor of n on the running time from the algorithm in [Pascal Koiran and Mateusz Skomra, 2021]. The price to be paid for this improvement is that the algorithm now has two-sided error. - For d > 3, we provide the first randomized blackbox algorithm for this problem that runs in time poly(n,d) (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem [Kayal, 2011] as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms. - For d > 3, when f has rational coefficients (i.e. K = ℚ), the running time of the blackbox algorithm is polynomial in n,d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over ℂ with polynomial running time in the bit model of computation. These results are true even when we replace ℂ by ℝ. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e., decompositions in more than n powers of linear forms.
Cite as

Pascal Koiran and Subhayan Saha. Black Box Absolute Reconstruction for Sums of Powers of Linear Forms. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{koiran_et_al:LIPIcs.FSTTCS.2022.24,
  author =	{Koiran, Pascal and Saha, Subhayan},
  title =	{{Black Box Absolute Reconstruction for Sums of Powers of Linear Forms}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.24},
  URN =		{urn:nbn:de:0030-drops-174163},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.24},
  annote =	{Keywords: reconstruction algorithms, tensor decomposition, sums of powers of linear forms, simultaneous diagonalisation, algebraic algorithm, black box}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.37
On the Complexity of Partial Derivatives

Authors: Ignacio Garcia-Marco, Pascal Koiran, Timothée Pecatte, and Stéphan Thomassé

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the "trace method", recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.
Cite as

Ignacio Garcia-Marco, Pascal Koiran, Timothée Pecatte, and Stéphan Thomassé. On the Complexity of Partial Derivatives. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{garciamarco_et_al:LIPIcs.STACS.2017.37,
  author =	{Garcia-Marco, Ignacio and Koiran, Pascal and Pecatte, Timoth\'{e}e and Thomass\'{e}, St\'{e}phan},
  title =	{{On the Complexity of Partial Derivatives}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{37:1--37:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.37},
  URN =		{urn:nbn:de:0030-drops-69964},
  doi =		{10.4230/LIPIcs.STACS.2017.37},
  annote =	{Keywords: counting complexity, simplicial complex, lower bounds, arithmetic circuits}
}
Document
DOI: 10.4230/DagRep.3.1.47
Computational Counting (Dagstuhl Seminar 13031)

Authors: Peter Bürgisser, Leslie Ann Goldberg, Mark Jerrum, and Pascal Koiran

Published in: Dagstuhl Reports, Volume 3, Issue 1 (2013)

Abstract
Dagstuhl Seminar 13031 "Computational Counting" was held from 13th to 18th January 2013, at Schloss Dagstuhl -- Leibnitz Center for Informatics. A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.
Cite as

Peter Bürgisser, Leslie Ann Goldberg, Mark Jerrum, and Pascal Koiran. Computational Counting (Dagstuhl Seminar 13031). In Dagstuhl Reports, Volume 3, Issue 1, pp. 47-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{burgisser_et_al:DagRep.3.1.47,
  author =	{B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark and Koiran, Pascal},
  title =	{{Computational Counting (Dagstuhl Seminar 13031)}},
  pages =	{47--66},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{3},
  number =	{1},
  editor =	{B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark and Koiran, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.1.47},
  URN =		{urn:nbn:de:0030-drops-40087},
  doi =		{10.4230/DagRep.3.1.47},
  annote =	{Keywords: Computational complexity, counting problems, graph polynomials, holographic algorithms, statistical physics, constraint satisfaction}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2011.127
The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Authors: Bruno Grenet, Pascal Koiran, Natacha Portier, and Yann Strozecki

Published in: LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Abstract
Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real tau-conjecture" which is inspired by this connection. The real tau-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real tau-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.
Cite as

Bruno Grenet, Pascal Koiran, Natacha Portier, and Yann Strozecki. The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 127-139, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{grenet_et_al:LIPIcs.FSTTCS.2011.127,
  author =	{Grenet, Bruno and Koiran, Pascal and Portier, Natacha and Strozecki, Yann},
  title =	{{The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent}},
  booktitle =	{IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
  pages =	{127--139},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-34-7},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{13},
  editor =	{Chakraborty, Supratik and Kumar, Amit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.127},
  URN =		{urn:nbn:de:0030-drops-33501},
  doi =		{10.4230/LIPIcs.FSTTCS.2011.127},
  annote =	{Keywords: Algebraic Complexity, Sparse Polynomials, Descartes' Rule of Signs, Lower Bound for the Permanent, Polynomial Identity Testing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2011.543
Symmetric Determinantal Representation of Weakly-Skew Circuits

Authors: Bruno Grenet, Erich L. Kaltofen, Pascal Koiran, and Natacha Portier

Published in: LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

Abstract
We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of weakly-skew circuits, which include formulas. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly-skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Buergisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.
Cite as

Bruno Grenet, Erich L. Kaltofen, Pascal Koiran, and Natacha Portier. Symmetric Determinantal Representation of Weakly-Skew Circuits. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 543-554, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{grenet_et_al:LIPIcs.STACS.2011.543,
  author =	{Grenet, Bruno and Kaltofen, Erich L. and Koiran, Pascal and Portier, Natacha},
  title =	{{Symmetric Determinantal Representation of Weakly-Skew Circuits}},
  booktitle =	{28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)},
  pages =	{543--554},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-25-5},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{9},
  editor =	{Schwentick, Thomas and D\"{u}rr, Christoph},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.543},
  URN =		{urn:nbn:de:0030-drops-30426},
  doi =		{10.4230/LIPIcs.STACS.2011.543},
  annote =	{Keywords: algebraic complexity, determinant and permanent of symmetric matrices, formulas, skew circuits, Valiant’s classes}
}
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