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Documents authored by Brand, Cornelius

Document
DOI: 10.4230/LIPIcs.CCC.2026.11
Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?

Authors: Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, and Jiaheng Wang

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)

Abstract
The complexity of bilinear maps (equivalently, of 3-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for 3-mode tensors, this correspondence breaks down for d ≥ 4 modes. As a result, the complexity of d-mode tensors for larger fixed d remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen’s 2ω/3 bound on the asymptotic rank exponent of 3-mode tensors. Our proof directly generalizes to an upper bound of (d-1)ω/3 for d-mode tensors. Using refined techniques available only for d ≥ 4 modes, we improve this bound beyond the current state of the art for ω. We also obtain a bound of d/2+1 on the asymptotic exponent of circuit complexity of generic d-mode tensors and optimized bounds for d ∈ {4,5}. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of T and U imply low complexity of their Kronecker product T ⊗ U. While this crucially holds for rank (and thus for circuit complexity in 3 modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for d = 8 modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors T_H, as employed by Christandl and Zuiddam (Comput. Complexity 28 (2019) 27-56) and Christandl, Vrana and Zuiddam (Comput. Complexity 28 (2019) 57-111), whose modes correspond to the vertices of undirected graphs H. We make the simple but conceptually crucial observation that Kronecker products T_G ⊗ T_H are isomorphic to T_{G+H}, and that G and H may also be fractional graphs. By asymptotically converting generic tensors to specific graph tensors, we can use nontrivial results from algorithmic graph theory to study the rank and complexity of d-mode tensors for fixed d.
Cite as

Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, and Jiaheng Wang. Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{brand_et_al:LIPIcs.CCC.2026.11,
  author =	{Brand, Cornelius and Curticapean, Radu and Kaski, Petteri and Li, Baitian and Orzel, Ian and Seppelt, Tim and Wang, Jiaheng},
  title =	{{Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{11:1--11:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.11},
  URN =		{urn:nbn:de:0030-drops-270530},
  doi =		{10.4230/LIPIcs.CCC.2026.11},
  annote =	{Keywords: arithmetic circuits, tensor rank, bilinear complexity, graph tensors}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.32
Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds

Authors: Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We provide several novel algorithms and lower bounds in central settings of mixed-integer (non-)linear optimization, shedding new light on classic results in the field. This includes an improvement on record running time bounds obtained from a slight extension of Lenstra’s 1983 algorithm [Math. Oper. Res. '83] to optimizing under few constraints with small coefficients. This is important for ubiquitous tasks like knapsack-, subset sum- or scheduling problems [Eisenbrand and Weismantel, SODA'18, Jansen and Rohwedder, ITCS'19]. Further, we extend our algorithm to an intermediate linear optimization problem when the matrix has many rows that exhibit 2-stage stochastic structure, which adds to a prominent line of recent results on this and similarly restricted cases [Jansen et al. ICALP'19, Cslovjecsek et al. SODA'21, Brand et al. AAAI'21, Klein, Reuter SODA'22, Cslovjecsek et al. SODA'24]. We also show that the generalization of two fundamental classes of structured constraints from these works (n-fold and 2-stage stochastic programs) to separable-convex mixed-integer optimization are harder than their mixed-integer, linear counterparts. This counters a widespread belief popularized initially by an influential paper of Hochbaum and Shanthikumar, namely that "convex separable optimization is not much harder than linear optimization" [J. ACM '90]. To obtain our algorithms, we employ the mixed Graver basis introduced by Hemmecke [Math. Prog. '03], and our work is the first to give bounds on the norm of its elements. Importantly, we use these bounds differently from how purely-integer Graver bounds are exploited in related approaches, and prove that, surprisingly, this cannot be avoided.
Cite as

Cornelius Brand, Martin Koutecký, Alexandra Lassota, and Sebastian Ordyniak. Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{brand_et_al:LIPIcs.ESA.2024.32,
  author =	{Brand, Cornelius and Kouteck\'{y}, Martin and Lassota, Alexandra and Ordyniak, Sebastian},
  title =	{{Separable Convex Mixed-Integer Optimization: Improved Algorithms and Lower Bounds}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.32},
  URN =		{urn:nbn:de:0030-drops-211033},
  doi =		{10.4230/LIPIcs.ESA.2024.32},
  annote =	{Keywords: Mixed-Integer Programming, Separable Convex Optimization, Parameterized Algorithms, Parameterized Complexity}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.16
Fast Convolutions for Near-Convex Sequences

Authors: Cornelius Brand and Alexandra Lassota

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
We develop algorithms for (min,+)-Convolution and related convolution problems such as Super Additivity Testing, Convolution 3-Sum and Minimum Consecutive Subsums which use the degree of convexity of the instance as a parameter. Assuming the min-plus conjecture (Künnemann-Paturi-Schneider, ICALP'17 and Cygan et al., ICALP'17), our results interpolate in an optimal manner between fully convex instances, which can be solved in near-linear time using Legendre transformations, and general non-convex sequences, where the trivial quadratic-time algorithm is conjectured to be best possible, up to subpolynomial factors.
Cite as

Cornelius Brand and Alexandra Lassota. Fast Convolutions for Near-Convex Sequences. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brand_et_al:LIPIcs.ISAAC.2023.16,
  author =	{Brand, Cornelius and Lassota, Alexandra},
  title =	{{Fast Convolutions for Near-Convex Sequences}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.16},
  URN =		{urn:nbn:de:0030-drops-193188},
  doi =		{10.4230/LIPIcs.ISAAC.2023.16},
  annote =	{Keywords: (min,+)-convolution, fine-grained complexity, convex sequences}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.25
Deterministic Constrained Multilinear Detection

Authors: Cornelius Brand, Viktoriia Korchemna, and Michael Skotnica

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
We extend the algebraic techniques of Brand and Pratt (ICALP'21) for deterministic detection of k-multilinear monomials in a given polynomial with non-negative coefficients to the more general situation of detecting colored k-multilinear monomials that satisfy additional constraints on the multiplicities of the colors appearing in them. Our techniques can be viewed as a characteristic-zero generalization of the algebraic tools developed by Guillemot and Sikora (MFCS'10) and Björklund, Kaski and Kowalik (STACS'13) As applications, we recover the state-of-the-art deterministic algorithms for the Graph Motif problem due to Pinter, Schachnai and Zehavi (MFCS'14), and give new deterministic algorithms for generalizations of certain questions on colored directed spanning trees or bipartite planar matchings running in deterministic time O^∗(4^k), studied originally by Gutin, Reidl, Wahlström and Zehavi (J. Comp. Sys. Sci. 95, '18). Finally, we give improved randomized algorithms for intersecting three and four matroids of rank k in characteristic zero, improving the record bounds of Brand and Pratt (ICALP'21) from O^∗(64^k) and O^∗(256^k), respectively, to O^∗(4^k).
Cite as

Cornelius Brand, Viktoriia Korchemna, and Michael Skotnica. Deterministic Constrained Multilinear Detection. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{brand_et_al:LIPIcs.MFCS.2023.25,
  author =	{Brand, Cornelius and Korchemna, Viktoriia and Skotnica, Michael},
  title =	{{Deterministic Constrained Multilinear Detection}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.25},
  URN =		{urn:nbn:de:0030-drops-185595},
  doi =		{10.4230/LIPIcs.MFCS.2023.25},
  annote =	{Keywords: Fixed-parameter algorithms, Algebraic algorithms, Motif discovery, Matroid intersection}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.38
Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials

Authors: Cornelius Brand and Kevin Pratt

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
We study the following problem and its applications: given a homogeneous degree-d polynomial g as an arithmetic circuit C, and a d × d matrix X whose entries are homogeneous linear polynomials, compute g(∂/∂ x₁, …, ∂/∂ x_n) det X. We show that this quantity can be computed using 2^{ω d}|C|poly(n,d) arithmetic operations, where ω is the exponent of matrix multiplication. In the case that C is skew, we improve this to 4^d|C| poly(n,d) operations, and if furthermore X is a Hankel matrix, to φ^{2d}|C| poly(n,d) operations, where φ = (1+√5)/2 is the golden ratio. Using these observations we give faster parameterized algorithms for the matroid k-parity and k-matroid intersection problems for linear matroids, and faster deterministic algorithms for several problems, including the first deterministic polynomial time algorithm for testing if a linear space of matrices of logarithmic dimension contains an invertible matrix. We also match the runtime of the fastest deterministic algorithm for detecting subgraphs of bounded pathwidth with a new and simple approach. Our approach generalizes several previous methods in parameterized algorithms and can be seen as a relaxation of Waring rank based methods [Pratt, FOCS19].
Cite as

Cornelius Brand and Kevin Pratt. Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brand_et_al:LIPIcs.ICALP.2021.38,
  author =	{Brand, Cornelius and Pratt, Kevin},
  title =	{{Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{38:1--38:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.38},
  URN =		{urn:nbn:de:0030-drops-141079},
  doi =		{10.4230/LIPIcs.ICALP.2021.38},
  annote =	{Keywords: Parameterized Algorithms, Algebraic Algorithms, Longest Cycle, Matroid Parity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.25
Patching Colors with Tensors

Authors: Cornelius Brand

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We describe a generic way of exponentially speeding up algorithms which rely on Color-Coding by using the recently introduced technique of Extensor-Coding (Brand, Dell and Husfeldt, STOC 2018). To demonstrate the usefulness of this "patching" of Color-Coding algorithms, we apply it ad hoc to the exponential-space algorithms given in Gutin et al. (Journal Comp. Sys. Sci. 2018) and obtain the fastest known deterministic algorithms for, among others, the k-internal out-branching and k-internal spanning tree problems. To realize these technical advances, we make qualitative progress in a special case of the detection of multilinear monomials in multivariate polynomials: We give the first deterministic fixed-parameter tractable algorithm for the k-multilinear detection problem on a class of arithmetic circuits that may involve cancellations, as long as the computed polynomial is promised to satisfy a certain natural condition. Furthermore, we explore the limitations of using this very approach to speed up algorithms by determining exactly the dimension of a crucial subalgebra of extensors that arises naturally in the instantiation of the technique: It is equal to F_{2k+1}, the kth odd term in the Fibonacci sequence. To determine this dimension, we use tools from the theory of Gröbner bases, and the studied algebraic object may be of independent interest. We note that the asymptotic bound of F_{2k+1} ~~ phi^(2k) = O(2.619^k) curiously coincides with the running time bound on one of the fastest algorithms for the k-path problem based on representative sets due to Fomin et al. (JACM 2016). Here, phi is the golden ratio.
Cite as

Cornelius Brand. Patching Colors with Tensors. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brand:LIPIcs.ESA.2019.25,
  author =	{Brand, Cornelius},
  title =	{{Patching Colors with Tensors}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{25:1--25:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.25},
  URN =		{urn:nbn:de:0030-drops-111467},
  doi =		{10.4230/LIPIcs.ESA.2019.25},
  annote =	{Keywords: Color-Coding, Extensor-Coding, internal out-branching, colorful problems, algebraic algorithms, multilinear detection, deterministic algorithms, exterior algebra}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2016.9
Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Authors: Cornelius Brand, Holger Dell, and Marc Roth

Published in: LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

Abstract
Jaeger, Vertigan, and Welsh (1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (2010) and Husfeldt and Taslaman (2010), in combination with the results of Curticapean (2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp(o(n)) unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases cannot be solved in time exp(o(n)) unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp(o(n)) unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean (2015) and transfer it to systems of linear equations that might not directly correspond to interpolation.
Cite as

Cornelius Brand, Holger Dell, and Marc Roth. Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{brand_et_al:LIPIcs.IPEC.2016.9,
  author =	{Brand, Cornelius and Dell, Holger and Roth, Marc},
  title =	{{Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP}},
  booktitle =	{11th International Symposium on Parameterized and Exact Computation (IPEC 2016)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-023-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{63},
  editor =	{Guo, Jiong and Hermelin, Danny},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.9},
  URN =		{urn:nbn:de:0030-drops-69426},
  doi =		{10.4230/LIPIcs.IPEC.2016.9},
  annote =	{Keywords: computational complexity, counting problems, Tutte polynomial, exponential time hypothesis, forests, independent sets}
}
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