8 Search Results for "Brand, Cornelius"

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-dev.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-dev.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-dev.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
Invited Talk
DOI: 10.4230/LIPIcs.CPM.2020.1
Algebraic Algorithms for Finding Patterns in Graphs (Invited Talk)

Authors: Thore Husfeldt

Published in: LIPIcs, Volume 161, 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)

Abstract
I will give a gentle introduction to algebraic graph algorithms by showing how to determine if a given graph contains a simple path of length k. This is a famous problem admitting a beautiful and widely-known algorithm, namely the colour-coding method of Alon, Yuster and Zwick (1995). Starting from this entirely combinatorial approach, I will carefully develop an algebraic perspective on the same problem. First, I will explain how the colour-coding algorithm can be understood as the evaluation of a well-known expression (sometimes called the "walk-sum" of the graph) in a commutative algebra called the zeon algebra. From there, I will introduce the exterior algebra and present the algebraic framework recently developed with Brand and Dell (2018). The presentation is aimed at a combinatorially-minded audience largely innocent of abstract algebra.
Cite as

Thore Husfeldt. Algebraic Algorithms for Finding Patterns in Graphs (Invited Talk). In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{husfeldt:LIPIcs.CPM.2020.1,
  author =	{Husfeldt, Thore},
  title =	{{Algebraic Algorithms for Finding Patterns in Graphs}},
  booktitle =	{31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-149-8},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{161},
  editor =	{G{\o}rtz, Inge Li and Weimann, Oren},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2020.1},
  URN =		{urn:nbn:de:0030-drops-121261},
  doi =		{10.4230/LIPIcs.CPM.2020.1},
  annote =	{Keywords: paths, exterior algebra, wedge product, color-coding, parameterized complexity}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2019.9
Fast Exact Algorithms Using Hadamard Product of Polynomials

Authors: V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016]. - (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. - k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f. Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial. - For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. - For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space. Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits.
Cite as

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Fast Exact Algorithms Using Hadamard Product of Polynomials. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2019.9,
  author =	{Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha},
  title =	{{Fast Exact Algorithms Using Hadamard Product of Polynomials}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{9:1--9:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.9},
  URN =		{urn:nbn:de:0030-drops-115711},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.9},
  annote =	{Keywords: Hadamard Product, Multilinear Monomial Detection and Counting, Rectangular Permanent, Symmetric Polynomial}
}
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-dev.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
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2019.113
Counting Answers to Existential Questions (Track B: Automata, Logic, Semantics, and Theory of Programming)

Authors: Holger Dell, Marc Roth, and Philip Wellnitz

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem’s parameterized and data complexity, where the query is considered to be small or even fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis (SETH) as well as a #W[2]-hardness result in parameterized complexity. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; in the language of parameterized complexity, this essentially establishes a #A[2]-completeness result. Using ideas stemming from Lovász (1967), we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that refines and generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number (and thus the treewidth) is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.
Cite as

Holger Dell, Marc Roth, and Philip Wellnitz. Counting Answers to Existential Questions (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 113:1-113:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dell_et_al:LIPIcs.ICALP.2019.113,
  author =	{Dell, Holger and Roth, Marc and Wellnitz, Philip},
  title =	{{Counting Answers to Existential Questions}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{113:1--113:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.113},
  URN =		{urn:nbn:de:0030-drops-106894},
  doi =		{10.4230/LIPIcs.ICALP.2019.113},
  annote =	{Keywords: Conjunctive queries, graph homomorphisms, counting complexity, parameterized complexity, fine-grained complexity}
}
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-dev.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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