41 Search Results for "K�bler, Johannes"


Document
On a Hierarchy of Spectral Invariants for Graphs

Authors: V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)


Abstract
We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer’s question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

Cite as

V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On a Hierarchy of Spectral Invariants for Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{arvind_et_al:LIPIcs.STACS.2024.6,
  author =	{Arvind, V. and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg},
  title =	{{On a Hierarchy of Spectral Invariants for Graphs}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.6},
  URN =		{urn:nbn:de:0030-drops-197166},
  doi =		{10.4230/LIPIcs.STACS.2024.6},
  annote =	{Keywords: Graph Isomorphism, spectra of graphs, combinatorial refinement, strongly regular graphs}
}
Document
Coloring and Recognizing Mixed Interval Graphs

Authors: Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, and Johannes Zink

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


Abstract
A mixed interval graph is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph G, an interval u receives a lower (different) color than an interval v if G contains arc (u,v) (edge {u,v}). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a min {ω(G), λ(G)+1}-approximation algorithm, where ω(G) is the size of a largest clique and λ(G) is the length of a longest directed path in G. For the subclass of bidirectional interval graphs (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call containment interval graphs. In such a graph, there is an arc (u,v) if interval u contains interval v, and there is an edge {u,v} if u and v overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.

Cite as

Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Felix Klesen, Paweł Rzążewski, Alexander Wolff, and Johannes Zink. Coloring and Recognizing Mixed Interval Graphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{gutowski_et_al:LIPIcs.ISAAC.2023.36,
  author =	{Gutowski, Grzegorz and Junosza-Szaniawski, Konstanty and Klesen, Felix and Rz\k{a}\.{z}ewski, Pawe{\l} and Wolff, Alexander and Zink, Johannes},
  title =	{{Coloring and Recognizing Mixed Interval Graphs}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{36:1--36:14},
  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.2023.36},
  URN =		{urn:nbn:de:0030-drops-193388},
  doi =		{10.4230/LIPIcs.ISAAC.2023.36},
  annote =	{Keywords: Interval Graphs, Mixed Graphs, Graph Coloring}
}
Document
Sliding Window String Indexing in Streams

Authors: Philip Bille, Johannes Fischer, Inge Li Gørtz, Max Rishøj Pedersen, and Tord Joakim Stordalen

Published in: LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)


Abstract
Given a string S over an alphabet Σ, the string indexing problem is to preprocess S to subsequently support efficient pattern matching queries, that is, given a pattern string P report all the occurrences of P in S. In this paper we study the streaming sliding window string indexing problem. Here the string S arrives as a stream, one character at a time, and the goal is to maintain an index of the last w characters, called the window, for a specified parameter w. At any point in time a pattern matching query for a pattern P may arrive, also streamed one character at a time, and all occurrences of P within the current window must be returned. The streaming sliding window string indexing problem naturally captures scenarios where we want to index the most recent data (i.e. the window) of a stream while supporting efficient pattern matching. Our main result is a simple O(w) space data structure that uses O(log w) time with high probability to process each character from both the input string S and any pattern string P. Reporting each occurrence of P uses additional constant time per reported occurrence. Compared to previous work in similar scenarios this result is the first to achieve an efficient worst-case time per character from the input stream with high probability. We also consider a delayed variant of the problem, where a query may be answered at any point within the next δ characters that arrive from either stream. We present an O(w + δ) space data structure for this problem that improves the above time bounds to O(log (w/δ)). In particular, for a delay of δ = ε w we obtain an O(w) space data structure with constant time processing per character. The key idea to achieve our result is a novel and simple hierarchical structure of suffix trees of independent interest, inspired by the classic log-structured merge trees.

Cite as

Philip Bille, Johannes Fischer, Inge Li Gørtz, Max Rishøj Pedersen, and Tord Joakim Stordalen. Sliding Window String Indexing in Streams. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bille_et_al:LIPIcs.CPM.2023.4,
  author =	{Bille, Philip and Fischer, Johannes and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Stordalen, Tord Joakim},
  title =	{{Sliding Window String Indexing in Streams}},
  booktitle =	{34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)},
  pages =	{4:1--4:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-276-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{259},
  editor =	{Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.4},
  URN =		{urn:nbn:de:0030-drops-179587},
  doi =		{10.4230/LIPIcs.CPM.2023.4},
  annote =	{Keywords: String indexing, pattern matching, sliding window, streaming}
}
Document
The Complexity of Translationally Invariant Problems Beyond Ground State Energies

Authors: James D. Watson, Johannes Bausch, and Sevag Gharibian

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)


Abstract
The physically motivated quantum generalisation of k-SAT, the k-Local Hamiltonian (k-LH) problem, is well-known to be QMA-complete ("quantum NP"-complete). What is surprising, however, is that while the former is easy on 1D Boolean formulae, the latter remains hard on 1D local Hamiltonians, even if all constraints are identical [Gottesman, Irani, FOCS 2009]. Such "translation-invariant" systems are much closer in structure to what one might see in Nature. Moving beyond k-LH, what is often more physically interesting is the computation of properties of the ground space (i.e. "solution space") itself. In this work, we focus on two such recent problems: Simulating local measurements on the ground space (APX-SIM, analogous to computing properties of optimal solutions to MAX-SAT formulae) [Ambainis, CCC 2014], and deciding if the low energy space has an energy barrier (GSCON, analogous to classical reconfiguration problems) [Gharibian, Sikora, ICALP 2015]. These problems are known to be P^{QMA[log]}- and QCMA-complete, respectively, in the general case. Yet, to date, it is not known whether they remain hard in such simple 1D translationally invariant systems. In this work, we show that the 1D translationally invariant versions of both APX-SIM and GSCON are intractable, namely are P^{QMA_{EXP}}- and QCMA^{EXP}-complete ("quantum P^{NEXP}" and "quantum NEXP"), respectively. Each of these results is attained by giving a respective generic "lifting theorem". For APX-SIM we give a framework for lifting any abstract local circuit-to-Hamiltonian mapping H satisfying mild assumptions to hardness of APX-SIM on the family of Hamiltonians produced by H, while preserving the structural properties of H (e.g. translation invariance, geometry, locality, etc). Each result also leverages counterintuitive properties of our constructions: for APX-SIM, we compress the answers to polynomially many parallel queries to a QMA oracle into a single qubit. For GSCON, we show strong robustness, i.e. soundness even against adversaries acting on all but a single qudit in the system.

Cite as

James D. Watson, Johannes Bausch, and Sevag Gharibian. The Complexity of Translationally Invariant Problems Beyond Ground State Energies. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 54:1-54:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{watson_et_al:LIPIcs.STACS.2023.54,
  author =	{Watson, James D. and Bausch, Johannes and Gharibian, Sevag},
  title =	{{The Complexity of Translationally Invariant Problems Beyond Ground State Energies}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{54:1--54:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.54},
  URN =		{urn:nbn:de:0030-drops-177065},
  doi =		{10.4230/LIPIcs.STACS.2023.54},
  annote =	{Keywords: Complexity, Quantum Computing, Physics, Constraint Satisfaction, Combinatorial Reconfiguration, Many-Body Physics}
}
Document
Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings

Authors: Frank Kammer and Johannes Meintrup

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)


Abstract
We present a novel space-efficient graph coarsening technique for n-vertex planar graphs G, called cloud partition, which partitions the vertices V(G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition 𝒫 we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/log n) vertices. The combination of (F, 𝒫) is referred to as a cloud decomposition. For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, 𝒫) constructed for a planar graph G we are able to find a balanced separator of G in O(n/log n) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H-minor-free graphs for any fixed graph H. This allows us to construct the succinct encoding scheme for H-minor-free graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of Θ(log n). As an additional application of our cloud decomposition we show that, for H-minor-free graphs, a tree decomposition of width O(n^{1/2 + ε}) for any ε > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k √n log n) for graphs of treewidth k ≤ √n in sublinear space and polynomial time.

Cite as

Frank Kammer and Johannes Meintrup. Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{kammer_et_al:LIPIcs.ISAAC.2022.62,
  author =	{Kammer, Frank and Meintrup, Johannes},
  title =	{{Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.62},
  URN =		{urn:nbn:de:0030-drops-173478},
  doi =		{10.4230/LIPIcs.ISAAC.2022.62},
  annote =	{Keywords: planar graph, H-minor-free, space-efficient, separator, tree decomposition}
}
Document
On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension

Authors: Johannes Blum, Yann Disser, Andreas Emil Feldmann, Siddharth Gupta, and Anna Zych-Pawlewicz

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)


Abstract
We consider the Sparse Hitting Set (Sparse-HS) problem, where we are given a set system (V,ℱ,ℬ) with two families ℱ,ℬ of subsets of the universe V. The task is to find a hitting set for ℱ that minimizes the maximum number of elements in any of the sets of ℬ. This generalizes several problems that have been studied in the literature. Our focus is on determining the complexity of some of these special cases of Sparse-HS with respect to the sparseness k, which is the optimum number of hitting set elements in any set of ℬ (i.e., the value of the objective function). For the Sparse Vertex Cover (Sparse-VC) problem, the universe is given by the vertex set V of a graph, and ℱ is its edge set. We prove NP-hardness for sparseness k ≥ 2 and polynomial time solvability for k = 1. We also provide a polynomial-time 2-approximation algorithm for any k. A special case of Sparse-VC is Fair Vertex Cover (Fair-VC), where the family ℬ is given by vertex neighbourhoods. For this problem it was open whether it is FPT (or even XP) parameterized by the sparseness k. We answer this question in the negative, by proving NP-hardness for constant k. We also provide a polynomial-time (2-1/k)-approximation algorithm for Fair-VC, which is better than any approximation algorithm possible for Sparse-VC or the Vertex Cover problem (under the Unique Games Conjecture). We then switch to a different set of problems derived from Sparse-HS related to the highway dimension, which is a graph parameter modelling transportation networks. In recent years a growing literature has shown interesting algorithms for graphs of low highway dimension. To exploit the structure of such graphs, most of them compute solutions to the r-Shortest Path Cover (r-SPC) problem, where r > 0, ℱ contains all shortest paths of length between r and 2r, and ℬ contains all balls of radius 2r. It is known that there is an XP algorithm that computes solutions to r-SPC of sparseness at most h if the input graph has highway dimension h. However it was not known whether a corresponding FPT algorithm exists as well. We prove that r-SPC and also the related r-Highway Dimension (r-HD) problem, which can be used to formally define the highway dimension of a graph, are both W[1]-hard. Furthermore, by the result of Abraham et al. [ICALP 2011] there is a polynomial-time O(log k)-approximation algorithm for r-HD, but for r-SPC such an algorithm is not known. We prove that r-SPC admits a polynomial-time O(log n)-approximation algorithm.

Cite as

Johannes Blum, Yann Disser, Andreas Emil Feldmann, Siddharth Gupta, and Anna Zych-Pawlewicz. On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{blum_et_al:LIPIcs.IPEC.2022.5,
  author =	{Blum, Johannes and Disser, Yann and Feldmann, Andreas Emil and Gupta, Siddharth and Zych-Pawlewicz, Anna},
  title =	{{On Sparse Hitting Sets: From Fair Vertex Cover to Highway Dimension}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{5:1--5:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.5},
  URN =		{urn:nbn:de:0030-drops-173612},
  doi =		{10.4230/LIPIcs.IPEC.2022.5},
  annote =	{Keywords: sparse hitting set, fair vertex cover, highway dimension}
}
Document
Proofs for Propositional Model Counting

Authors: Johannes K. Fichte, Markus Hecher, and Valentin Roland

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)


Abstract
Although propositional model counting (#SAT) was long considered too hard to be practical, today’s highly efficient solvers facilitate applications in probabilistic reasoning, reliability estimation, quantitative design space exploration, and more. The current trend of solvers growing more capable every year is likely to continue as a diverse range of algorithms are explored in the field. However, to establish model counters as reliable tools like SAT-solvers, correctness is as critical as speed. As in the nature of complex systems, bugs emerge as soon as the tools are widely used. To identify and avoid bugs, explain decisions, and provide trustworthy results, we need verifiable results. We propose a novel system for certifying model counts. We show how proof traces can be generated for exact model counters based on dynamic programming, counting CDCL with component caching, and knowledge compilation to Decision-DNNF, which are the predominant techniques in today’s exact implementations. We provide proof-of-concepts for emitting proofs and a parallel trace checker. Based on this, we show the feasibility of using certified model counting in an empirical experiment.

Cite as

Johannes K. Fichte, Markus Hecher, and Valentin Roland. Proofs for Propositional Model Counting. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 30:1-30:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{fichte_et_al:LIPIcs.SAT.2022.30,
  author =	{Fichte, Johannes K. and Hecher, Markus and Roland, Valentin},
  title =	{{Proofs for Propositional Model Counting}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{30:1--30:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.30},
  URN =		{urn:nbn:de:0030-drops-167043},
  doi =		{10.4230/LIPIcs.SAT.2022.30},
  annote =	{Keywords: Model Counting, Verification, Certified Counting}
}
Document
A Parallel Framework for Approximate Max-Dicut in Partitionable Graphs

Authors: Nico Bertram, Jonas Ellert, and Johannes Fischer

Published in: LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)


Abstract
Computing a maximum cut in undirected and weighted graphs is a well studied problem and has many practical solutions that also scale well in shared memory (despite its NP-completeness). For its counterpart in directed graphs, however, we are not aware of practical solutions that also utilize parallelism. We engineer a framework that computes a high quality approximate cut in directed and weighted graphs by using a graph partitioning approach. The general idea is to partition a graph into k subgraphs using a parallel partitioning algorithm of our choice (the first ingredient of our framework). Then, for each subgraph in parallel, we compute a cut using any polynomial time approximation algorithm (the second ingredient). In a final step, we merge the locally computed solutions using a high-quality or exact parallel Max-Dicut algorithm (the third ingredient). On graphs that can be partitioned well, the quality of the computed cut is significantly better than the best cut achieved by any linear time algorithm. This is particularly relevant for large graphs, where linear time algorithms used to be the only feasible option.

Cite as

Nico Bertram, Jonas Ellert, and Johannes Fischer. A Parallel Framework for Approximate Max-Dicut in Partitionable Graphs. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bertram_et_al:LIPIcs.SEA.2022.10,
  author =	{Bertram, Nico and Ellert, Jonas and Fischer, Johannes},
  title =	{{A Parallel Framework for Approximate Max-Dicut in Partitionable Graphs}},
  booktitle =	{20th International Symposium on Experimental Algorithms (SEA 2022)},
  pages =	{10:1--10:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-251-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{233},
  editor =	{Schulz, Christian and U\c{c}ar, Bora},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.10},
  URN =		{urn:nbn:de:0030-drops-165441},
  doi =		{10.4230/LIPIcs.SEA.2022.10},
  annote =	{Keywords: maximum directed cut, graph partitioning, algorithm engineering, approximation, parallel algorithms}
}
Document
Edge Partitions of Complete Geometric Graphs

Authors: Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)


Abstract
In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

Cite as

Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber. Edge Partitions of Complete Geometric Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.6,
  author =	{Aichholzer, Oswin and Obenaus, Johannes and Orthaber, Joachim and Paul, Rosna and Schnider, Patrick and Steiner, Raphael and Taubner, Tim and Vogtenhuber, Birgit},
  title =	{{Edge Partitions of Complete Geometric Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.6},
  URN =		{urn:nbn:de:0030-drops-160141},
  doi =		{10.4230/LIPIcs.SoCG.2022.6},
  annote =	{Keywords: edge partition, complete geometric graph, plane spanning tree, wheel set}
}
Document
On Polynomially Many Queries to NP or QMA Oracles

Authors: Sevag Gharibian and Dorian Rudolph

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


Abstract
We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as P^NP and P^QMA, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes P^NP[log] and P^QMA[log], defined identically to P^NP and P^QMA, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a P^NP machine have a "query graph" which is a tree, then this computation can be simulated in P^NP[log]. In this work, we first show that for any verification class C ∈ {NP, MA, QCMA, QMA, QMA(2), NEXP, QMA_exp}, any P^C machine with a query graph of "separator number" s can be simulated using deterministic time exp(slog n) and slog n queries to a C-oracle. When s ∈ O(1) (which includes the case of O(1)-treewidth, and thus also of trees), this gives an upper bound of P^C[log], and when s ∈ O(log^k(n)), this yields bound QP^{C[log^{k+1}]} (QP meaning quasi-polynomial time). We next show how to combine Gottlob’s "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding P^C computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial p specified via an arithmetic circuit, if one can "weakly compress" p so that its optimal value requires m bits to represent, then P^NP can be decided with only m queries to an NP-oracle.

Cite as

Sevag Gharibian and Dorian Rudolph. On Polynomially Many Queries to NP or QMA Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 75:1-75:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{gharibian_et_al:LIPIcs.ITCS.2022.75,
  author =	{Gharibian, Sevag and Rudolph, Dorian},
  title =	{{On Polynomially Many Queries to NP or QMA Oracles}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{75:1--75:27},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.75},
  URN =		{urn:nbn:de:0030-drops-156717},
  doi =		{10.4230/LIPIcs.ITCS.2022.75},
  annote =	{Keywords: admissible weighting function, oracle complexity class, quantum complexity theory, Quantum Merlin Arthur (QMA), simulation of local measurement}
}
Document
Parallel Model Counting with CUDA: Algorithm Engineering for Efficient Hardware Utilization

Authors: Johannes K. Fichte, Markus Hecher, and Valentin Roland

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)


Abstract
Propositional model counting (MC) and its extensions as well as applications in the area of probabilistic reasoning have received renewed attention in recent years. As a result, also the need for quickly solving counting-based problems with automated solvers is critical for certain areas. In this paper, we present experiments evaluating various techniques in order to improve the performance of parallel model counting on general purpose graphics processing units (GPGPUs). Thereby, we mainly consider engineering efficient algorithms for model counting on GPGPUs that utilize the treewidth of a propositional formula by means of dynamic programming. The combination of our techniques results in the solver GPUSAT3, which is based on the programming framework Cuda that -compared to other frameworks- shows superior extensibility and driver support. When combining all findings of this work, we show that GPUSAT3 not only solves more instances of the recent Model Counting Competition 2020 (MCC 2020) than existing GPGPU-based systems, but also solves those significantly faster. A portfolio with one of the best solvers of MCC 2020 and GPUSAT3 solves 19% more instances than the former alone in less than half of the runtime.

Cite as

Johannes K. Fichte, Markus Hecher, and Valentin Roland. Parallel Model Counting with CUDA: Algorithm Engineering for Efficient Hardware Utilization. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{fichte_et_al:LIPIcs.CP.2021.24,
  author =	{Fichte, Johannes K. and Hecher, Markus and Roland, Valentin},
  title =	{{Parallel Model Counting with CUDA: Algorithm Engineering for Efficient Hardware Utilization}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.24},
  URN =		{urn:nbn:de:0030-drops-153150},
  doi =		{10.4230/LIPIcs.CP.2021.24},
  annote =	{Keywords: Propositional Satisfiability, GPGPU, Model Counting, Treewidth, Tree Decomposition}
}
Document
Complications for Computational Experiments from Modern Processors

Authors: Johannes K. Fichte, Markus Hecher, Ciaran McCreesh, and Anas Shahab

Published in: LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)


Abstract
In this paper, we revisit the approach to empirical experiments for combinatorial solvers. We provide a brief survey on tools that can help to make empirical work easier. We illustrate origins of uncertainty in modern hardware and show how strong the influence of certain aspects of modern hardware and its experimental setup can be in an actual experimental evaluation. More specifically, there can be situations where (i) two different researchers run a reasonable-looking experiment comparing the same solvers and come to different conclusions and (ii) one researcher runs the same experiment twice on the same hardware and reaches different conclusions based upon how the hardware is configured and used. We investigate these situations from a hardware perspective. Furthermore, we provide an overview on standard measures, detailed explanations on effects, potential errors, and biased suggestions for useful tools. Alongside the tools, we discuss their feasibility as experiments often run on clusters to which the experimentalist has only limited access. Our work sheds light on a number of benchmarking-related issues which could be considered to be folklore or even myths.

Cite as

Johannes K. Fichte, Markus Hecher, Ciaran McCreesh, and Anas Shahab. Complications for Computational Experiments from Modern Processors. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 25:1-25:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{fichte_et_al:LIPIcs.CP.2021.25,
  author =	{Fichte, Johannes K. and Hecher, Markus and McCreesh, Ciaran and Shahab, Anas},
  title =	{{Complications for Computational Experiments from Modern Processors}},
  booktitle =	{27th International Conference on Principles and Practice of Constraint Programming (CP 2021)},
  pages =	{25:1--25:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-211-2},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{210},
  editor =	{Michel, Laurent D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.25},
  URN =		{urn:nbn:de:0030-drops-153165},
  doi =		{10.4230/LIPIcs.CP.2021.25},
  annote =	{Keywords: Experimenting, Combinatorial Solving, Empirical Work}
}
Document
Parameterized (Modular) Counting and Cayley Graph Expanders

Authors: Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)


Abstract
We study the problem #EdgeSub(Φ) of counting k-edge subgraphs satisfying a given graph property Φ in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients. Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field 𝔽_p which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(Φ) for minor-closed properties Φ, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21). Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts. In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p.

Cite as

Norbert Peyerimhoff, Marc Roth, Johannes Schmitt, Jakob Stix, and Alina Vdovina. Parameterized (Modular) Counting and Cayley Graph Expanders. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 84:1-84:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{peyerimhoff_et_al:LIPIcs.MFCS.2021.84,
  author =	{Peyerimhoff, Norbert and Roth, Marc and Schmitt, Johannes and Stix, Jakob and Vdovina, Alina},
  title =	{{Parameterized (Modular) Counting and Cayley Graph Expanders}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{84:1--84:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.84},
  URN =		{urn:nbn:de:0030-drops-145246},
  doi =		{10.4230/LIPIcs.MFCS.2021.84},
  annote =	{Keywords: Cayley graphs, counting complexity, expander graphs, fine-grained complexity, parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

Authors: Marc Roth, Johannes Schmitt, and Philip Wellnitz

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


Abstract
Given a graph property Φ, we consider the problem EdgeSub(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies Φ. Specifically, we study the parameterized complexity of EdgeSub(Φ) and of its counting problem #EdgeSub(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ: the decision problem EdgeSub(Φ) always admits an FPT ("fixed-parameter tractable") algorithm and the counting problem #EdgeSub(Φ) always admits an FPTRAS ("fixed-parameter tractable randomized approximation scheme"). For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #W[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #EdgeSub(Φ) that run in time f(k)⋅ |G|^{o(k/log k)} for any computable function f. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #EdgeSub(Φ). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). This approach does not only apply to #EdgeSub(Φ) but also to the more general problem of computing weighted linear combinations of subgraph counts. As a special case of such a linear combination, we introduce a parameterized variant of the Tutte Polynomial T^k_G of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T^k_G(2,1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parameterized complexity of computing the evaluation of T^k_G at every pair of rational coordinates (x,y). In particular, our results give a new proof for the #W[1]-hardness of the problem of counting k-forests in a graph.

Cite as

Marc Roth, Johannes Schmitt, and Philip Wellnitz. Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 108:1-108:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{roth_et_al:LIPIcs.ICALP.2021.108,
  author =	{Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
  title =	{{Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{108:1--108:16},
  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.2021.108},
  URN =		{urn:nbn:de:0030-drops-141778},
  doi =		{10.4230/LIPIcs.ICALP.2021.108},
  annote =	{Keywords: Counting complexity, parameterized complexity, Tutte polynomial}
}
Document
Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines

Authors: Martin Koutecký and Johannes Zink

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)


Abstract
The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector, are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general NP-hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines (Q|HM|C_max) is NP-hard already with 6 job types, and that the related Cutting Stock problem is NP-hard already with 8 item types. For the more general unrelated machines model (R|HM|C_max), we show that if either the largest job size p_max, or the number of jobs n are polynomially bounded in the instance size |I|, there are algorithms with complexity |I|^poly(k). Our main result is that this is unlikely to be improved, because Q||C_max is W[1]-hard parameterized by k already when n, p_max, and the numbers describing the speeds are polynomial in |I|; the same holds for R|HM|C_max (without speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives 𝓁₂-norm minimization of the load vector and, partially, sum of weighted completion times ∑ w_j C_j. Along the way, we answer affirmatively the question whether makespan minimization on identical machines (P||C_max) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for Q||C_max this implies that the complexity of P|HM|C_max is the only remaining open case.

Cite as

Martin Koutecký and Johannes Zink. Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{koutecky_et_al:LIPIcs.ISAAC.2020.18,
  author =	{Kouteck\'{y}, Martin and Zink, Johannes},
  title =	{{Complexity of Scheduling Few Types of Jobs on Related and Unrelated Machines}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.18},
  URN =		{urn:nbn:de:0030-drops-133620},
  doi =		{10.4230/LIPIcs.ISAAC.2020.18},
  annote =	{Keywords: Scheduling, cutting stock, hardness, parameterized complexity}
}
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