7 Search Results for "Mori, Ryuhei"

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.17
Spectral Refutations of Semirandom k-LIN over Larger Fields

Authors: Nicholas Kocurek and Peter Manohar

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We study the problem of strongly refuting semirandom k-LIN(𝔽) instances: systems of k-sparse inhomogeneous linear equations over a finite field 𝔽. For the case of 𝔽 = 𝔽₂, this is the well-studied problem of refuting semirandom instances of k-XOR, where the works of [Venkatesan Guruswami et al., 2022; Jun-Ting Hsieh et al., 2023] establish a tight trade-off between runtime and clause density for refutation: for any choice of a parameter 𝓁, they give an n^{O(𝓁)}-time algorithm to certify that there is no assignment that can satisfy more than 1/2 + ε-fraction of constraints in a semirandom k-XOR instance, provided that the instance has O(n)⋅(n/𝓁)^{k/2 - 1} log n/ε⁴ constraints, and the work of [Pravesh K. Kothari et al., 2017] provides good evidence that this tight up to a polylog(n) factor via lower bounds for the Sum-of-Squares hierarchy. However, for larger fields, the only known results for this problem are established via black-box reductions to the case of 𝔽₂, resulting in a |𝔽|^{3k} gap between the current best upper and lower bounds. In this paper, we give an algorithm for refuting semirandom k-LIN(𝔽) instances with the "correct" dependence on the field size |𝔽|. For any choice of a parameter 𝓁, our algorithm runs in (|𝔽|)^O(𝓁)-time and strongly refutes semirandom k-LIN(𝔽) instances with at least O(n) ⋅ (|𝔽^*| n/𝓁) ^{k/2 - 1} log(n|𝔽^*|)/ε⁴ constraints. We give good evidence that this dependence on the field size |𝔽| is optimal by proving a lower bound for the Sum-of-Squares hierarchy that matches this threshold up to a polylog(n |𝔽^*|) factor. Our results also extend beyond finite fields to the more general case of ℤ_m and arbitrary finite Abelian groups. Our key technical innovation is a generalization of the "𝔽₂ Kikuchi matrices" of [Alexander S. Wein et al., 2019; Venkatesan Guruswami et al., 2022] to larger fields, and finite Abelian groups more generally.
Cite as

Nicholas Kocurek and Peter Manohar. Spectral Refutations of Semirandom k-LIN over Larger Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kocurek_et_al:LIPIcs.APPROX/RANDOM.2025.17,
  author =	{Kocurek, Nicholas and Manohar, Peter},
  title =	{{Spectral Refutations of Semirandom k-LIN over Larger Fields}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.17},
  URN =		{urn:nbn:de:0030-drops-243834},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.17},
  annote =	{Keywords: Spectral Algorithms, CSP Refutation, Kikuchi Matrices}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.14
Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

Authors: Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, and Martin Nöllenburg

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the {dependency digraph} G_𝒫(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V(G_𝒫(I))| √δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√{nm}) time, which improves the best known classical upper bound when m ∈ Ω(n^{1.4}).
Cite as

Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, and Martin Nöllenburg. Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{caroppo_et_al:LIPIcs.WADS.2025.14,
  author =	{Caroppo, Susanna and Da Lozzo, Giordano and Di Battista, Giuseppe and Goodrich, Michael T. and N\"{o}llenburg, Martin},
  title =	{{Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{14:1--14:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.14},
  URN =		{urn:nbn:de:0030-drops-242454},
  doi =		{10.4230/LIPIcs.WADS.2025.14},
  annote =	{Keywords: Dynamic Programming, Quantum Algorithms, Quantum Random Access Memory}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.32
A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion

Authors: Anastasia Sofronova and Dmitry Sokolov

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Random Δ-CNF formulas are one of the few candidates that are expected to be hard for proof systems and SAT algotirhms. Assume we sample m clauses over n variables. Here, the main complexity parameter is clause density, χ := m/n. For a fixed Δ, there exists a satisfiability threshold c_Δ such that for χ > c_Δ a formula is unsatisfiable with high probability. and for χ < c_Δ it is satisfiable with high probability. Near satisfiability threshold, there are various lower bounds for algorithms and proof systems [Eli Ben-Sasson, 2001; Eli Ben-Sasson and Russell Impagliazzo, 1999; Michael Alekhnovich and Alexander A. Razborov, 2003; Dima Grigoriev, 2001; Grant Schoenebeck, 2008; Pavel Hrubes and Pavel Pudlák, 2017; Noah Fleming et al., 2017; Dmitry Sokolov, 2024], and for high-density regimes, there exist upper bounds [Uriel Feige et al., 2006; Sebastian Müller and Iddo Tzameret, 2014; Jackson Abascal et al., 2021; Venkatesan Guruswami et al., 2022]. One of the frontiers in the direction of proving lower bounds on these formulas is the k-DNF Resolution proof system (aka Res(k)). There are several known results for k = 𝒪(√{log n}/{log log n}}) [Nathan Segerlind et al., 2004; Michael Alekhnovich, 2011], that are applicable only for density regime near the threshold. In this paper, we show the first Res(k) lower bound that is applicable in higher-density regimes. Our results work for slightly larger k = 𝒪(√{log n}).
Cite as

Anastasia Sofronova and Dmitry Sokolov. A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 32:1-32:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sofronova_et_al:LIPIcs.CCC.2025.32,
  author =	{Sofronova, Anastasia and Sokolov, Dmitry},
  title =	{{A Lower Bound for k-DNF Resolution on Random CNF Formulas via Expansion}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{32:1--32:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.32},
  URN =		{urn:nbn:de:0030-drops-237269},
  doi =		{10.4230/LIPIcs.CCC.2025.32},
  annote =	{Keywords: proof complexity, random CNFs}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.99
Parameterized Quantum Query Algorithms for Graph Problems

Authors: Tatsuya Terao and Ryuhei Mori

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

Abstract
In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.
Cite as

Tatsuya Terao and Ryuhei Mori. Parameterized Quantum Query Algorithms for Graph Problems. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 99:1-99:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{terao_et_al:LIPIcs.ESA.2024.99,
  author =	{Terao, Tatsuya and Mori, Ryuhei},
  title =	{{Parameterized Quantum Query Algorithms for Graph Problems}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{99:1--99:16},
  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.2024.99},
  URN =		{urn:nbn:de:0030-drops-211707},
  doi =		{10.4230/LIPIcs.ESA.2024.99},
  annote =	{Keywords: Quantum query complexity, parameterized algorithms, vertex cover, matching, kernelization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.69
Quantum Algorithms for Graph Coloring and Other Partitioning, Covering, and Packing Problems

Authors: Serge Gaspers and Jerry Zirui Li

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Let U be a universe on n elements, let k be a positive integer, and let ℱ be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from ℱ, covering U with k sets from ℱ, and packing k non-intersecting sets from ℱ into U. Classically, these problems can be solved via inclusion-exclusion in 2ⁿ n^O(1) time [Andreas Björklund et al., 2009]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140ⁿ) [Kazuya Shimizu and Ryuhei Mori, 2022] and for Set Cover with a small number of sets with running time O(1.7274ⁿ |ℱ|^O(1)) [Andris Ambainis et al., 2019]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a set X ⊆ U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in c^|X| n^O(1) time with c < 2. Our bounded-error quantum algorithm runs in time (2+c)^{n/2} n^O(1) for Set Partition, Set Cover, and Set Packing. It is obtained by combining three algorithms that have the best running time for some values of c. When c ≤ 1.147899, our algorithm is slightly faster than (2+c)^{n/2} n^O(1); when c approaches 1, it matches the O(1.7274ⁿ |ℱ|^O(1)) running time of [Andris Ambainis et al., 2019] for Set Cover when |ℱ| is subexponential in n. For covering, packing, and partitioning into maximal independent sets, maximal cliques, maximal bicliques, maximal cluster graphs, maximal triangle-free graphs, maximal cographs, maximal claw-free graphs, maximal trivially-perfect graphs, maximal threshold graphs, maximal split graphs, maximal line graphs, and maximal induced forests, we obtain bounded-error quantum algorithms with running times ranging from O(1.8554ⁿ) to O(1.9629ⁿ). Packing and covering by maximal induced matchings can be done quantumly in O(1.8934ⁿ) time. For Graph Coloring (covering with k maximal independent sets), we further improve the running time to O(1.7956ⁿ) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number (packing k minimal dominating sets), we obtain a O((2-ε)ⁿ) running time for some ε > 0. Several of our results should be of interest to proponents of classical computing: - We present an inclusion-exclusion algorithm with running time O^*(∑_{i=0}^⌊αn⌋ binom(n,i)), which determines, for each X ⊆ U of size at most α n, 0 ≤ α ≤ 1, whether (X,ℱ) has a k-cover, k-partition, or k-packing. This running time is best-possible, up to polynomial factors. - We prove that for any linear-sized vertex subset X ⊆ V of a graph G = (V,E), the number of minimal dominating sets of G that are subsets of X is O((2-ε)^|X|) for some ε > 0.
Cite as

Serge Gaspers and Jerry Zirui Li. Quantum Algorithms for Graph Coloring and Other Partitioning, Covering, and Packing Problems. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 69:1-69:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gaspers_et_al:LIPIcs.ICALP.2024.69,
  author =	{Gaspers, Serge and Li, Jerry Zirui},
  title =	{{Quantum Algorithms for Graph Coloring and Other Partitioning, Covering, and Packing Problems}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{69:1--69:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.69},
  URN =		{urn:nbn:de:0030-drops-202124},
  doi =		{10.4230/LIPIcs.ICALP.2024.69},
  annote =	{Keywords: Graph algorithms, quantum algorithms, graph coloring, domatic number, set cover, set partition, set packing}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.50
Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs

Authors: Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs

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

Abstract
Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the n-dimensional lattice graph Q(D,n) with vertices in {0,1,…,D}ⁿ. We study the complexity of the following problem: given a subgraph G of Q(D,n) via query access to the edges, determine whether there is a path from 0ⁿ to Dⁿ. While the classical query complexity is Θ̃((D+1)ⁿ), we show a quantum algorithm with complexity Õ(T_Dⁿ), where T_D < D+1. The first few values of T_D are T₁ ≈ 1.817, T₂ ≈ 2.660, T₃ ≈ 3.529, T₄ ≈ 4.421, T₅ ≈ 5.332. We also prove that T_D ≥ (D+1)/e (here, e ≈ 2.718 is the Euler’s number), thus for general D, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for D = 1 by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time and space complexity poly(n)^{log n} T_Dⁿ in the QRAM model, and apply it to the Set Multicover problem. In this problem, m subsets of [n] are given, and the task is to find the smallest number of these subsets that cover each element of [n] at least D times. While the time complexity of the best known classical algorithm is O(m(D+1)ⁿ), the time complexity of our quantum algorithm is poly(m,n)^{log n} T_Dⁿ.
Cite as

Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs. Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 50:1-50:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{glos_et_al:LIPIcs.MFCS.2021.50,
  author =	{Glos, Adam and Kokainis, Martins and Mori, Ryuhei and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{50:1--50:23},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.50},
  URN =		{urn:nbn:de:0030-drops-144901},
  doi =		{10.4230/LIPIcs.MFCS.2021.50},
  annote =	{Keywords: Quantum query complexity, Dynamic programming, Lattice graphs}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2016.41
Lower Bounds for CSP Refutation by SDP Hierarchies

Authors: Ryuhei Mori and David Witmer

Published in: LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Abstract
For a k-ary predicate P, a random instance of CSP(P) with n variables and m constraints is unsatisfiable with high probability when m >= O(n). The natural algorithmic task in this regime is refutation: finding a proof that a given random instance is unsatisfiable. Recent work of Allen et al. suggests that the difficulty of refuting CSP(P) using an SDP is determined by a parameter cmplx(P), the smallest t for which there does not exist a t-wise uniform distribution over satisfying assignments to P. In particular they show that random instances of CSP(P) with m >> n^{cmplx(P)/2} can be refuted efficiently using an SDP. In this work, we give evidence that n^{cmplx(P)/2} constraints are also necessary for refutation using SDPs. Specifically, we show that if P supports a (t-1)-wise uniform distribution over satisfying assignments, then the Sherali-Adams_+ and Lovasz-Schrijver_+ SDP hierarchies cannot refute a random instance of CSP(P) in polynomial time for any m <= n^{t/2-epsilon}.
Cite as

Ryuhei Mori and David Witmer. Lower Bounds for CSP Refutation by SDP Hierarchies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 41:1-41:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{mori_et_al:LIPIcs.APPROX-RANDOM.2016.41,
  author =	{Mori, Ryuhei and Witmer, David},
  title =	{{Lower Bounds for CSP Refutation by SDP Hierarchies}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
  pages =	{41:1--41:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-018-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{60},
  editor =	{Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.41},
  URN =		{urn:nbn:de:0030-drops-66645},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2016.41},
  annote =	{Keywords: constraint satisfaction problems, LP and SDP relaxations, average-case complexity}
}
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