11 Search Results for "Zhou, Leo"

Document
DOI: 10.4230/LIPIcs.SEA.2024.19
Algorithms for Gradual Polyline Simplification

Authors: Nick Krumbholz, Stefan Funke, Peter Schäfer, and Sabine Storandt

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Displaying line data is important in many visualization applications, and especially in the context of interactive geographical and cartographic visualization. When rendering linear features as roads, rivers or movement data on zoomable maps, the challenge is to display the data in an appropriate level of detail. A too detailed representation results in slow rendering and cluttered maps, while a too coarse representation might miss important data aspects. In this paper, we propose the gradual line simplification (GLS) problem, which aims to compute a fine-grained succession of consistent simplifications of a given input polyline with certain quality guarantees. The core concept of gradual simplification is to iteratively remove points from the polyline to obtain increasingly coarser representations. We devise two objective functions to guide this simplification process and present dynamic programs that compute the optimal solutions in 𝒪(n³) for an input line with n points. For practical application to large inputs, we also devise significantly faster greedy algorithms that provide constant factor guarantees for both problem variants at once. In an extensive experimental study on real-world data, we demonstrate that our algorithms are capable of producing simplification sequences of high quality within milliseconds on polylines consisting of over half a million points.
Cite as

Nick Krumbholz, Stefan Funke, Peter Schäfer, and Sabine Storandt. Algorithms for Gradual Polyline Simplification. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{krumbholz_et_al:LIPIcs.SEA.2024.19,
  author =	{Krumbholz, Nick and Funke, Stefan and Sch\"{a}fer, Peter and Storandt, Sabine},
  title =	{{Algorithms for Gradual Polyline Simplification}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{19:1--19:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.19},
  URN =		{urn:nbn:de:0030-drops-203847},
  doi =		{10.4230/LIPIcs.SEA.2024.19},
  annote =	{Keywords: Polyline simplification, Progressive simplification, Fr\'{e}chet distance}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.20
Targeted Branching for the Maximum Independent Set Problem Using Graph Neural Networks

Authors: Kenneth Langedal, Demian Hespe, and Peter Sanders

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Identifying a maximum independent set is a fundamental NP-hard problem. This problem has several real-world applications and requires finding the largest possible set of vertices not adjacent to each other in an undirected graph. Over the past few years, branch-and-bound and branch-and-reduce algorithms have emerged as some of the most effective methods for solving the problem exactly. Specifically, the branch-and-reduce approach, which combines branch-and-bound principles with reduction rules, has proven particularly successful in tackling previously unmanageable real-world instances. This progress was largely made possible by the development of more effective reduction rules. Nevertheless, other key components that can impact the efficiency of these algorithms have not received the same level of interest. Among these is the branching strategy, which determines which vertex to branch on next. Until recently, the most widely used strategy was to choose the vertex of the highest degree. In this work, we present a graph neural network approach for selecting the next branching vertex. The intricate nature of current branch-and-bound solvers makes supervised and reinforcement learning difficult. Therefore, we use a population-based genetic algorithm to evolve the model’s parameters instead. Our proposed approach results in a speedup on 73% of the benchmark instances with a median speedup of 24%.
Cite as

Kenneth Langedal, Demian Hespe, and Peter Sanders. Targeted Branching for the Maximum Independent Set Problem Using Graph Neural Networks. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 20:1-20:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{langedal_et_al:LIPIcs.SEA.2024.20,
  author =	{Langedal, Kenneth and Hespe, Demian and Sanders, Peter},
  title =	{{Targeted Branching for the Maximum Independent Set Problem Using Graph Neural Networks}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{20:1--20:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.20},
  URN =		{urn:nbn:de:0030-drops-203853},
  doi =		{10.4230/LIPIcs.SEA.2024.20},
  annote =	{Keywords: Graphs, Independent Set, Vertex Cover, Graph Neural Networks, Branch-and-Reduce}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.21
SPIDER: Improved Succinct Rank and Select Performance

Authors: Matthew D. Laws, Jocelyn Bliven, Kit Conklin, Elyes Laalai, Samuel McCauley, and Zach S. Sturdevant

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
Rank and select data structures seek to preprocess a bit vector to quickly answer two kinds of queries: Rank(i) gives the number of 1 bits in slots 0 through i, and Select(j) gives the first slot s with Rank(s) = j. A succinct data structure can answer these queries while using space much smaller than the size of the original bit vector. State of the art succinct rank and select data structures use as little as 4% extra space (over the underlying bit vector) while answering rank and select queries very quickly. Rank queries can be answered using only a handful of array accesses. Select queries can be answered by starting with similar array accesses, followed by a linear scan through the bit vector. Nonetheless, a tradeoff remains: data structures that use under 4% space are significantly slower at answering rank and select queries than less-space-efficient data structures (using, say, over 20% extra space). In this paper we make significantly progress towards closing this gap. We give a new data structure, SPIDER, which uses 3.82% extra space. SPIDER gives the best known rank query time for data sets of 8 billion or more bits, even compared to much less space-efficient data structures. For select queries, SPIDER outperforms all data structures that use less than 4% space, and significantly closes the gap in select performance between data structures with less than 4% space, and those that use more (over 20% for both rank and select) space. SPIDER makes two main technical contributions. For rank queries, it improves performance by interleaving the metadata with the bit vector to improve cache efficiency. For select queries, it uses predictions to almost eliminate the cost of the linear scan. These predictions are inspired by recent results on data structures with machine-learned predictions, adapted to the succinct data structure setting. Our results hold on both real and synthetic data, showing that these predictions are effective in practice.
Cite as

Matthew D. Laws, Jocelyn Bliven, Kit Conklin, Elyes Laalai, Samuel McCauley, and Zach S. Sturdevant. SPIDER: Improved Succinct Rank and Select Performance. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{laws_et_al:LIPIcs.SEA.2024.21,
  author =	{Laws, Matthew D. and Bliven, Jocelyn and Conklin, Kit and Laalai, Elyes and McCauley, Samuel and Sturdevant, Zach S.},
  title =	{{SPIDER: Improved Succinct Rank and Select Performance}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{21:1--21:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.21},
  URN =		{urn:nbn:de:0030-drops-203865},
  doi =		{10.4230/LIPIcs.SEA.2024.21},
  annote =	{Keywords: Rank and Select, Succinct Data Structures, Data Structres, Cache Performance, Predictions}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.26
Scalable Hard Instances for Independent Set Reconfiguration

Authors: Takehide Soh, Takumu Watanabe, Jun Kawahara, Akira Suzuki, and Takehiro Ito

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
The Token Jumping problem, also known as the independent set reconfiguration problem under the token jumping model, is defined as follows: Given a graph and two same-sized independent sets, determine whether one can be transformed into the other via a sequence of independent sets. Token Jumping has been extensively studied, mainly from the viewpoint of algorithmic theory, but its practical study has just begun. To develop a practically good solver, it is important to construct benchmark datasets that are scalable and hard. Here, "scalable" means the ability to change the scale of the instance while maintaining its characteristics by adjusting the given parameters; and "hard" means that the instance can become so difficult that it cannot be solved within a practical time frame by a solver. In this paper, we propose four types of instance series for Token Jumping. Our instance series is scalable in the sense that instance scales are controlled by the number of vertices. To establish their hardness, we focus on the numbers of transformation steps; our instance series requires exponential numbers of steps with respect to the number of vertices. Interestingly, three types of instance series are constructed by importing theories developed by algorithmic research. We experimentally evaluate the scalability and hardness of the proposed instance series, using the SAT solver and award-winning solvers of the international competition for Token Jumping.
Cite as

Takehide Soh, Takumu Watanabe, Jun Kawahara, Akira Suzuki, and Takehiro Ito. Scalable Hard Instances for Independent Set Reconfiguration. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{soh_et_al:LIPIcs.SEA.2024.26,
  author =	{Soh, Takehide and Watanabe, Takumu and Kawahara, Jun and Suzuki, Akira and Ito, Takehiro},
  title =	{{Scalable Hard Instances for Independent Set Reconfiguration}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.26},
  URN =		{urn:nbn:de:0030-drops-203913},
  doi =		{10.4230/LIPIcs.SEA.2024.26},
  annote =	{Keywords: Combinatorial reconfiguration, Benckmark dataset, Graph Algorithm, PSPACE-complete}
}
Document
DOI: 10.4230/LIPIcs.SEA.2024.27
Improved Cut Strategy for Tensor Network Contraction Orders

Authors: Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract
In the field of quantum computing, simulating quantum systems on classical computers is crucial. Tensor networks are fundamental in simulating quantum systems. A tensor network is a collection of tensors, that need to be contracted into a result tensor. Tensor contraction is a generalization of matrix multiplication to higher order tensors. The contractions can be performed in different orders, and the order has a significant impact on the number of floating point operations (flops) needed to get the result tensor. It is known that finding an optimal contraction order is NP-hard. The current state-of-the-art approach for finding efficient contraction orders is to combinine graph partitioning with a greedy strategy. Although heavily used in practice, the current approach ignores so-called free indices, chooses node weights without regarding previous computations, and requires numerous hyperparameters that need to be tuned at runtime. In this paper, we address these shortcomings by developing a novel graph cut strategy. The proposed modifications yield contraction orders that significantly reduce the number of flops in the tensor contractions compared to the current state of the art. Moreover, by removing the need for hyperparameter tuning at runtime, our approach converges to an efficient solution faster, which reduces the required optimization time by at least an order of magnitude.
Cite as

Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen. Improved Cut Strategy for Tensor Network Contraction Orders. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{staudt_et_al:LIPIcs.SEA.2024.27,
  author =	{Staudt, Christoph and Blacher, Mark and Klaus, Julien and Lippmann, Farin and Giesen, Joachim},
  title =	{{Improved Cut Strategy for Tensor Network Contraction Orders}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.27},
  URN =		{urn:nbn:de:0030-drops-203924},
  doi =		{10.4230/LIPIcs.SEA.2024.27},
  annote =	{Keywords: tensor network, contraction order, graph partitioniong, quantum simulation}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2024.33
State Canonization and Early Pruning in Width-Based Automated Theorem Proving

Authors: Mateus de Oliveira Oliveira and Farhad Vadiee

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract
Width-based automated theorem proving is a framework where counter-examples for graph theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time 2^{2^{k^{O(1)}}} on the class of graphs of treewidth at most k. In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counter-example of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form ℙ₁ → ℙ₂, for graph properties ℙ₁ and ℙ₂, where ℙ₁ is a property closed under subgraphs. As a concrete application, we use our framework in the study of graph theoretic conjectures related to coloring triangle free graphs. In particular, our algorithm is able to show that Reed’s conjecture for triangle free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counter-examples for non-valid strengthenings of Reed’s conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.
Cite as

Mateus de Oliveira Oliveira and Farhad Vadiee. State Canonization and Early Pruning in Width-Based Automated Theorem Proving. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deoliveiraoliveira_et_al:LIPIcs.FSCD.2024.33,
  author =	{de Oliveira Oliveira, Mateus and Vadiee, Farhad},
  title =	{{State Canonization and Early Pruning in Width-Based Automated Theorem Proving}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.33},
  URN =		{urn:nbn:de:0030-drops-203622},
  doi =		{10.4230/LIPIcs.FSCD.2024.33},
  annote =	{Keywords: Width-Based Automated Theorem Proving, Dynamic Programming, Parameterized Complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.103
Towards an Analysis of Quadratic Probing

Authors: William Kuszmaul and Zoe Xi

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

Abstract
Since 1968, one of the simplest open questions in the theory of hash tables has been to prove anything nontrivial about the correctness of quadratic probing. We make the first tangible progress towards this goal, showing that there exists a positive-constant load factor at which quadratic probing is a constant-expected-time hash table. Our analysis applies more generally to any fixed-offset open-addressing hash table, and extends to higher load factors in the case where the hash table examines blocks of some size B = ω(1).
Cite as

William Kuszmaul and Zoe Xi. Towards an Analysis of Quadratic Probing. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 103:1-103:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kuszmaul_et_al:LIPIcs.ICALP.2024.103,
  author =	{Kuszmaul, William and Xi, Zoe},
  title =	{{Towards an Analysis of Quadratic Probing}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{103:1--103:19},
  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.2024.103},
  URN =		{urn:nbn:de:0030-drops-202463},
  doi =		{10.4230/LIPIcs.ICALP.2024.103},
  annote =	{Keywords: quadratic probing, hashing, open addressing, witness trees}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.5
Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations

Authors: Anurag Anshu and Tony Metger

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from [De Palma et al., 2022]; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form e^{ιH^{(p)}} ⋯ e^{ιH^{(1)}} |ψ₀⟩ for any n-qubit product state |ψ₀⟩, where each H^{(i)} can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates. An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level p = o(log log n), assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [Basso et al., 2022].
Cite as

Anurag Anshu and Tony Metger. Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 5:1-5:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{anshu_et_al:LIPIcs.ITCS.2023.5,
  author =	{Anshu, Anurag and Metger, Tony},
  title =	{{Concentration Bounds for Quantum States and Limitations on the QAOA from Polynomial Approximations}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{5:1--5:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.5},
  URN =		{urn:nbn:de:0030-drops-175085},
  doi =		{10.4230/LIPIcs.ITCS.2023.5},
  annote =	{Keywords: quantum computing, polynomial approximation, quantum optimization algorithm, QAOA, overlap gap property}
}
Document
DOI: 10.4230/LIPIcs.TQC.2022.7
The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model

Authors: Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou

Published in: LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Abstract
The Quantum Approximate Optimization Algorithm (QAOA) finds approximate solutions to combinatorial optimization problems. Its performance monotonically improves with its depth p. We apply the QAOA to MaxCut on large-girth D-regular graphs. We give an iterative formula to evaluate performance for any D at any depth p. Looking at random D-regular graphs, at optimal parameters and as D goes to infinity, we find that the p = 11 QAOA beats all classical algorithms (known to the authors) that are free of unproven conjectures. While the iterative formula for these D-regular graphs is derived by looking at a single tree subgraph, we prove that it also gives the ensemble-averaged performance of the QAOA on the Sherrington-Kirkpatrick (SK) model defined on the complete graph. We also generalize our formula to Max-q-XORSAT on large-girth regular hypergraphs. Our iteration is a compact procedure, but its computational complexity grows as O(p² 4^p). This iteration is more efficient than the previous procedure for analyzing QAOA performance on the SK model, and we are able to numerically go to p = 20. Encouraged by our findings, we make the optimistic conjecture that the QAOA, as p goes to infinity, will achieve the Parisi value. We analyze the performance of the quantum algorithm, but one needs to run it on a quantum computer to produce a string with the guaranteed performance.
Cite as

Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou. The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{basso_et_al:LIPIcs.TQC.2022.7,
  author =	{Basso, Joao and Farhi, Edward and Marwaha, Kunal and Villalonga, Benjamin and Zhou, Leo},
  title =	{{The Quantum Approximate Optimization Algorithm at High Depth for MaxCut on Large-Girth Regular Graphs and the Sherrington-Kirkpatrick Model}},
  booktitle =	{17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)},
  pages =	{7:1--7:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-237-2},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{232},
  editor =	{Le Gall, Fran\c{c}ois and Morimae, Tomoyuki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.7},
  URN =		{urn:nbn:de:0030-drops-165144},
  doi =		{10.4230/LIPIcs.TQC.2022.7},
  annote =	{Keywords: Quantum algorithm, Max-Cut, spin glass, approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.14
Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs

Authors: Boaz Barak and Kunal Marwaha

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

Abstract
We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of randomized classical algorithms. 1) We prove that every (quantum or classical) one-local algorithm (where the value of a vertex only depends on its and its neighbors' state) achieves on D-regular graphs of girth > 5 a maximum cut of at most 1/2 + C/√D for C = 1/√2 ≈ 0.7071. This is the first such result showing that one-local algorithms achieve a value that is bounded away from the true optimum for random graphs, which is 1/2 + P_*/√D + o(1/√D) for P_* ≈ 0.7632 [Dembo et al., 2017]. 2) We show that there is a classical k-local algorithm that achieves a value of 1/2 + C/√D - O(1/√k) for D-regular graphs of girth > 2k+1, where C = 2/π ≈ 0.6366. This is an algorithmic version of the existential bound of [Lyons, 2017] and is related to the algorithm of [Aizenman et al., 1987] (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-local versions of QAOA on high-girth graphs [M. B. Hastings, 2019; Marwaha, 2021]. 3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constant-locality QAOA for random D-regular graphs, as well as other natural instances, including graphs that do have short cycles. While our theoretical bounds require the locality and girth assumptions, our experimental work suggests that it could be possible to extend them beyond these constraints. This points at the tantalizing possibility that O(1)-local quantum maximum-cut algorithms might be pointwise dominated by polynomial-time classical algorithms, in the sense that there is a classical algorithm outputting cuts of equal or better quality on every possible instance. This is in contrast to the evidence that polynomial-time algorithms cannot simulate the probability distributions induced by local quantum algorithms.
Cite as

Boaz Barak and Kunal Marwaha. Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{barak_et_al:LIPIcs.ITCS.2022.14,
  author =	{Barak, Boaz and Marwaha, Kunal},
  title =	{{Classical Algorithms and Quantum Limitations for Maximum Cut on High-Girth Graphs}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{14:1--14:21},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.14},
  URN =		{urn:nbn:de:0030-drops-156105},
  doi =		{10.4230/LIPIcs.ITCS.2022.14},
  annote =	{Keywords: approximation algorithms, QAOA, maximum cut, local distributions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2019.2
Hamiltonian Sparsification and Gap-Simulation

Authors: Dorit Aharonov and Leo Zhou

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
Analog quantum simulation - simulation of one Hamiltonian by another - is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian H~ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian H (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where H~ is only required to simulate the groundstate(s) and spectral gap of H instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves as a good starting point for future research of these topics.
Cite as

Dorit Aharonov and Leo Zhou. Hamiltonian Sparsification and Gap-Simulation. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{aharonov_et_al:LIPIcs.ITCS.2019.2,
  author =	{Aharonov, Dorit and Zhou, Leo},
  title =	{{Hamiltonian Sparsification and Gap-Simulation}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{2:1--2:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.2},
  URN =		{urn:nbn:de:0030-drops-100956},
  doi =		{10.4230/LIPIcs.ITCS.2019.2},
  annote =	{Keywords: quantum simulation, quantum Hamiltonian complexity, sparsification, quantum PCP}
}
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