48 Search Results for "Buhrman, Harry"

Document
DOI: 10.4230/LIPIcs.ESA.2024.3
From Donkeys to Kings in Tournaments

Authors: Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon

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

Abstract
A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 3 kings is O(n), and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes O(n^{3/2}) queries). On the other hand, we show that finding k ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (the time for matrix multiplication). We provide evidence that this is optimal for large k by suggesting a fine-grained reduction from a variant of the triangle detection problem.
Cite as

Amir Abboud, Tomer Grossman, Moni Naor, and Tomer Solomon. From Donkeys to Kings in Tournaments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{abboud_et_al:LIPIcs.ESA.2024.3,
  author =	{Abboud, Amir and Grossman, Tomer and Naor, Moni and Solomon, Tomer},
  title =	{{From Donkeys to Kings in Tournaments}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{3:1--3:14},
  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.2024.3},
  URN =		{urn:nbn:de:0030-drops-210740},
  doi =		{10.4230/LIPIcs.ESA.2024.3},
  annote =	{Keywords: Tournament Graphs, Kings, Query Complexity, Fine Grained Complexity}
}
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
DOI: 10.4230/LIPIcs.ESA.2024.101
Time-Efficient Quantum Entropy Estimator via Samplizer

Authors: Qisheng Wang and Zhicheng Zhang

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

Abstract
Entropy is a measure of the randomness of a system. Estimating the entropy of a quantum state is a basic problem in quantum information. In this paper, we introduce a time-efficient quantum approach to estimating the von Neumann entropy S(ρ) and Rényi entropy S_α(ρ) of an N-dimensional quantum state ρ, given access to independent samples of ρ. Specifically, we provide the following quantum estimators. - A quantum estimator for S(ρ) with time complexity Õ(N²), improving the prior best time complexity Õ(N⁶) by Acharya, Issa, Shende, and Wagner (2020) and Bavarian, Mehraba, and Wright (2016). - A quantum estimator for S_α(ρ) with time complexity Õ(N^{4/α-2}) for 0 < α < 1 and Õ(N^{4-2/α}) for α > 1, improving the prior best time complexity Õ(N^{6/α}) for 0 < α < 1 and Õ(N⁶) for α > 1 by Acharya, Issa, Shende, and Wagner (2020), though at a cost of a slightly larger sample complexity. Moreover, these estimators are naturally extensible to the low-rank case. We also provide a sample lower bound Ω(max{N/ε, N^{1/α-1}/ε^{1/α}}) for estimating S_α(ρ). Technically, our method is quite different from the previous ones that are based on weak Schur sampling and Young diagrams. At the heart of our construction, is a novel tool called samplizer, which can "samplize" a quantum query algorithm to a quantum algorithm with similar behavior using only samples of quantum states; this suggests a unified framework for estimating quantum entropies. Specifically, when a quantum oracle U block-encodes a mixed quantum state ρ, any quantum query algorithm using Q queries to U can be samplized to a δ-close (in the diamond norm) quantum algorithm using Θ~(Q²/δ) samples of ρ. Moreover, this samplization is proven to be optimal, up to a polylogarithmic factor.
Cite as

Qisheng Wang and Zhicheng Zhang. Time-Efficient Quantum Entropy Estimator via Samplizer. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 101:1-101:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wang_et_al:LIPIcs.ESA.2024.101,
  author =	{Wang, Qisheng and Zhang, Zhicheng},
  title =	{{Time-Efficient Quantum Entropy Estimator via Samplizer}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{101:1--101:15},
  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.2024.101},
  URN =		{urn:nbn:de:0030-drops-211722},
  doi =		{10.4230/LIPIcs.ESA.2024.101},
  annote =	{Keywords: Quantum computing, entropy estimation, von Neumann entropy, R\'{e}nyi entropy, sample complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.38
Approximating the Number of Relevant Variables in a Parity Implies Proper Learning

Authors: Nader H. Bshouty and George Haddad

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
Consider the model where we can access a parity function through random uniform labeled examples in the presence of random classification noise. In this paper, we show that approximating the number of relevant variables in the parity function is as hard as properly learning parities. More specifically, let γ:ℝ^+ → ℝ^+, where γ(x) ≥ x, be any strictly increasing function. In our first result, we show that from any polynomial-time algorithm that returns a γ-approximation, D (i.e., γ^{-1}(d(f)) ≤ D ≤ γ(d(f))), of the number of relevant variables d(f) for any parity f, we can, in polynomial time, construct a solution to the long-standing open problem of polynomial-time learning k(n)-sparse parities (parities with k(n) ≤ n relevant variables), where k(n) = ω_n(1). In our second result, we show that from any T(n)-time algorithm that, for any parity f, returns a γ-approximation of the number of relevant variables d(f) of f, we can, in polynomial time, construct a poly(Γ(n))T(Γ(n)²)-time algorithm that properly learns parities, where Γ(x) = γ(γ(x)). If T(Γ(n)²) = exp({o(n/log n)}), this would resolve another long-standing open problem of properly learning parities in the presence of random classification noise in time exp(o(n/log n)).
Cite as

Nader H. Bshouty and George Haddad. Approximating the Number of Relevant Variables in a Parity Implies Proper Learning. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bshouty_et_al:LIPIcs.APPROX/RANDOM.2024.38,
  author =	{Bshouty, Nader H. and Haddad, George},
  title =	{{Approximating the Number of Relevant Variables in a Parity Implies Proper Learning}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.38},
  URN =		{urn:nbn:de:0030-drops-210316},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.38},
  annote =	{Keywords: PAC Learning, Random Classification Noise, Uniform Distribution, Parity, Sparcity Approximation}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.42
Matrix Multiplication Verification Using Coding Theory

Authors: Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n × n matrices A, B, and C as input, to decide whether AB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in Õ(n²) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(n^ω) time). To that end, we give two algorithms for MMV in the case where AB - C is sparse. Specifically, when AB - C has at most O(n^δ) non-zero entries for a constant 0 ≤ δ < 2, we give (1) a deterministic O(n^(ω-ε))-time algorithm for constant ε = ε(δ) > 0, and (2) a randomized Õ(n²)-time algorithm using δ/2 ⋅ log₂ n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of Künnemann (ESA, 2018) when δ ≥ 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log₂ n + O(1) random bits (in turn fewer than Freivalds’s algorithm). Our algorithms are simple and use techniques from coding theory. Let H be a parity-check matrix of a Maximum Distance Separable (MDS) code, and let G = (I | G') be a generator matrix of a (possibly different) MDS code in systematic form. Our deterministic algorithm uses fast rectangular matrix multiplication to check whether HAB = HC and H(AB)^T = H(C^T), and our randomized algorithm samples a uniformly random row g' from G' and checks whether g'AB = g'C and g'(AB)^T = g'C^T. We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(n^ω) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic Õ(n²)-time reductions).
Cite as

Huck Bennett, Karthik Gajulapalli, Alexander Golovnev, and Evelyn Warton. Matrix Multiplication Verification Using Coding Theory. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bennett_et_al:LIPIcs.APPROX/RANDOM.2024.42,
  author =	{Bennett, Huck and Gajulapalli, Karthik and Golovnev, Alexander and Warton, Evelyn},
  title =	{{Matrix Multiplication Verification Using Coding Theory}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  URN =		{urn:nbn:de:0030-drops-210352},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.42},
  annote =	{Keywords: Matrix Multiplication Verification, Derandomization, Sparse Matrices, Error-Correcting Codes, Hardness Barriers, Reductions}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.51
Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity

Authors: Halley Goldberg and Valentine Kabanets

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
A central open question within meta-complexity is that of NP-hardness of problems such as MCSP and MK^{t}P. Despite a large body of work giving consequences of and barriers for NP-hardness of these problems under (restricted) deterministic reductions, very little is known in the setting of randomized reductions. In this work, we give consequences of randomized NP-hardness reductions for both approximating and exactly computing time-bounded and time-unbounded Kolmogorov complexity. In the setting of approximate K^{poly} complexity, our results are as follows. 1) Under a derandomization assumption, for any constant δ > 0, if approximating K^t complexity within n^{δ} additive error is hard for SAT under an honest randomized non-adaptive Turing reduction running in time polynomially less than t, then NP = coNP. 2) Under the same assumptions, the worst-case hardness of NP is equivalent to the existence of one-way functions. Item 1 above may be compared with a recent work of Saks and Santhanam [Michael E. Saks and Rahul Santhanam, 2022], which makes the same assumptions except with ω(log n) additive error, obtaining the conclusion NE = coNE. In the setting of exact K^{poly} complexity, where the barriers of Item 1 and [Michael E. Saks and Rahul Santhanam, 2022] do not apply, we show: 3) If computing K^t complexity is hard for SAT under reductions as in Item 1, then the average-case hardness of NP is equivalent to the existence of one-way functions. That is, "Pessiland" is excluded. Finally, we give consequences of NP-hardness of exact time-unbounded Kolmogorov complexity under randomized reductions. 4) If computing Kolmogorov complexity is hard for SAT under a randomized many-one reduction running in time t_R and with failure probability at most 1/(t_R)^16, then coNP is contained in non-interactive statistical zero-knowledge; thus NP ⊆ coAM. Also, the worst-case hardness of NP is equivalent to the existence of one-way functions. We further exploit the connection to NISZK along with a previous work of Allender et al. [Eric Allender et al., 2023] to show that hardness of K complexity under randomized many-one reductions is highly robust with respect to failure probability, approximation error, output length, and threshold parameter.
Cite as

Halley Goldberg and Valentine Kabanets. Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 51:1-51:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goldberg_et_al:LIPIcs.APPROX/RANDOM.2024.51,
  author =	{Goldberg, Halley and Kabanets, Valentine},
  title =	{{Consequences of Randomized Reductions from SAT to Time-Bounded Kolmogorov Complexity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{51:1--51:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.51},
  URN =		{urn:nbn:de:0030-drops-210444},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.51},
  annote =	{Keywords: Meta-complexity, Randomized reductions, NP-hardness, Worst-case complexity, Time-bounded Kolmogorov complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2024.64
On the Communication Complexity of Finding a King in a Tournament

Authors: Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh

Published in: LIPIcs, Volume 317, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)

Abstract
A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: - We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament. - The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n). - The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges. One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.
Cite as

Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mande_et_al:LIPIcs.APPROX/RANDOM.2024.64,
  author =	{Mande, Nikhil S. and Paraashar, Manaswi and Sanyal, Swagato and Saurabh, Nitin},
  title =	{{On the Communication Complexity of Finding a King in a Tournament}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)},
  pages =	{64:1--64:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-348-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{317},
  editor =	{Kumar, Amit and Ron-Zewi, Noga},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.64},
  URN =		{urn:nbn:de:0030-drops-210571},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2024.64},
  annote =	{Keywords: Communication complexity, tournaments, query complexity}
}
Document
DOI: 10.4230/LIPIcs.TQC.2024.8
(Quantum) Complexity of Testing Signed Graph Clusterability

Authors: Kuo-Chin Chen, Simon Apers, and Min-Hsiu Hsieh

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity Õ(N^{1/3}) for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [Adriaens and Apers, 2023]. Second, we prove an Ω̃(√N) classical query lower bound for testing clusterability, which nearly matches the upper bound from [Adriaens and Apers, 2023]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.
Cite as

Kuo-Chin Chen, Simon Apers, and Min-Hsiu Hsieh. (Quantum) Complexity of Testing Signed Graph Clusterability. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.TQC.2024.8,
  author =	{Chen, Kuo-Chin and Apers, Simon and Hsieh, Min-Hsiu},
  title =	{{(Quantum) Complexity of Testing Signed Graph Clusterability}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{8:1--8:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.8},
  URN =		{urn:nbn:de:0030-drops-206786},
  doi =		{10.4230/LIPIcs.TQC.2024.8},
  annote =	{Keywords: Quantum Algorithm, classical Query lower Bound, Graph Property testing}
}
Document
DOI: 10.4230/LIPIcs.TQC.2024.10
Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture

Authors: Jordi Weggemans, Marten Folkertsma, and Chris Cade

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
We study "Merlinized" versions of the recently defined Guided Local Hamiltonian problem, which we call "Guidable Local Hamiltonian" problems. Unlike their guided counterparts, these problems do not have a guiding state provided as a part of the input, but merely come with the promise that one exists. We consider in particular two classes of guiding states: those that can be prepared efficiently by a quantum circuit; and those belonging to a class of quantum states we call classically evaluatable, for which it is possible to efficiently compute expectation values of local observables classically. We show that guidable local Hamiltonian problems for both classes of guiding states are QCMA-complete in the inverse-polynomial precision setting, but lie within NP (or NqP) in the constant precision regime when the guiding state is classically evaluatable. Our completeness results show that, from a complexity-theoretic perspective, classical Ansätze selected by classical heuristics are just as powerful as quantum Ansätze prepared by quantum heuristics, as long as one has access to quantum phase estimation. In relation to the quantum PCP conjecture, we (i) define a complexity class capturing quantum-classical probabilistically checkable proof systems and show that it is contained in BQP^NP[1] for constant proof queries; (ii) give a no-go result on "dequantizing" the known quantum reduction which maps a QPCP-verification circuit to a local Hamiltonian with constant promise gap; (iii) give several no-go results for the existence of quantum gap amplification procedures that preserve certain ground state properties; and (iv) propose two conjectures that can be viewed as stronger versions of the NLTS theorem. Finally, we show that many of our results can be directly modified to obtain similar results for the class MA.
Cite as

Jordi Weggemans, Marten Folkertsma, and Chris Cade. Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{weggemans_et_al:LIPIcs.TQC.2024.10,
  author =	{Weggemans, Jordi and Folkertsma, Marten and Cade, Chris},
  title =	{{Guidable Local Hamiltonian Problems with Implications to Heuristic Ansatz State Preparation and the Quantum PCP Conjecture}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.10},
  URN =		{urn:nbn:de:0030-drops-206804},
  doi =		{10.4230/LIPIcs.TQC.2024.10},
  annote =	{Keywords: Quantum complexity theory, local Hamiltonian problem, quantum state ansatzes, QCMA, quantum PCP conjecture}
}
Document
DOI: 10.4230/LIPIcs.TQC.2024.11
A Direct Reduction from the Polynomial to the Adversary Method

Authors: Aleksandrs Belovs

Published in: LIPIcs, Volume 310, 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)

Abstract
The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.
Cite as

Aleksandrs Belovs. A Direct Reduction from the Polynomial to the Adversary Method. In 19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 310, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{belovs:LIPIcs.TQC.2024.11,
  author =	{Belovs, Aleksandrs},
  title =	{{A Direct Reduction from the Polynomial to the Adversary Method}},
  booktitle =	{19th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2024)},
  pages =	{11:1--11:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-328-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{310},
  editor =	{Magniez, Fr\'{e}d\'{e}ric and Grilo, Alex Bredariol},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2024.11},
  URN =		{urn:nbn:de:0030-drops-206814},
  doi =		{10.4230/LIPIcs.TQC.2024.11},
  annote =	{Keywords: Polynomials, Quantum Adversary Bound}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.9
Quantum Algorithms for Hopcroft’s Problem

Authors: Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given n points and n lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in O(n^{4/3}) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity Õ(n^{5/6}). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.
Cite as

Vladimirs Andrejevs, Aleksandrs Belovs, and Jevgēnijs Vihrovs. Quantum Algorithms for Hopcroft’s Problem. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{andrejevs_et_al:LIPIcs.MFCS.2024.9,
  author =	{Andrejevs, Vladimirs and Belovs, Aleksandrs and Vihrovs, Jevg\={e}nijs},
  title =	{{Quantum Algorithms for Hopcroft’s Problem}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.9},
  URN =		{urn:nbn:de:0030-drops-205653},
  doi =		{10.4230/LIPIcs.MFCS.2024.9},
  annote =	{Keywords: Quantum algorithms, Quantum walks, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.SAT.2024.29
Quantum Circuit Mapping Based on Incremental and Parallel SAT Solving

Authors: Jiong Yang, Yaroslav A. Kharkov, Yunong Shi, Marijn J. H. Heule, and Bruno Dutertre

Published in: LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

Abstract
Quantum Computing (QC) is a new computational paradigm that promises significant speedup over classical computing in various domains. However, near-term QC faces numerous challenges, including limited qubit connectivity and noisy quantum operations. To address the qubit connectivity constraint, circuit mapping is required for executing quantum circuits on quantum computers. This process involves performing initial qubit placement and using the quantum SWAP operations to relocate non-adjacent qubits for nearest-neighbor interaction. Reducing the SWAP count in circuit mapping is essential for improving the success rate of quantum circuit execution as SWAPs are costly and error-prone. In this work, we introduce a novel circuit mapping method by combining incremental and parallel solving for Boolean Satisfiability (SAT). We present an innovative SAT encoding for circuit mapping problems, which significantly improves solver-based mapping methods and provides a smooth trade-off between compilation quality and compilation time. Through comprehensive benchmarking of 78 instances covering 3 quantum algorithms on 2 distinct quantum computer topologies, we demonstrate that our method is 26× faster than state-of-the-art solver-based methods, reducing the compilation time from hours to minutes for important quantum applications. Our method also surpasses the existing heuristics algorithm by 26% in SWAP count.
Cite as

Jiong Yang, Yaroslav A. Kharkov, Yunong Shi, Marijn J. H. Heule, and Bruno Dutertre. Quantum Circuit Mapping Based on Incremental and Parallel SAT Solving. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{yang_et_al:LIPIcs.SAT.2024.29,
  author =	{Yang, Jiong and Kharkov, Yaroslav A. and Shi, Yunong and Heule, Marijn J. H. and Dutertre, Bruno},
  title =	{{Quantum Circuit Mapping Based on Incremental and Parallel SAT Solving}},
  booktitle =	{27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-334-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{305},
  editor =	{Chakraborty, Supratik and Jiang, Jie-Hong Roland},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.29},
  URN =		{urn:nbn:de:0030-drops-205517},
  doi =		{10.4230/LIPIcs.SAT.2024.29},
  annote =	{Keywords: Quantum computing, Quantum compilation, SAT solving, Incremental solving, Parallel solving}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.4
Derandomizing Logspace with a Small Shared Hard Drive

Authors: Edward Pyne

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We obtain new catalytic algorithms for space-bounded derandomization. In the catalytic computation model introduced by (Buhrman, Cleve, Koucký, Loff, and Speelman STOC 2013), we are given a small worktape, and a larger catalytic tape that has an arbitrary initial configuration. We may edit this tape, but it must be exactly restored to its initial configuration at the completion of the computation. We prove that BPSPACE[S] ⊆ CSPACE[S,S²] where BPSPACE[S] corresponds to randomized space S computation, and CSPACE[S,C] corresponds to catalytic algorithms that use O(S) bits of workspace and O(C) bits of catalytic space. Previously, only BPSPACE[S] ⊆ CSPACE[S,2^O(S)] was known. In fact, we prove a general tradeoff, that for every α ∈ [1,1.5], BPSPACE[S] ⊆ CSPACE[S^α,S^(3-α)]. We do not use the algebraic techniques of prior work on catalytic computation. Instead, we develop an algorithm that branches based on if the catalytic tape is conditionally random, and instantiate this primitive in a recursive framework. Our result gives an alternate proof of the best known time-space tradeoff for BPSPACE[S], due to (Cai, Chakaravarthy, and van Melkebeek, Theory Comput. Sys. 2006). As a final application, we extend our results to solve search problems in CSPACE[S,S²]. As far as we are aware, this constitutes the first study of search problems in the catalytic computing model.
Cite as

Edward Pyne. Derandomizing Logspace with a Small Shared Hard Drive. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 4:1-4:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{pyne:LIPIcs.CCC.2024.4,
  author =	{Pyne, Edward},
  title =	{{Derandomizing Logspace with a Small Shared Hard Drive}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{4:1--4:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.4},
  URN =		{urn:nbn:de:0030-drops-204006},
  doi =		{10.4230/LIPIcs.CCC.2024.4},
  annote =	{Keywords: Catalytic computation, space-bounded computation, derandomization}
}
Document
DOI: 10.4230/LIPIcs.CCC.2024.6
The Entangled Quantum Polynomial Hierarchy Collapses

Authors: Sabee Grewal and Justin Yirka

Published in: LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Abstract
We introduce the entangled quantum polynomial hierarchy, QEPH, as the class of problems that are efficiently verifiable given alternating quantum proofs that may be entangled with each other. We prove QEPH collapses to its second level. In fact, we show that a polynomial number of alternations collapses to just two. As a consequence, QEPH = QRG(1), the class of problems having one-turn quantum refereed games, which is known to be contained in PSPACE. This is in contrast to the unentangled quantum polynomial hierarchy, QPH, which contains QMA(2). We also introduce DistributionQCPH, a generalization of the quantum-classical polynomial hierarchy QCPH where the provers send probability distributions over strings (instead of strings). We prove DistributionQCPH = QCPH, suggesting that only quantum superposition (not classical probability) increases the computational power of these hierarchies. To prove this equality, we generalize a game-theoretic result of Lipton and Young (1994) which says that, without loss of generality, the provers can send uniform distributions over a polynomial-size support. We also prove the analogous result for the polynomial hierarchy, i.e., DistributionPH = PH. Finally, we show that PH and QCPH are contained in QPH, resolving an open question of Gharibian et al. (2022).
Cite as

Sabee Grewal and Justin Yirka. The Entangled Quantum Polynomial Hierarchy Collapses. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{grewal_et_al:LIPIcs.CCC.2024.6,
  author =	{Grewal, Sabee and Yirka, Justin},
  title =	{{The Entangled Quantum Polynomial Hierarchy Collapses}},
  booktitle =	{39th Computational Complexity Conference (CCC 2024)},
  pages =	{6:1--6:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-331-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{300},
  editor =	{Santhanam, Rahul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.6},
  URN =		{urn:nbn:de:0030-drops-204028},
  doi =		{10.4230/LIPIcs.CCC.2024.6},
  annote =	{Keywords: Polynomial hierarchy, Entangled proofs, Correlated proofs, Minimax}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.21
Oracle Separation of QMA and QCMA with Bounded Adaptivity

Authors: Shalev Ben-David and Srijita Kundu

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

Abstract
We give an oracle separation between QMA and QCMA for quantum algorithms that have bounded adaptivity in their oracle queries; that is, the number of rounds of oracle calls is small, though each round may involve polynomially many queries in parallel. Our oracle construction is a simplified version of the construction used recently by Li, Liu, Pelecanos, and Yamakawa (2023), who showed an oracle separation between QMA and QCMA when the quantum algorithms are only allowed to access the oracle classically. To prove our results, we introduce a property of relations called slipperiness, which may be useful for getting a fully general classical oracle separation between QMA and QCMA.
Cite as

Shalev Ben-David and Srijita Kundu. Oracle Separation of QMA and QCMA with Bounded Adaptivity. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bendavid_et_al:LIPIcs.ICALP.2024.21,
  author =	{Ben-David, Shalev and Kundu, Srijita},
  title =	{{Oracle Separation of QMA and QCMA with Bounded Adaptivity}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{21:1--21:18},
  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.2024.21},
  URN =		{urn:nbn:de:0030-drops-201642},
  doi =		{10.4230/LIPIcs.ICALP.2024.21},
  annote =	{Keywords: Quantum computing, computational complexity}
}
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