30 Search Results for "New, Max S."


Document
New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines

Authors: Sebastian Berndt, Hauke Brinkop, Klaus Jansen, Matthias Mnich, and Tobias Stamm

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


Abstract
Mixed-integer linear programming (MILP) is at the core of many advanced algorithms for solving fundamental problems in combinatorial optimization. The complexity of solving MILPs directly correlates with their support size, which is the minimum number of non-zero integer variables in an optimal solution. A hallmark result by Eisenbrand and Shmonin (Oper. Res. Lett. , 2006) shows that any feasible integer linear program (ILP) has a solution with support size s ≤ 2m⋅log(4mΔ), where m is the number of constraints, and Δ is the largest absolute coefficient in any constraint. Our main combinatorial result are improved support size bounds for ILPs. We show that any ILP has a solution with support size s ≤ m⋅(log(3A_max)+√{log(A_max)}), where A_max≔ ‖A‖₁ denotes the 1-norm of the constraint matrix A. Furthermore, we show support bounds in the linearized form s ≤ 2m⋅log(1.46 A_max). Our upper bounds also hold with A_max replaced by √mΔ, which improves on the previously best constants in the linearized form. Our main algorithmic result are the fastest known approximation schemes for fundamental scheduling problems, which use the improved support bounds as one ingredient. We design an efficient approximation scheme (EPTAS) for makespan minimization on uniformly related machines (Q||C_{max}). Our EPTAS yields a (1+ε)-approximation for Q||C_{max} on N jobs in time 2^𝒪(1/ε log³(1/ε)log(log(1/ε))) + 𝒪(N), which improves over the previously fastest algorithm by Jansen, Klein and Verschae (Math. Oper. Res., 2020) with run time 2^𝒪(1/ε log⁴(1/ε)) + N^𝒪(1). Arguably, our approximation scheme is also simpler than all previous EPTASes for Q||C_max, as we reduce the problem to a novel MILP formulation which greatly benefits from the small support.

Cite as

Sebastian Berndt, Hauke Brinkop, Klaus Jansen, Matthias Mnich, and Tobias Stamm. New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{berndt_et_al:LIPIcs.ISAAC.2023.13,
  author =	{Berndt, Sebastian and Brinkop, Hauke and Jansen, Klaus and Mnich, Matthias and Stamm, Tobias},
  title =	{{New Support Size Bounds for Integer Programming, Applied to Makespan Minimization on Uniformly Related Machines}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.13},
  URN =		{urn:nbn:de:0030-drops-193155},
  doi =		{10.4230/LIPIcs.ISAAC.2023.13},
  annote =	{Keywords: Integer programming, scheduling algorithms, uniformly related machines, makespan minimization}
}
Document
Censorship Resistance in On-Chain Auctions

Authors: Elijah Fox, Mallesh M. Pai, and Max Resnick

Published in: LIPIcs, Volume 282, 5th Conference on Advances in Financial Technologies (AFT 2023)


Abstract
Modern blockchains guarantee that submitted transactions will be included eventually; a property formally known as liveness. But financial activity requires transactions to be included in a timely manner. Classical liveness does not guarantee this, particularly in the presence of a motivated adversary who benefits from censoring transactions. We define censorship resistance as the amount it would cost the adversary to censor a transaction for a fixed interval of time as a function of the associated tip. This definition has two advantages, first it captures the fact that transactions with a higher miner tip can be more costly to censor, and therefore are more likely to swiftly make their way onto the chain. Second, it applies to a finite time window, so it can be used to assess whether a blockchain is capable of hosting financial activity that relies on timely inclusion. We apply this definition in the context of auctions. Auctions are a building block for many financial applications, and censoring competing bids offers an easy-to-model motivation for our adversary. Traditional proof-of-stake blockchains have poor enough censorship resistance that it is difficult to retain the integrity of an auction when bids can only be submitted in a single block. As the number of bidders n in a single block auction increases, the probability that the winner is not the adversary, and the economic efficiency of the auction, both decrease faster than 1/n. Running the auction over multiple blocks, each with a different proposer, alleviates the problem only if the number of blocks grows faster than the number of bidders. We argue that blockchains with more than one concurrent proposer can have strong censorship resistance. We achieve this by setting up a prisoner’s dilemma among the proposers using conditional tips.

Cite as

Elijah Fox, Mallesh M. Pai, and Max Resnick. Censorship Resistance in On-Chain Auctions. In 5th Conference on Advances in Financial Technologies (AFT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 282, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fox_et_al:LIPIcs.AFT.2023.19,
  author =	{Fox, Elijah and Pai, Mallesh M. and Resnick, Max},
  title =	{{Censorship Resistance in On-Chain Auctions}},
  booktitle =	{5th Conference on Advances in Financial Technologies (AFT 2023)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-303-4},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{282},
  editor =	{Bonneau, Joseph and Weinberg, S. Matthew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.AFT.2023.19},
  URN =		{urn:nbn:de:0030-drops-192089},
  doi =		{10.4230/LIPIcs.AFT.2023.19},
  annote =	{Keywords: Censorship Resistance, Auctions, Blockchain, MEV}
}
Document
Approximating Connected Maximum Cuts via Local Search

Authors: Baruch Schieber and Soroush Vahidi

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)


Abstract
The Connected Max Cut (CMC) problem takes in an undirected graph G(V,E) and finds a subset S ⊆ V such that the induced subgraph G[S] is connected and the number of edges connecting vertices in S to vertices in V⧵S is maximized. This problem is closely related to the Max Leaf Degree (MLD) problem. The input to the MLD problem is an undirected graph G(V,E) and the goal is to find a subtree of G that maximizes the degree (in G) of its leaves. [Gandhi et al. 2018] observed that an α-approximation for the MLD problem induces an 𝒪(α)-approximation for the CMC problem. We present an 𝒪(log log |V|)-approximation algorithm for the MLD problem via local search. This implies an 𝒪(log log |V|)-approximation algorithm for the CMC problem. Thus, improving (exponentially) the best known 𝒪(log |V|) approximation of the Connected Max Cut problem [Hajiaghayi et al. 2015].

Cite as

Baruch Schieber and Soroush Vahidi. Approximating Connected Maximum Cuts via Local Search. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 93:1-93:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{schieber_et_al:LIPIcs.ESA.2023.93,
  author =	{Schieber, Baruch and Vahidi, Soroush},
  title =	{{Approximating Connected Maximum Cuts via Local Search}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{93:1--93:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.93},
  URN =		{urn:nbn:de:0030-drops-187466},
  doi =		{10.4230/LIPIcs.ESA.2023.93},
  annote =	{Keywords: approximation algorithms, graph theory, max-cut, local search}
}
Document
On Flipping the Fréchet Distance

Authors: Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk

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


Abstract
The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min - the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Cite as

Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk. On Flipping the Fréchet Distance. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.51,
  author =	{Filtser, Omrit and Goswami, Mayank and Mitchell, Joseph S. B. and Polishchuk, Valentin},
  title =	{{On Flipping the Fr\'{e}chet Distance}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{51:1--51:22},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.51},
  URN =		{urn:nbn:de:0030-drops-175548},
  doi =		{10.4230/LIPIcs.ITCS.2023.51},
  annote =	{Keywords: curves, polygons, distancing measure}
}
Document
Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

Authors: Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size n which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized α-approximation algorithm that runs in c^k⋅n^𝒪(1) time, where k is the solution size, can be used to derive an α-approximation randomized algorithm that runs in dⁿ⋅n^𝒪(1) time, where d is the unique value in (1, 1+{c-1}/α) such that 𝒟(1/α‖{d-1}/{c-1}) = {ln c}/α and 𝒟(a‖b) is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for α = 1, and is strictly better when α > 1, for any c > 1. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We use an approximate variant of the exhaustive search as a benchmark for our algorithm. We show that the classic 2ⁿ⋅n^𝒪(1) exhaustive search can be adapted to an α-approximate exhaustive search that runs in time (1+exp(-α⋅ℋ(1/(α))))ⁿ⋅n^𝒪(1), where ℋ is the entropy function. Furthermore, we provide a lower bound stating that the running time of this α-approximate exhaustive search is the best achievable running time in an oracle model. When compared to approximate exhaustive search, and to other techniques, the running times obtained by approximate monotone local search are strictly better for any α ≥ 1, c > 1. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, 3-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a 1.1-approximation algorithm for Vertex Cover with running time 1.114ⁿ⋅n^𝒪(1), improving upon the previously best known 1.1-approximation running in time 1.127ⁿ⋅n^𝒪(1) by Bourgeois et al. [DAM 2011].

Cite as

Barış Can Esmer, Ariel Kulik, Dániel Marx, Daniel Neuen, and Roohani Sharma. Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 50:1-50:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{esmer_et_al:LIPIcs.ESA.2022.50,
  author =	{Esmer, Bar{\i}\c{s} Can and Kulik, Ariel and Marx, D\'{a}niel and Neuen, Daniel and Sharma, Roohani},
  title =	{{Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{50:1--50:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.50},
  URN =		{urn:nbn:de:0030-drops-169887},
  doi =		{10.4230/LIPIcs.ESA.2022.50},
  annote =	{Keywords: parameterized approximations, exponential approximations, monotone local search}
}
Document
A New Exact Solver for (Weighted) Max#SAT

Authors: Gilles Audemard, Jean-Marie Lagniez, and Marie Miceli

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


Abstract
We present and evaluate d4Max, an exact approach for solving the Weighted Max#SAT problem. The Max#SAT problem extends the model counting problem (#SAT) by considering a tripartition of the variables {X, Y, Z}, and consists in maximizing over X the number of assignments to Y that can be extended to a solution with some assignment to Z. The Weighted Max#SAT problem is an extension of the Max#SAT problem with weights associated on each interpretation. We test and compare our approach with other state-of-the-art solvers on the challenging task in probabilistic inference of finding the marginal maximum a posteriori probability (MMAP) of a given subset of the variables in a Bayesian network and on exist-random quantified SSAT benchmarks. The results clearly show the overall superiority of d4Max in term of speed and number of instances solved. Moreover, we experimentally show that, in general, d4Max is able to quickly spot a solution that is close to optimal, thereby opening the door to an efficient anytime approach.

Cite as

Gilles Audemard, Jean-Marie Lagniez, and Marie Miceli. A New Exact Solver for (Weighted) Max#SAT. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 28:1-28:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{audemard_et_al:LIPIcs.SAT.2022.28,
  author =	{Audemard, Gilles and Lagniez, Jean-Marie and Miceli, Marie},
  title =	{{A New Exact Solver for (Weighted) Max#SAT}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{28:1--28:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.28},
  URN =		{urn:nbn:de:0030-drops-167022},
  doi =		{10.4230/LIPIcs.SAT.2022.28},
  annote =	{Keywords: Max#SAT, EMaj-SAT, Weighted Projected Model Counting, SSAT}
}
Document
On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

Authors: Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang

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


Abstract
We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics.

Cite as

Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{afshani_et_al:LIPIcs.SoCG.2022.2,
  author =	{Afshani, Peyman and de Berg, Mark and Buchin, Kevin and Gao, Jie and L\"{o}ffler, Maarten and Nayyeri, Amir and Raichel, Benjamin and Sarkar, Rik and Wang, Haotian and Yang, Hao-Tsung},
  title =	{{On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.2},
  URN =		{urn:nbn:de:0030-drops-160109},
  doi =		{10.4230/LIPIcs.SoCG.2022.2},
  annote =	{Keywords: Approximation, Motion Planning, Scheduling}
}
Document
Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves

Authors: Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, Dániel Marx, and André Nusser

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


Abstract
The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in ℝ^d, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of π and σ under arbitrary translations, to compare the curves' shape irrespective of their absolute location? There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and k-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm. For the L₁ norm in ℝ^d, we provide an 𝒪(n^{2(d+1)})-time algorithm, i.e., an exact polynomial-time algorithm for constant d. Here and below, n bounds the curves' complexities. For the Euclidean norm in ℝ², we show that a simple problem-specific insight leads to a (1+ε)-approximation in time 𝒪(n³/ε²). We then show how to obtain a subcubic 𝒪̃(n^{2.5}/ε²) time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure. We hope that our results will facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.

Cite as

Karl Bringmann, Sándor Kisfaludi‑Bak, Marvin Künnemann, Dániel Marx, and André Nusser. Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bringmann_et_al:LIPIcs.SoCG.2022.20,
  author =	{Bringmann, Karl and Kisfaludi‑Bak, S\'{a}ndor and K\"{u}nnemann, Marvin and Marx, D\'{a}niel and Nusser, Andr\'{e}},
  title =	{{Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{20:1--20:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.20},
  URN =		{urn:nbn:de:0030-drops-160287},
  doi =		{10.4230/LIPIcs.SoCG.2022.20},
  annote =	{Keywords: Dynamic Time Warping, Sequence Similarity Measures}
}
Document
Correlation Detection in Trees for Planted Graph Alignment

Authors: Luca Ganassali, Laurent Massoulié, and Marc Lelarge

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


Abstract
Motivated by alignment of correlated sparse random graphs, we study a hypothesis problem of deciding whether two random trees are correlated or not. Based on this correlation detection problem, we propose MPAlign, a message-passing algorithm for graph alignment, which we prove to succeed in polynomial time at partial alignment whenever tree detection is feasible. As a result our analysis of tree detection reveals new ranges of parameters for which partial alignment of sparse random graphs is feasible in polynomial time. We conjecture that the connection between partial graph alignment and tree detection runs deeper, and that the parameter range where tree detection is impossible, which we partially characterize, corresponds to a region where partial graph alignment is hard (not polytime feasible).

Cite as

Luca Ganassali, Laurent Massoulié, and Marc Lelarge. Correlation Detection in Trees for Planted Graph Alignment. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 74:1-74:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{ganassali_et_al:LIPIcs.ITCS.2022.74,
  author =	{Ganassali, Luca and Massouli\'{e}, Laurent and Lelarge, Marc},
  title =	{{Correlation Detection in Trees for Planted Graph Alignment}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{74:1--74:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.74},
  URN =		{urn:nbn:de:0030-drops-156709},
  doi =		{10.4230/LIPIcs.ITCS.2022.74},
  annote =	{Keywords: inference on graphs, hypothesis testing, Erd\H{o}s-R\'{e}nyi random graphs}
}
Document
Tight Chang’s-Lemma-Type Bounds for Boolean Functions

Authors: Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)


Abstract
Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f) := Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold: - the Fourier sparsity of f, denoted k(f), - the Fourier max-supp-entropy of f, denoted k'(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient, - the Fourier max-rank-entropy of f, denoted k''(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω(r(f)²/(k(f) log² k(f))), subsumes and refines the previously best known upper bound r(f) = O(√{k(f)}log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(√{k(f)δ(f)}log k(f)). Another lower bound, δ(f) = Ω(r(f)/(k''(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of 𝔽₂-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f). Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications.

Cite as

Sourav Chakraborty, Nikhil S. Mande, Rajat Mittal, Tulasimohan Molli, Manaswi Paraashar, and Swagato Sanyal. Tight Chang’s-Lemma-Type Bounds for Boolean Functions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2021.10,
  author =	{Chakraborty, Sourav and Mande, Nikhil S. and Mittal, Rajat and Molli, Tulasimohan and Paraashar, Manaswi and Sanyal, Swagato},
  title =	{{Tight Chang’s-Lemma-Type Bounds for Boolean Functions}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{10:1--10:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.10},
  URN =		{urn:nbn:de:0030-drops-155215},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.10},
  annote =	{Keywords: Analysis of Boolean functions, Chang’s lemma, Parity decision trees, Fourier dimension}
}
Document
Efficient Duration-Based Workload Balancing for Interdependent Vehicle Routes

Authors: Carlo S. Sartori, Pieter Smet, and Greet Vanden Berghe

Published in: OASIcs, Volume 96, 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)


Abstract
Vehicle routing and scheduling problems with interdependent routes arise when some services must be performed by at least two vehicles and temporal synchronization is thus required between the starting times of these services. These problems are often coupled with time window constraints in order to model various real-world applications such as pickup and delivery with transfers, cross-docking and home care scheduling. Interdependent routes in these applications can lead to large idle times for some drivers, unnecessarily lengthening their working hours. To remedy this unfairness, it is necessary to balance the duration of the drivers' routes. However, quickly evaluating duration-based equity functions for interdependent vehicle routes with time windows poses a significant computational challenge, particularly when the departure time of routes is flexible. This paper introduces models and algorithms to compute two well-known equity functions in flexible departure time settings: min-max and range minimization. We explore the challenges and algorithmic complexities of evaluating these functions both from a theoretical and an experimental viewpoint. The results of this paper enable the development of new heuristic methods to balance the workload of interdependent vehicle routes with time windows.

Cite as

Carlo S. Sartori, Pieter Smet, and Greet Vanden Berghe. Efficient Duration-Based Workload Balancing for Interdependent Vehicle Routes. In 21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021). Open Access Series in Informatics (OASIcs), Volume 96, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{sartori_et_al:OASIcs.ATMOS.2021.1,
  author =	{Sartori, Carlo S. and Smet, Pieter and Vanden Berghe, Greet},
  title =	{{Efficient Duration-Based Workload Balancing for Interdependent Vehicle Routes}},
  booktitle =	{21st Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2021)},
  pages =	{1:1--1:15},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-213-6},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{96},
  editor =	{M\"{u}ller-Hannemann, Matthias and Perea, Federico},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2021.1},
  URN =		{urn:nbn:de:0030-drops-148703},
  doi =		{10.4230/OASIcs.ATMOS.2021.1},
  annote =	{Keywords: Vehicle scheduling, Workload balancing, Route duration, Interdependent routes, Time windows}
}
Document
Gapped Indexing for Consecutive Occurrences

Authors: Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, and Teresa Anna Steiner

Published in: LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)


Abstract
The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient pattern matching queries. Typical queries include existential queries (decide if the pattern occurs in S), reporting queries (return all positions where the pattern occurs), and counting queries (return the number of occurrences of the pattern). In this paper we consider a variant of string indexing, where the goal is to compactly represent the string such that given two patterns P₁ and P₂ and a gap range [α, β] we can quickly find the consecutive occurrences of P₁ and P₂ with distance in [α, β], i.e., pairs of subsequent occurrences with distance within the range. We present data structures that use Õ(n) space and query time Õ(|P₁|+|P₂|+n^{2/3}) for existence and counting and Õ(|P₁|+|P₂|+n^{2/3}occ^{1/3}) for reporting. We complement this with a conditional lower bound based on the set intersection problem showing that any solution using Õ(n) space must use Ω̃(|P₁| + |P₂| + √n) query time. To obtain our results we develop new techniques and ideas of independent interest including a new suffix tree decomposition and hardness of a variant of the set intersection problem.

Cite as

Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, and Teresa Anna Steiner. Gapped Indexing for Consecutive Occurrences. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bille_et_al:LIPIcs.CPM.2021.10,
  author =	{Bille, Philip and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Steiner, Teresa Anna},
  title =	{{Gapped Indexing for Consecutive Occurrences}},
  booktitle =	{32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)},
  pages =	{10:1--10:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-186-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{191},
  editor =	{Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.10},
  URN =		{urn:nbn:de:0030-drops-139615},
  doi =		{10.4230/LIPIcs.CPM.2021.10},
  annote =	{Keywords: String indexing, two patterns, consecutive occurrences, conditional lower bound}
}
Document
String Indexing for Top-k Close Consecutive Occurrences

Authors: Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, Eva Rotenberg, and Teresa Anna Steiner

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)


Abstract
The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i,j), i < j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j-i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give two time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ε ∈ (0,1]. Our first result achieves O(nlog n) space and optimal query time of O(m+k), and our second result achieves linear space and query time O(m+k^{1+ε}). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.

Cite as

Philip Bille, Inge Li Gørtz, Max Rishøj Pedersen, Eva Rotenberg, and Teresa Anna Steiner. String Indexing for Top-k Close Consecutive Occurrences. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{bille_et_al:LIPIcs.FSTTCS.2020.14,
  author =	{Bille, Philip and G{\o}rtz, Inge Li and Pedersen, Max Rish{\o}j and Rotenberg, Eva and Steiner, Teresa Anna},
  title =	{{String Indexing for Top-k Close Consecutive Occurrences}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.14},
  URN =		{urn:nbn:de:0030-drops-132558},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.14},
  annote =	{Keywords: String indexing, pattern matching, consecutive occurrences}
}
Document
A Quasi-Polynomial Algorithm for Well-Spaced Hyperbolic TSP

Authors: Sándor Kisfaludi-Bak

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


Abstract
We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature -1. Let α denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an n^{O(log² n)max(1,1/α)} algorithm for Hyperbolic TSP. This is quasi-polynomial time if α is at least some absolute constant, and it grows to n^O(√n) as α decreases to log² n/√n. (For even smaller values of α, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of n^O(√n).)

Cite as

Sándor Kisfaludi-Bak. A Quasi-Polynomial Algorithm for Well-Spaced Hyperbolic TSP. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kisfaludibak:LIPIcs.SoCG.2020.55,
  author =	{Kisfaludi-Bak, S\'{a}ndor},
  title =	{{A Quasi-Polynomial Algorithm for Well-Spaced Hyperbolic TSP}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{55:1--55:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.55},
  URN =		{urn:nbn:de:0030-drops-122135},
  doi =		{10.4230/LIPIcs.SoCG.2020.55},
  annote =	{Keywords: Computational geometry, Hyperbolic geometry, Traveling salesman}
}
Document
Oracle Complexity Classes and Local Measurements on Physical Hamiltonians

Authors: Sevag Gharibian, Stephen Piddock, and Justin Yirka

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)


Abstract
The canonical hard problems for NP and its quantum analogue, Quantum Merlin-Arthur (QMA), are MAX-k-SAT and the k-local Hamiltonian problem (k-LH), the quantum generalization of MAX-k-SAT, respectively. In recent years, however, an arguably even more physically motivated problem than k-LH has been formalized - the problem of simulating local measurements on ground states of local Hamiltonians (APX-SIM). Perhaps surprisingly, [Ambainis, CCC 2014] showed that APX-SIM is likely harder than QMA. Indeed, [Ambainis, CCC 2014] showed that APX-SIM is P^{QMA[log]}-complete, for P^{QMA[log]} the class of languages decidable by a P machine making a logarithmic number of adaptive queries to a QMA oracle. In this work, we show that APX-SIM is P^{QMA[log]}-complete even when restricted to physically motivated Hamiltonians, obtaining as intermediate steps a variety of related complexity-theoretic results. Specifically, we first give a sequence of results which together yield P^{QMA[log]}-hardness for APX-SIM on well-motivated Hamiltonians such as the 2D Heisenberg model: - We show that for NP, StoqMA, and QMA oracles, a logarithmic number of adaptive queries is equivalent to polynomially many parallel queries. Formally, P^{NP[log]}=P^{||NP}, P^{StoqMA[log]}=P^{||StoqMA}, and P^{QMA[log]}=P^{||QMA}. (The result for NP was previously shown using a different proof technique.) These equalities simplify the proofs of our subsequent results. - Next, we show that the hardness of APX-SIM is preserved under Hamiltonian simulations (à la [Cubitt, Montanaro, Piddock, 2017]) by studying a seemingly weaker problem, ∀-APX-SIM. As a byproduct, we obtain a full complexity classification of APX-SIM, showing it is complete for P, P^{||NP},P^{||StoqMA}, or P^{||QMA} depending on the Hamiltonians employed. - Leveraging the above, we show that APX-SIM is P^{QMA[log]}-complete for any family of Hamiltonians which can efficiently simulate spatially sparse Hamiltonians. This implies APX-SIM is P^{QMA[log]}-complete even on physically motivated models such as the 2D Heisenberg model. Our second focus considers 1D systems: We show that APX-SIM remains P^{QMA[log]}-complete even for local Hamiltonians on a 1D line of 8-dimensional qudits. This uses a number of ideas from above, along with replacing the "query Hamiltonian" of [Ambainis, CCC 2014] with a new "sifter" construction.

Cite as

Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle Complexity Classes and Local Measurements on Physical Hamiltonians. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 20:1-20:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{gharibian_et_al:LIPIcs.STACS.2020.20,
  author =	{Gharibian, Sevag and Piddock, Stephen and Yirka, Justin},
  title =	{{Oracle Complexity Classes and Local Measurements on Physical Hamiltonians}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{20:1--20:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.20},
  URN =		{urn:nbn:de:0030-drops-118818},
  doi =		{10.4230/LIPIcs.STACS.2020.20},
  annote =	{Keywords: Quantum Merlin Arthur (QMA), simulation of local measurement, local Hamiltonian, oracle complexity class, physical Hamiltonians}
}
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