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Documents authored by Frei, Fabian

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.31
From Chinese Postman to Salesman and Beyond: Shortest Tour δ-Covering All Points on All Edges

Authors: Fabian Frei, Ahmed Ghazy, Tim A. Hartmann, Florian Hörsch, and Dániel Marx

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
A well-studied continuous model of graphs, introduced by Dearing and Francis [Transportation Science, 1974], considers each edge as a continuous unit-length interval of points. For δ ≥ 0, we introduce the problem δ-Tour, where the objective is to find the shortest tour that comes within a distance of δ of every point on every edge. It can be observed that 0-Tour is essentially equivalent to the Chinese Postman Problem, which is solvable in polynomial time. In contrast, 1/2-Tour is essentially equivalent to the graphic Traveling Salesman Problem (TSP), which is NP-hard but admits a constant-factor approximation in polynomial time. We investigate δ-Tour for other values of δ, noting that the problem’s behavior and the insights required to understand it differ significantly across various δ regimes. On the one hand, we first examine the approximability of the problem for every fixed δ > 0: 1) For every fixed 0 < δ < 3/2, the problem δ-Tour admits a constant-factor approximation and is APX-hard, while for every fixed δ ≥ 3/2, the problem admits an O(log n)-approximation in polynomial time and has no polynomial-time o(log n)-approximation, unless P = NP. Our techniques also yield a new APX-hardness result for graphic TSP on cubic bipartite graphs. When parameterizing by the length of a shortest tour, it is relatively easy to show that 3/2 is the threshold of fixed-parameter tractability: 2) For every fixed 0 < δ < 3/2, the problem δ-Tour is fixed-parameter tractable (FPT) when parameterized by the length of a shortest tour, while it is W[2]-hard for every fixed δ ≥ 3/2. On the other hand, if δ is considered to be part of the input, then an interesting nontrivial phenomenon appears when δ is a constant fraction of the number of vertices: 3) If δ is part of the input, then the problem can be solved in time f(k)n^O(k), where k = ⌈n/δ⌉; however, assuming the Exponential-Time Hypothesis (ETH), there is no algorithm that solves the problem and runs in time f(k)n^o(k/log k).
Cite as

Fabian Frei, Ahmed Ghazy, Tim A. Hartmann, Florian Hörsch, and Dániel Marx. From Chinese Postman to Salesman and Beyond: Shortest Tour δ-Covering All Points on All Edges. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{frei_et_al:LIPIcs.ISAAC.2024.31,
  author =	{Frei, Fabian and Ghazy, Ahmed and Hartmann, Tim A. and H\"{o}rsch, Florian and Marx, D\'{a}niel},
  title =	{{From Chinese Postman to Salesman and Beyond: Shortest Tour \delta-Covering All Points on All Edges}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{31:1--31:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.31},
  URN =		{urn:nbn:de:0030-drops-221582},
  doi =		{10.4230/LIPIcs.ISAAC.2024.31},
  annote =	{Keywords: Chinese Postman Problem, Traveling Salesman Problem, Continuous Graphs, Approximation Algorithms, Inapproximability, Parameterized Complexity}
}
Document
DOI: 10.4230/LIPIcs.DISC.2024.26
Content-Oblivious Leader Election on Rings

Authors: Fabian Frei, Ran Gelles, Ahmed Ghazy, and Alexandre Nolin

Published in: LIPIcs, Volume 319, 38th International Symposium on Distributed Computing (DISC 2024)

Abstract
In content-oblivious computation, n nodes wish to compute a given task over an asynchronous network that suffers from an extremely harsh type of noise, which corrupts the content of all messages across all channels. In a recent work, Censor-Hillel, Cohen, Gelles, and Sela (Distributed Computing, 2023) showed how to perform arbitrary computations in a content-oblivious way in 2-edge connected networks but only if the network has a distinguished node (called root) to initiate the computation. Our goal is to remove this assumption, which was conjectured to be necessary. Achieving this goal essentially reduces to performing a content-oblivious leader election since an elected leader can then serve as the root required to perform arbitrary content-oblivious computations. We focus on ring networks, which are the simplest 2-edge connected graphs. On oriented rings, we obtain a leader election algorithm with message complexity O(n ⋅ ID_max), where ID_max is the maximal assigned ID. As it turns out, this dependency on ID_max is inherent: we show a lower bound of Ω(n log(ID_max/n)) messages for content-oblivious leader election algorithms. We also extend our results to non-oriented rings, where nodes cannot tell which channel leads to which neighbor. In this case, however, the algorithm does not terminate but only reaches quiescence.
Cite as

Fabian Frei, Ran Gelles, Ahmed Ghazy, and Alexandre Nolin. Content-Oblivious Leader Election on Rings. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{frei_et_al:LIPIcs.DISC.2024.26,
  author =	{Frei, Fabian and Gelles, Ran and Ghazy, Ahmed and Nolin, Alexandre},
  title =	{{Content-Oblivious Leader Election on Rings}},
  booktitle =	{38th International Symposium on Distributed Computing (DISC 2024)},
  pages =	{26:1--26:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-352-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{319},
  editor =	{Alistarh, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.26},
  URN =		{urn:nbn:de:0030-drops-212527},
  doi =		{10.4230/LIPIcs.DISC.2024.26},
  annote =	{Keywords: Content-Oblivious Computation, Faulty Communication, Leader Election, Ring Networks, Ring Orientation}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2024.48
Brief Announcement: Distinct Gathering Under Round Robin

Authors: Fabian Frei and Koichi Wada

Published in: LIPIcs, Volume 319, 38th International Symposium on Distributed Computing (DISC 2024)

Abstract
We resolve one of the longest-standing questions about autonomous mobile robots in a surprising way. Distinct Gathering is the fundamental cooperation task of letting robots, initially scattered across the plane in distinct locations, gather in an arbitrary single point. The scheduler Round Robin cyclically activates the robots one by one in a fixed order. When activated, a robot perceives all robot locations and moves wherever it wants based only on this information. For n = 2 robots, the task is trivial. What happens for n ≥ 3 has remained an open problem for decades by now. The established conjecture declares the task to be impossible in this case. We prove that it is indeed impossible for n = 3 but, to great surprise, possible again for any n ≥ 4. We go beyond the standard requirements by providing a very robust algorithm that does not require any consistency or self-consistency for the local Cartesian maps perceived by the robots and works even for non-rigid movement, that is, if robots may be unpredictably stopped and deactivated during a movement.
Cite as

Fabian Frei and Koichi Wada. Brief Announcement: Distinct Gathering Under Round Robin. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 48:1-48:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{frei_et_al:LIPIcs.DISC.2024.48,
  author =	{Frei, Fabian and Wada, Koichi},
  title =	{{Brief Announcement: Distinct Gathering Under Round Robin}},
  booktitle =	{38th International Symposium on Distributed Computing (DISC 2024)},
  pages =	{48:1--48:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-352-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{319},
  editor =	{Alistarh, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.48},
  URN =		{urn:nbn:de:0030-drops-212768},
  doi =		{10.4230/LIPIcs.DISC.2024.48},
  annote =	{Keywords: Autonomous mobile robots, Distinct Gathering, Round Robin}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.55
Hitting Meets Packing: How Hard Can It Be?

Authors: Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki

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

Abstract
We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of 𝒳-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing 𝓁 vertex-disjoint objects of type 𝒳? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination 𝒳-HitPack can be significantly harder than these two base problems. Already for one particular choice of 𝒳, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain at the intersection of the areas of hitting and packing problems. At a high level, we present two case studies: (1) 𝒳 being all cycles, and (2) 𝒳 being all copies of a fixed graph H. In each, we explore the classical complexity as well as the parameterized complexity with the natural parameters k+𝓁 and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph on at least 3 vertices, the problem is Σ₂^𝖯-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For 𝒳 being all cycles, we establish a 2^{poly(k+𝓁)}⋅ n^{𝒪(1)} algorithm using an involved branching method, for example. Also, for 𝒳 being all edges (i.e., H = K₂; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^{poly(tw)}⋅ n^{𝒪(1)} on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.
Cite as

Jacob Focke, Fabian Frei, Shaohua Li, Dániel Marx, Philipp Schepper, Roohani Sharma, and Karol Węgrzycki. Hitting Meets Packing: How Hard Can It Be?. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 55:1-55:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{focke_et_al:LIPIcs.ESA.2024.55,
  author =	{Focke, Jacob and Frei, Fabian and Li, Shaohua and Marx, D\'{a}niel and Schepper, Philipp and Sharma, Roohani and W\k{e}grzycki, Karol},
  title =	{{Hitting Meets Packing: How Hard Can It Be?}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{55:1--55:21},
  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.2024.55},
  URN =		{urn:nbn:de:0030-drops-211261},
  doi =		{10.4230/LIPIcs.ESA.2024.55},
  annote =	{Keywords: Hitting, Packing, Covering, Parameterized Algorithms, Lower Bounds, Treewidth}
}
Document
DOI: 10.4230/LIPIcs.STACS.2024.18
Removable Online Knapsack and Advice

Authors: Hans-Joachim Böckenhauer, Fabian Frei, and Peter Rossmanith

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

Abstract
In the proportional knapsack problem, we are given a knapsack of some capacity and a set of variably sized items. The goal is to pack a selection of these items that fills the knapsack as much as possible. The online version of this problem reveals the items and their sizes not all at once but one by one. For each item, the algorithm has to decide immediately whether to pack it or not. We consider a natural variant of this online knapsack problem, which has been coined removable knapsack. It differs from the classical variant by allowing the removal of any packed item from the knapsack. Repacking is impossible, however: Once an item is removed, it is gone for good. We analyze the advice complexity of this problem. It measures how many advice bits an omniscient oracle needs to provide for an online algorithm to reach any given competitive ratio, which is - understood in its strict sense - just the algorithm’s approximation factor. The online knapsack problem is known for its peculiar advice behavior involving three jumps in competitivity. We show that the advice complexity of the version with removability is quite different but just as interesting: The competitivity starts from the golden ratio when no advice is given. It then drops down to 1+ε for a constant amount of advice already, which requires logarithmic advice in the classical version. Removability comes as no relief to the perfectionist, however: Optimality still requires linear advice as before. These results are particularly noteworthy from a structural viewpoint for the exceptionally slow transition from near-optimality to optimality. Our most important and demanding result shows that the general knapsack problem, which allows an item’s value to differ from its size, exhibits a similar behavior for removability, but with an even more pronounced jump from an unbounded competitive ratio to near-optimality within just constantly many advice bits. This is a unique behavior among the problems considered in the literature so far. An advice analysis is interesting in its own right, as it allows us to measure the information content of a problem and leads to structural insights. But it also provides insurmountable lower bounds, applicable to any kind of additional information about the instances, including predictions provided by machine-learning algorithms and artificial intelligence. Unexpectedly, advice algorithms are useful in various real-life situations, too. For example, they provide smart strategies for cooperation in winner-take-all competitions, where several participants pool together to implement different strategies and share the obtained prize. Further illustrating the versatility of our advice-complexity bounds, our results automatically improve some of the best known lower bounds on the competitive ratio for removable knapsack with randomization. The presented advice algorithms also automatically yield deterministic algorithms for established deterministic models such as knapsack with a resource buffer and various problems with more than one knapsack. In their seminal paper introducing removability to the knapsack problem, Iwama and Taketomi have indeed proposed a multiple knapsack problem for which we can establish a one-to-one correspondence with the advice model; this paper therefore even provides a comprehensive analysis for this up until now neglected problem.
Cite as

Hans-Joachim Böckenhauer, Fabian Frei, and Peter Rossmanith. Removable Online Knapsack and Advice. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bockenhauer_et_al:LIPIcs.STACS.2024.18,
  author =	{B\"{o}ckenhauer, Hans-Joachim and Frei, Fabian and Rossmanith, Peter},
  title =	{{Removable Online Knapsack and Advice}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.18},
  URN =		{urn:nbn:de:0030-drops-197283},
  doi =		{10.4230/LIPIcs.STACS.2024.18},
  annote =	{Keywords: Removable Online Knapsack, Multiple Knapsack, Advice Analysis, Advice Applications, Machine Learning and AI}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.19
Complexity of Stability

Authors: Fabian Frei, Edith Hemaspaandra, and Jörg Rothe

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

Abstract
Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for Θ₂ᵖ, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP."
Cite as

Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of Stability. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{frei_et_al:LIPIcs.ISAAC.2020.19,
  author =	{Frei, Fabian and Hemaspaandra, Edith and Rothe, J\"{o}rg},
  title =	{{Complexity of Stability}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{19:1--19:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.19},
  URN =		{urn:nbn:de:0030-drops-133631},
  doi =		{10.4230/LIPIcs.ISAAC.2020.19},
  annote =	{Keywords: Stability, Robustness, Complexity, Local Modifications, Colorability, Vertex Cover, Clique, Independent Set, Satisfiability, Unfrozenness, Criticality, DP, coDP, Parallel Access to NP}
}
Document
DOI: 10.4230/LIPIcs.FUN.2021.14
An Open Pouring Problem

Authors: Fabian Frei, Peter Rossmanith, and David Wehner

Published in: LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Abstract
We analyze a little riddle that has challenged mathematicians for half a century. Imagine three clubs catering to people with some niche interest. Everyone willing to join a club has done so and nobody new will pick up this eccentric hobby for the foreseeable future, thus the mutually exclusive clubs compete for a common constituency. Members are highly invested in their chosen club; only a targeted campaign plus prolonged personal persuasion can convince them to consider switching. Even then, they will never be enticed into a bigger group as they naturally pride themselves in avoiding the mainstream. Therefore each club occasionally starts a campaign against a larger competitor and sends its own members out on a recommendation program. Each will win one person over; the small club can thus effectively double its own numbers at the larger one’s expense. Is there always a risk for one club to wind up with zero members, forcing it out of business? If so, how many campaign cycles will this take?
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Fabian Frei, Peter Rossmanith, and David Wehner. An Open Pouring Problem. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 14:1-14:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{frei_et_al:LIPIcs.FUN.2021.14,
  author =	{Frei, Fabian and Rossmanith, Peter and Wehner, David},
  title =	{{An Open Pouring Problem}},
  booktitle =	{10th International Conference on Fun with Algorithms (FUN 2021)},
  pages =	{14:1--14:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-145-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{157},
  editor =	{Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.14},
  URN =		{urn:nbn:de:0030-drops-127751},
  doi =		{10.4230/LIPIcs.FUN.2021.14},
  annote =	{Keywords: Pitcher Pouring Problem, Water Jug Riddle, Water Bucket Problem, Vessel Puzzle, Complexity, Die Hard}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2019.52
Efficient Circuit Simulation in MapReduce

Authors: Fabian Frei and Koichi Wada

Published in: LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Abstract
The MapReduce framework has firmly established itself as one of the most widely used parallel computing platforms for processing big data on tera- and peta-byte scale. Approaching it from a theoretical standpoint has proved to be notoriously difficult, however. In continuation of Goodrich et al.’s early efforts, explicitly espousing the goal of putting the MapReduce framework on footing equal to that of long-established models such as the PRAM, we investigate the obvious complexity question of how the computational power of MapReduce algorithms compares to that of combinational Boolean circuits commonly used for parallel computations. Relying on the standard MapReduce model introduced by Karloff et al. a decade ago, we develop an intricate simulation technique to show that any problem in NC (i.e., a problem solved by a logspace-uniform family of Boolean circuits of polynomial size and a depth polylogarithmic in the input size) can be solved by a MapReduce computation in O(T(n)/log n) rounds, where n is the input size and T(n) is the depth of the witnessing circuit family. Thus, we are able to closely relate the standard, uniform NC hierarchy modeling parallel computations to the deterministic MapReduce hierarchy DMRC by proving that NC^{i+1} subseteq DMRC^i for all i in N. Besides the theoretical significance, this result has important applied aspects as well. In particular, we show for all problems in NC^1 - many practically relevant ones, such as integer multiplication and division and the parity function, being among these - how to solve them in a constant number of deterministic MapReduce rounds.
Cite as

Fabian Frei and Koichi Wada. Efficient Circuit Simulation in MapReduce. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 52:1-52:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{frei_et_al:LIPIcs.ISAAC.2019.52,
  author =	{Frei, Fabian and Wada, Koichi},
  title =	{{Efficient Circuit Simulation in MapReduce}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{52:1--52:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Lu, Pinyan and Zhang, Guochuan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.52},
  URN =		{urn:nbn:de:0030-drops-115487},
  doi =		{10.4230/LIPIcs.ISAAC.2019.52},
  annote =	{Keywords: MapReduce, Circuit Complexity, Parallel Algorithms, Nick’s Class NC}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.78
Finding Optimal Solutions With Neighborly Help

Authors: Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.
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Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner. Finding Optimal Solutions With Neighborly Help. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 78:1-78:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{burjons_et_al:LIPIcs.MFCS.2019.78,
  author =	{Burjons, Elisabet and Frei, Fabian and Hemaspaandra, Edith and Komm, Dennis and Wehner, David},
  title =	{{Finding Optimal Solutions With Neighborly Help}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{78:1--78:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.78},
  URN =		{urn:nbn:de:0030-drops-110221},
  doi =		{10.4230/LIPIcs.MFCS.2019.78},
  annote =	{Keywords: Critical Graphs, Computational Complexity, Structural Self-Reducibility, Minimality Problems, Colorability, Vertex Cover, Satisfiability, Reoptimization, Advice}
}
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