Volume
LIPIcs, Volume 204
29th Annual European Symposium on Algorithms (ESA 2021)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: European Symposium on Algorithms (ESA)
Publication Details
- published at: 2021-08-31
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-204-4
- DBLP: db/conf/esa/esa2021
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Complete Volume
LIPIcs, Volume 204, ESA 2021, Complete Volume
Abstract
LIPIcs, Volume 204, ESA 2021, Complete Volume
Cite as
29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 1-1340, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@Proceedings{mutzel_et_al:LIPIcs.ESA.2021,
title = {{LIPIcs, Volume 204, ESA 2021, Complete Volume}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {1--1340},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021},
URN = {urn:nbn:de:0030-drops-145808},
doi = {10.4230/LIPIcs.ESA.2021},
annote = {Keywords: LIPIcs, Volume 204, ESA 2021, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{mutzel_et_al:LIPIcs.ESA.2021.0,
author = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {0:i--0:xx},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.0},
URN = {urn:nbn:de:0030-drops-145816},
doi = {10.4230/LIPIcs.ESA.2021.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Network Planning and Routing Problems over Time: Models, Complexity and Algorithms (Invited Talk)
Abstract
In this invited contribution for ESA 2021, we will study the complexity of and algorithms for network optimization tasks with a timing component. They occur, for example, in planning or routing problems that need to be solved repeatedly over time. Typically, already simplified versions of such problems are NP-hard. In addition, the instances typically are too large to be solved straight-forwardly on a time-expanded graph. After an introduction into the area, we state the problem of determining best possible non-stop trajectories in a network that are not allowed to cross at any point in time. For simplified settings, polynomial-time solution approaches are presented whereas already for restricted settings the problems are shown to be NP-hard. When moving to more complex and more realistic settings as they occur, for example, in determining non-stop disjoint trajectories for a set of aircraft, we present heuristic algorithms that adaptively refine coarse disjoint trajectories in the timing dimension. In order to be able to solve the non-stop disjoint trajectories problem over time, the method is integrated in a rolling-horizon algorithm. We present computational results for realistic settings. Motivated by the fact that rolling-horizon approaches are often applied in practice without knowledge on the quality of the obtained solutions, we study this problem from an abstract point of view. In fact, we more abstractly analyze the solution quality of general rolling-horizon algorithms for optimization tasks that show a timing component. We apply it to different planning problems. We end by pointing out some challenges and possibilities for future research.
Cite as
Lukas Glomb, Benno Hoch, Frauke Liers, and Florian Rösel. Network Planning and Routing Problems over Time: Models, Complexity and Algorithms (Invited Talk). In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 1:1-1:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{glomb_et_al:LIPIcs.ESA.2021.1,
author = {Glomb, Lukas and Hoch, Benno and Liers, Frauke and R\"{o}sel, Florian},
title = {{Network Planning and Routing Problems over Time: Models, Complexity and Algorithms}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {1:1--1:3},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.1},
URN = {urn:nbn:de:0030-drops-145822},
doi = {10.4230/LIPIcs.ESA.2021.1},
annote = {Keywords: network problems over time, rolling-horizon, complexity, approximation}
}
Document
Invited Talk
A User Friendly Power Tool for Deriving Online Learning Algorithms (Invited Talk)
Abstract
In this talk, we overview a simple and user friendly framework developed in [Noarov et al., 2021] that can be used to derive online learning algorithms in a number of settings. In the core framework, at every round, an adaptive adversary introduces a new game, consisting of an action space for the learner, an action space for the adversary, and a vector valued objective function that is concave-convex in every coordinate. The learner and the adversary then play in this game. The learner’s goal is to play so as to minimize the maximum coordinate of the cumulative vector-valued loss. The resulting one-shot game is not concave-convex, and so the minimax theorem does not apply. Nevertheless we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret.
We demonstrate the power of our simple framework by using it to derive optimal bounds and algorithms across a variety of domains. This includes no regret learning: we can recover optimal algorithms and bounds for minimizing exernal regret, internal regret, adaptive regret, multigroup regret, subsequence regret, and permutation regret in the sleeping experts setting. It also includes (multi)calibration [Hébert-Johnson et al., 2018] and related notions: we are able to recover recently derived algorithms and bounds for online adversarial multicalibration [Gupta et al., 2021], mean conditioned moment multicalibration [Jung et al., 2021], and prediction interval multivalidity [Gupta et al., 2021]. Finally we use it to derive a new variant of Blackwell’s Approachability Theorem, which we term "Fast Polytope Approachability".
Cite as
Aaron Roth. A User Friendly Power Tool for Deriving Online Learning Algorithms (Invited Talk). In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{roth:LIPIcs.ESA.2021.2,
author = {Roth, Aaron},
title = {{A User Friendly Power Tool for Deriving Online Learning Algorithms}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.2},
URN = {urn:nbn:de:0030-drops-145835},
doi = {10.4230/LIPIcs.ESA.2021.2},
annote = {Keywords: Online Learning, Multicalibration, Multivalidity, Blackwell Approachability}
}
Document
Bi-Objective Search with Bi-Directional A*
Abstract
Bi-objective search is a well-known algorithmic problem, concerned with finding a set of optimal solutions in a two-dimensional domain. This problem has a wide variety of applications such as planning in transport systems or optimal control in energy systems. Recently, bi-objective A*-based search (BOA*) has shown state-of-the-art performance in large networks. This paper develops a bi-directional and parallel variant of BOA*, enriched with several speed-up heuristics. Our experimental results on 1,000 benchmark cases show that our bi-directional A* algorithm for bi-objective search (BOBA*) can optimally solve all of the benchmark cases within the time limit, outperforming the state of the art BOA*, bi-objective Dijkstra and bi-directional bi-objective Dijkstra by an average runtime improvement of a factor of five over all of the benchmark instances.
Cite as
Saman Ahmadi, Guido Tack, Daniel Harabor, and Philip Kilby. Bi-Objective Search with Bi-Directional A*. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ahmadi_et_al:LIPIcs.ESA.2021.3,
author = {Ahmadi, Saman and Tack, Guido and Harabor, Daniel and Kilby, Philip},
title = {{Bi-Objective Search with Bi-Directional A*}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {3:1--3:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.3},
URN = {urn:nbn:de:0030-drops-145849},
doi = {10.4230/LIPIcs.ESA.2021.3},
annote = {Keywords: Bi-objective search, heuristic search, shortest path problem}
}
Document
A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs
Abstract
Let G = (V,E) be a weighted undirected graph with n vertices and m edges, and let d_G(u,v) be the length of the shortest path between u and v in G. In this paper we present a unified approach for obtaining algorithms for all pairs approximate shortest paths in weighted undirected graphs. For every integer k ≥ 2 we show that there is an Õ(n²+kn^{2-3/k}m^{2/k}) expected running time algorithm that computes a matrix M such that for every u,v ∈ V:
d_G(u,v) ≤ M[u,v] ≤ (2+(k-2)/k)d_G(u,v).
Previous algorithms obtained only specific approximation factors. Baswana and Kavitha [FOCS 2006, SICOMP 2010] presented a 2-approximation algorithm with expected running time of Õ(n²+m√ n) and a 7/3-approximation algorithm with expected running time of Õ(n²+m^{2/3}n). Their results improved upon the results of Cohen and Zwick [SODA 1997, JoA 2001] for graphs with m = o(n²). Kavitha [FSTTCS 2007, Algorithmica 2012] presented a 5/2-approximation algorithm with expected running time of Õ(n^{9/4}).
For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we get a 5/2-approximation algorithm with Õ(n^{5/4}m^{1/2}) expected running time. This improves upon the running time of Õ(n^{9/4}) due to Kavitha, when m = o(n²).
Our unified approach reveals that all previous algorithms are a part of a family of algorithms that exhibit a smooth tradeoff between approximation of 2 and 3, and are not sporadic unrelated results. Moreover, our new algorithm uses, among other ideas, the celebrated approximate distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] in a non standard way, which we believe is of independent interest, due to their extensive usage in a variety of applications.
Cite as
Maor Akav and Liam Roditty. A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{akav_et_al:LIPIcs.ESA.2021.4,
author = {Akav, Maor and Roditty, Liam},
title = {{A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {4:1--4:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.4},
URN = {urn:nbn:de:0030-drops-145858},
doi = {10.4230/LIPIcs.ESA.2021.4},
annote = {Keywords: Graph algorithms, Approximate All Pairs of Shortest Paths, Distance Oracles}
}
Document
The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination
Abstract
We introduce the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays, and the distance function is the counterclockwise angular distance between a point and a ray-site. This novel Voronoi diagram is motivated by illumination and coverage problems, where a domain has to be covered by floodlights (wedges) of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the ray Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle. This last algorithm improves upon previous results, settling an interesting open problem.
Cite as
Carlos Alegría, Ioannis Mantas, Evanthia Papadopoulou, Marko Savić, Hendrik Schrezenmaier, Carlos Seara, and Martin Suderland. The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{alegria_et_al:LIPIcs.ESA.2021.5,
author = {Alegr{\'\i}a, Carlos and Mantas, Ioannis and Papadopoulou, Evanthia and Savi\'{c}, Marko and Schrezenmaier, Hendrik and Seara, Carlos and Suderland, Martin},
title = {{The Voronoi Diagram of Rotating Rays With applications to Floodlight Illumination}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {5:1--5:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.5},
URN = {urn:nbn:de:0030-drops-145865},
doi = {10.4230/LIPIcs.ESA.2021.5},
annote = {Keywords: rotating rays, Voronoi diagram, oriented angular distance, Brocard angle, floodlight illumination, coverage problems, art gallery problems}
}
Document
Parallel Computation of Combinatorial Symmetries
Abstract
In practice symmetries of combinatorial structures are computed by transforming the structure into an annotated graph whose automorphisms correspond exactly to the desired symmetries. An automorphism solver is then employed to compute the automorphism group of the constructed graph. Such solvers have been developed for over 50 years, and highly efficient sequential, single core tools are available. However no competitive parallel tools are available for the task.
We introduce a new parallel randomized algorithm that is based on a modification of the individualization-refinement paradigm used by sequential solvers. The use of randomization crucially enables parallelization.
We report extensive benchmark results that show that our solver is competitive to state-of-the-art solvers on a single thread, while scaling remarkably well with the use of more threads. This results in order-of-magnitude improvements on many graph classes over state-of-the-art solvers. In fact, our tool is the first parallel graph automorphism tool that outperforms current sequential tools.
Cite as
Markus Anders and Pascal Schweitzer. Parallel Computation of Combinatorial Symmetries. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{anders_et_al:LIPIcs.ESA.2021.6,
author = {Anders, Markus and Schweitzer, Pascal},
title = {{Parallel Computation of Combinatorial Symmetries}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.6},
URN = {urn:nbn:de:0030-drops-145875},
doi = {10.4230/LIPIcs.ESA.2021.6},
annote = {Keywords: graph isomorphism, automorphism groups, algorithm engineering, parallel algorithms}
}
Document
Graph Connectivity and Single Element Recovery via Linear and OR Queries
Abstract
We study the problem of finding a spanning forest in an undirected, n-vertex multi-graph under two basic query models. One are Linear queries which are linear measurements on the incidence vector induced by the edges; the other are the weaker OR queries which only reveal whether a given subset of plausible edges is empty or not. At the heart of our study lies a fundamental problem which we call the single element recovery problem: given a non-negative vector x ∈ ℝ^{N}_{≥ 0}, the objective is to return a single element x_j > 0 from the support. Queries can be made in rounds, and our goals is to understand the trade-offs between the query complexity and the rounds of adaptivity needed to solve these problems, for both deterministic and randomized algorithms. These questions have connections and ramifications to multiple areas such as sketching, streaming, graph reconstruction, and compressed sensing. Our main results are as follows:
- For the single element recovery problem, it is easy to obtain a deterministic, r-round algorithm which makes (N^{1/r}-1)-queries per-round. We prove that this is tight: any r-round deterministic algorithm must make ≥ (N^{1/r} - 1) Linear queries in some round. In contrast, a 1-round O(polylog)-query randomized algorithm is known to exist.
- We design a deterministic O(r)-round, Õ(n^{1+1/r})-OR query algorithm for graph connectivity. We complement this with an Ω̃(n^{1 + 1/r})-lower bound for any r-round deterministic algorithm in the OR-model.
- We design a randomized, 2-round algorithm for the graph connectivity problem which makes Õ(n)-OR queries. In contrast, we prove that any 1-round algorithm (possibly randomized) requires Ω̃(n²)-OR queries. A randomized, 1-round algorithm making Õ(n)-Linear queries is already known. All our algorithms, in fact, work with more natural graph query models which are special cases of the above, and have been extensively studied in the literature. These are Cross queries (cut-queries) and BIS (bipartite independent set) queries.
Cite as
Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph Connectivity and Single Element Recovery via Linear and OR Queries. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{assadi_et_al:LIPIcs.ESA.2021.7,
author = {Assadi, Sepehr and Chakrabarty, Deeparnab and Khanna, Sanjeev},
title = {{Graph Connectivity and Single Element Recovery via Linear and OR Queries}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {7:1--7:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.7},
URN = {urn:nbn:de:0030-drops-145880},
doi = {10.4230/LIPIcs.ESA.2021.7},
annote = {Keywords: Query Models, Graph Connectivity, Group Testing, Duality}
}
Document
Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach
Abstract
In the (fully) dynamic set cover problem, we have a collection of m sets from a universe of size n that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an O(f²) update time algorithm for this problem that achieves an f-approximation, where f is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed f. This is the first algorithm for dynamic set cover with approximation ratio that exactly matches f (as opposed to almost f in prior work), as well as the first one with runtime independent of n,m (for any approximation factor of o(f³)).
Prior to our work, the state-of-the-art algorithms for this problem were O(f²) update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et al. [IPCO'17] with O(f³) approximation, and the recent algorithm of Bhattacharya {et al. } [FOCS'19] with O(f⋅log{n}/ε²) update time and (1+ε)⋅f approximation, improving the O(f²⋅log{n}/ε⁵) bound of Abboud et al. [STOC'19].
The key technical ingredient of our work is an algorithm for maintaining a maximal matching in a dynamic hypergraph of rank r - where each hyperedge has at most r vertices - that undergoes hyperedge insertions and deletions in O(r²) amortized update time; our algorithm is randomized, and the bound on the update time holds in expectation and with high probability. This result generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant update time in ordinary graphs to hypergraphs, and is of independent merit; the previous state-of-the-art algorithms for set cover do not translate to (integral) matchings for hypergraphs, let alone a maximal one. Our quantitative result for the set cover problem is translated directly from this qualitative result for maximal matching using standard reductions.
An important advantage of our approach over the previous ones for approximation (1+ε)⋅f (by Abboud et al. [STOC'19] and Bhattacharya et al. [FOCS'19]) is that it is inherently local and can thus be distributed efficiently to achieve low amortized round and message complexities.
Cite as
Sepehr Assadi and Shay Solomon. Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{assadi_et_al:LIPIcs.ESA.2021.8,
author = {Assadi, Sepehr and Solomon, Shay},
title = {{Fully Dynamic Set Cover via Hypergraph Maximal Matching: An Optimal Approximation Through a Local Approach}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {8:1--8:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.8},
URN = {urn:nbn:de:0030-drops-145899},
doi = {10.4230/LIPIcs.ESA.2021.8},
annote = {Keywords: dynamic graph algorithms, hypergraph, maximal matching, matching, set cover}
}
Document
The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential
Abstract
The weighted k-server problem is a natural generalization of the k-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood even on the simplest class of metric spaces - the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted k-server, is known. In this paper, we make significant progress towards understanding the randomized competitive ratio of weighted k-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in k, substantially improving on the previously known lower bound of about ln k.
Cite as
Nikhil Ayyadevara and Ashish Chiplunkar. The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ayyadevara_et_al:LIPIcs.ESA.2021.9,
author = {Ayyadevara, Nikhil and Chiplunkar, Ashish},
title = {{The Randomized Competitive Ratio of Weighted k-Server Is at Least Exponential}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {9:1--9:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.9},
URN = {urn:nbn:de:0030-drops-145904},
doi = {10.4230/LIPIcs.ESA.2021.9},
annote = {Keywords: weighted k-server, competitive analysis}
}
Document
Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty
Abstract
Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time f(α)-competitive algorithm, where f(α) ∈ [1.618+ε,2] depends on the approximation ratio α for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than 1.5-competitive. Furthermore, we give polynomial-time 4/3-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds.
Cite as
Evripidis Bampis, Christoph Dürr, Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bampis_et_al:LIPIcs.ESA.2021.10,
author = {Bampis, Evripidis and D\"{u}rr, Christoph and Erlebach, Thomas and de Lima, Murilo Santos and Megow, Nicole and Schl\"{o}ter, Jens},
title = {{Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {10:1--10:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.10},
URN = {urn:nbn:de:0030-drops-145910},
doi = {10.4230/LIPIcs.ESA.2021.10},
annote = {Keywords: Explorable uncertainty, queries, stochastic optimization, graph orientation, selection problems}
}
Document
k-Distinct Branchings Admits a Polynomial Kernel
Abstract
Unlike the problem of deciding whether a digraph D = (V,A) has 𝓁 in-branchings (or 𝓁 out-branchings) is polynomial time solvable, the problem of deciding whether a digraph D = (V,A) has an in-branching B^- and an out-branching B^+ which are arc-disjoint is NP-complete. Motivated by this, a natural optimization question that has been studied in the realm of Parameterized Complexity is called Rooted k-Distinct Branchings. In this problem, a digraph D = (V,A) with two prescribed vertices s,t are given as input and the question is whether D has an in-branching rooted at t and an out-branching rooted at s such that they differ on at least k arcs. Bang-Jensen et al. [Algorithmica, 2016 ] showed that the problem is fixed parameter tractable (FPT) on strongly connected digraphs. Gutin et al. [ICALP, 2017; JCSS, 2018 ] completely resolved this problem by designing an algorithm with running time 2^{𝒪(k² log² k)}n^{𝒪(1)}. Here, n denotes the number of vertices of the input digraph. In this paper, answering an open question of Gutin et al., we design a polynomial kernel for Rooted k-Distinct Branchings. In particular, we obtain the following: Given an instance (D,k,s,t) of Rooted k-Distinct Branchings, in polynomial time we obtain an equivalent instance (D',k',s,t) of Rooted k-Distinct Branchings such that |V(D')| ≤ 𝒪(k²) and the treewidth of the underlying undirected graph is at most 𝒪(k). This result immediately yields an FPT algorithm with running time 2^{𝒪(klog k)}+ n^{𝒪(1)}; improving upon the previous running time of Gutin et al. For our algorithms, we prove a structural result about paths avoiding many arcs in a given in-branching or out-branching. This result might turn out to be useful for getting other results for problems concerning in-and out-branchings.
Cite as
Jørgen Bang-Jensen, Kristine Vitting Klinkby, and Saket Saurabh. k-Distinct Branchings Admits a Polynomial Kernel. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bangjensen_et_al:LIPIcs.ESA.2021.11,
author = {Bang-Jensen, J{\o}rgen and Klinkby, Kristine Vitting and Saurabh, Saket},
title = {{k-Distinct Branchings Admits a Polynomial Kernel}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {11:1--11:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.11},
URN = {urn:nbn:de:0030-drops-145925},
doi = {10.4230/LIPIcs.ESA.2021.11},
annote = {Keywords: Digraphs, Polynomial Kernel, In-branching, Out-Branching}
}
Document
Incremental Edge Orientation in Forests
Abstract
For any forest G = (V, E) it is possible to orient the edges E so that no vertex in V has out-degree greater than 1. This paper considers the incremental edge-orientation problem, in which the edges E arrive over time and the algorithm must maintain a low-out-degree edge orientation at all times. We give an algorithm that maintains a maximum out-degree of 3 while flipping at most O(log log n) edge orientations per edge insertion, with high probability in n. The algorithm requires worst-case time O(log n log log n) per insertion, and takes amortized time O(1). The previous state of the art required up to O(log n / log log n) edge flips per insertion.
We then apply our edge-orientation results to the problem of dynamic Cuckoo hashing. The problem of designing simple families ℋ of hash functions that are compatible with Cuckoo hashing has received extensive attention. These families ℋ are known to satisfy static guarantees, but do not come typically with dynamic guarantees for the running time of inserts and deletes. We show how to transform static guarantees (for 1-associativity) into near-state-of-the-art dynamic guarantees (for O(1)-associativity) in a black-box fashion. Rather than relying on the family ℋ to supply randomness, as in past work, we instead rely on randomness within our table-maintenance algorithm.
Cite as
Michael A. Bender, Tsvi Kopelowitz, William Kuszmaul, Ely Porat, and Clifford Stein. Incremental Edge Orientation in Forests. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bender_et_al:LIPIcs.ESA.2021.12,
author = {Bender, Michael A. and Kopelowitz, Tsvi and Kuszmaul, William and Porat, Ely and Stein, Clifford},
title = {{Incremental Edge Orientation in Forests}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {12:1--12:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.12},
URN = {urn:nbn:de:0030-drops-145933},
doi = {10.4230/LIPIcs.ESA.2021.12},
annote = {Keywords: edge orientation, graph algorithms, Cuckoo hashing, hash functions}
}
Document
k-Center Clustering with Outliers in the Sliding-Window Model
Abstract
The k-center problem for a point set P asks for a collection of k congruent balls (that is, balls of equal radius) that together cover all the points in P and whose radius is minimized. The k-center problem with outliers is defined similarly, except that z of the points in P do need not to be covered, for a given parameter z. We study the k-center problem with outliers in data streams in the sliding-window model. In this model we are given a possibly infinite stream P = ⟨ p₁,p₂,p₃,…⟩ of points and a time window of length W, and we want to maintain a small sketch of the set P(t) of points currently in the window such that using the sketch we can approximately solve the problem on P(t).
We present the first algorithm for the k-center problem with outliers in the sliding-window model. The algorithm works for the case where the points come from a space of bounded doubling dimension and it maintains a set S(t) such that an optimal solution on S(t) gives a (1+ε)-approximate solution on P(t). The algorithm uses O((kz/ε^d)log σ) storage, where d is the doubling dimension of the underlying space and σ is the spread of the points in the stream. Algorithms providing a (1+ε)-approximation were not even known in the setting without outliers or in the insertion-only setting with outliers. We also present a lower bound showing that any algorithm that provides a (1+ε)-approximation must use Ω((kz/ε)log σ) storage.
Cite as
Mark de Berg, Morteza Monemizadeh, and Yu Zhong. k-Center Clustering with Outliers in the Sliding-Window Model. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{deberg_et_al:LIPIcs.ESA.2021.13,
author = {de Berg, Mark and Monemizadeh, Morteza and Zhong, Yu},
title = {{k-Center Clustering with Outliers in the Sliding-Window Model}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {13:1--13:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.13},
URN = {urn:nbn:de:0030-drops-145945},
doi = {10.4230/LIPIcs.ESA.2021.13},
annote = {Keywords: Streaming algorithms, k-center problem, sliding window, bounded doubling dimension}
}
Document
Incremental SCC Maintenance in Sparse Graphs
Abstract
In the incremental cycle detection problem, edges are added to a directed graph (initially empty), and the algorithm has to report the presence of the first cycle, once it is formed. A closely related problem is the incremental topological sort problem, where edges are added to an acyclic graph, and the algorithm is required to maintain a valid topological ordering. Since these problems arise naturally in many applications such as scheduling tasks, pointer analysis, and circuit evaluation, they have been studied extensively in the last three decades. Motivated by the fact that in many of these applications, the presence of a cycle is not fatal, we study a generalization of these problems, incremental maintenance of strongly connected components (incremental SCC).
Several incremental algorithms in the literature which do cycle detection and topological sort in directed acyclic graphs, such as those by [Michael A. Bender et al., 2016] and [Haeupler et al., 2012], also generalize to maintain strongly connected components and their topological sort in general directed graphs. The algorithms of [Haeupler et al., 2012] and [Michael A. Bender et al., 2016] have a total update time of O(m^{3/2}) and O(m⋅ min{m^{1/2},n^{2/3}}) respectively, and this is the state of the art for incremental SCC. But the most recent algorithms for incremental cycle detection and topological sort ([Bernstein and Chechik, 2018] and [Bhattacharya and Kulkarni, 2020]), which yield total (randomized) update time Õ(min{m^{4/3}, n²}), do not extend to incremental SCC. Thus, there is a gap between the best known algorithms for these two closely related problems.
In this paper, we bridge this gap by extending the framework of [Bhattacharya and Kulkarni, 2020] to general directed graphs. More concretely, we give a Las Vegas algorithm for incremental SCCs with an expected total update time of Õ(m^{4/3}). A key ingredient in the algorithm of [Bhattacharya and Kulkarni, 2020] is a structural theorem (first introduced in [Bernstein and Chechik, 2018]) that bounds the number of "equivalent" vertices. Unfortunately, this theorem only applies to DAGs. We show a natural way to extend this structural theorem to general directed graphs, and along the way we develop a significantly simpler and more intuitive proof of this theorem.
Cite as
Aaron Bernstein, Aditi Dudeja, and Seth Pettie. Incremental SCC Maintenance in Sparse Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bernstein_et_al:LIPIcs.ESA.2021.14,
author = {Bernstein, Aaron and Dudeja, Aditi and Pettie, Seth},
title = {{Incremental SCC Maintenance in Sparse Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {14:1--14:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.14},
URN = {urn:nbn:de:0030-drops-145950},
doi = {10.4230/LIPIcs.ESA.2021.14},
annote = {Keywords: Directed Graphs, Strongly Connected Components, Dynamic Graph Algorithms}
}
Document
Lyndon Words Accelerate Suffix Sorting
Abstract
Suffix sorting is arguably the most fundamental building block in string algorithmics, like regular sorting in the broader field of algorithms. It is thus not surprising that the literature is full of algorithms for suffix sorting, in particular focusing on their practicality. However, the advances on practical suffix sorting stalled with the emergence of the DivSufSort algorithm more than 10 years ago, which, up to date, has remained the fastest suffix sorter. This article shows how properties of Lyndon words can be exploited algorithmically to accelerate suffix sorting again. Our new algorithm is 6-19% faster than DivSufSort on real-world texts, and up to three times as fast on artificial repetitive texts. It can also be parallelized, where similar speedups can be observed. Thus, we make the first advances in practical suffix sorting after more than a decade of standstill.
Cite as
Nico Bertram, Jonas Ellert, and Johannes Fischer. Lyndon Words Accelerate Suffix Sorting. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bertram_et_al:LIPIcs.ESA.2021.15,
author = {Bertram, Nico and Ellert, Jonas and Fischer, Johannes},
title = {{Lyndon Words Accelerate Suffix Sorting}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {15:1--15:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.15},
URN = {urn:nbn:de:0030-drops-145961},
doi = {10.4230/LIPIcs.ESA.2021.15},
annote = {Keywords: Suffix array, suffix sorting, Lyndon words, string algorithms}
}
Document
Online Euclidean Spanners
Abstract
In this paper, we study the online Euclidean spanners problem for points in ℝ^d. Given a set S of n points in ℝ^d, a t-spanner on S is a subgraph of the underlying complete graph G = (S,binom(S,2)), that preserves the pairwise Euclidean distances between points in S to within a factor of t, that is the stretch factor. Suppose we are given a sequence of n points (s₁,s₂,…, s_n) in ℝ^d, where point s_i is presented in step i for i = 1,…, n. The objective of an online algorithm is to maintain a geometric t-spanner on S_i = {s₁,…, s_i} for each step i. The algorithm is allowed to add new edges to the spanner when a new point is presented, but cannot remove any edge from the spanner. The performance of an online algorithm is measured by its competitive ratio, which is the supremum, over all sequences of points, of the ratio between the weight of the spanner constructed by the algorithm and the weight of an optimum spanner. Here the weight of a spanner is the sum of all edge weights.
First, we establish a lower bound of Ω(ε^{-1}log n / log ε^{-1}) for the competitive ratio of any online (1+ε)-spanner algorithm, for a sequence of n points in 1-dimension. We show that this bound is tight, and there is an online algorithm that can maintain a (1+ε)-spanner with competitive ratio O(ε^{-1}log n / log ε^{-1}). Next, we design online algorithms for sequences of points in ℝ^d, for any constant d ≥ 2, under the L₂ norm. We show that previously known incremental algorithms achieve a competitive ratio O(ε^{-(d+1)}log n). However, if the algorithm is allowed to use additional points (Steiner points), then it is possible to substantially improve the competitive ratio in terms of ε. We describe an online Steiner (1+ε)-spanner algorithm with competitive ratio O(ε^{(1-d)/2} log n). As a counterpart, we show that the dependence on n cannot be eliminated in dimensions d ≥ 2. In particular, we prove that any online spanner algorithm for a sequence of n points in ℝ^d under the L₂ norm has competitive ratio Ω(f(n)), where lim_{n → ∞}f(n) = ∞. Finally, we provide improved lower bounds under the L₁ norm: Ω(ε^{-2}/log ε^{-1}) in the plane and Ω(ε^{-d}) in ℝ^d for d ≥ 3.
Cite as
Sujoy Bhore and Csaba D. Tóth. Online Euclidean Spanners. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bhore_et_al:LIPIcs.ESA.2021.16,
author = {Bhore, Sujoy and T\'{o}th, Csaba D.},
title = {{Online Euclidean Spanners}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {16:1--16:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.16},
URN = {urn:nbn:de:0030-drops-145974},
doi = {10.4230/LIPIcs.ESA.2021.16},
annote = {Keywords: Geometric spanner, (1+\epsilon)-spanner, minimum weight, online algorithm}
}
Document
Distant Representatives for Rectangles in the Plane
Abstract
The input to the distant representatives problem is a set of n objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the L₁, L₂, and L_∞ distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP).
Cite as
Therese Biedl, Anna Lubiw, Anurag Murty Naredla, Peter Dominik Ralbovsky, and Graeme Stroud. Distant Representatives for Rectangles in the Plane. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{biedl_et_al:LIPIcs.ESA.2021.17,
author = {Biedl, Therese and Lubiw, Anna and Naredla, Anurag Murty and Ralbovsky, Peter Dominik and Stroud, Graeme},
title = {{Distant Representatives for Rectangles in the Plane}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {17:1--17:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.17},
URN = {urn:nbn:de:0030-drops-145982},
doi = {10.4230/LIPIcs.ESA.2021.17},
annote = {Keywords: Distant representatives, blocker shapes, matching, approximation algorithm, APX-hardness}
}
Document
Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles
Abstract
Given a graph with a distinguished source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute and output, for any target vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids a failing edge e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a compact data structure that answers queries of the form (t,e) by returning the distance d(s,t,e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1,M] into a Single-Source DSO that has size O(M^{1/2} n^{3/2}) and query time Õ(1). We prove that the space requirement is optimal (up to the word size). Our techniques can also handle vertex failures within the same bounds.
Chechik and Cohen [SODA 2019] presented a combinatorial, randomized Õ(m√n+n²) time SSRP algorithm for undirected and unweighted graphs. We derandomize their algorithm with the same asymptotic running time and apply our compression to obtain a deterministic Single-Source DSO with Õ(m√n+n²) preprocessing time, O(n^{3/2}) space, and Õ(1) query time. Our combinatorial Single-Source DSO has near-optimal space, preprocessing and query time for unweighted graphs, improving the preprocessing time by a √n-factor compared to previous results with o(n²) space.
Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized Õ(Mn^ω) time SSRP algorithm for (undirected and directed) graphs with integer edge weights in the range [1,M], where ω < 2.373 is the matrix multiplication exponent. We derandomize it for undirected graphs and apply our compression to obtain an algebraic Single-Source DSO with Õ(Mn^ω) preprocessing time, O(M^{1/2} n^{3/2}) space, and Õ(1) query time. This improves the preprocessing time of algebraic Single-Source DSOs by polynomial factors compared to previous o(n²)-space oracles.
We also present further improvements of our Single-Source DSOs. We show that the query time can be reduced to a constant at the cost of increasing the size of the oracle to O(M^{1/3} n^{5/3}) and that all our oracles can be made path-reporting. On sparse graphs with m = O(n^{5/4-ε}/M^{7/4}) edges, for any constant ε > 0, we reduce the preprocessing to randomized Õ(M^{7/8} m^{1/2} n^{11/8}) = O(n^{2-ε/2}) time. To the best of our knowledge, this is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.
Cite as
Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{bilo_et_al:LIPIcs.ESA.2021.18,
author = {Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin},
title = {{Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {18:1--18:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.18},
URN = {urn:nbn:de:0030-drops-145999},
doi = {10.4230/LIPIcs.ESA.2021.18},
annote = {Keywords: derandomization, distance sensitivity oracle, single-source replacement paths, space lower bound}
}
Document
Synchronized Planarity with Applications to Constrained Planarity Problems
Abstract
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(n⁸).
Cite as
Thomas Bläsius, Simon D. Fink, and Ignaz Rutter. Synchronized Planarity with Applications to Constrained Planarity Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.19,
author = {Bl\"{a}sius, Thomas and Fink, Simon D. and Rutter, Ignaz},
title = {{Synchronized Planarity with Applications to Constrained Planarity Problems}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {19:1--19:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.19},
URN = {urn:nbn:de:0030-drops-146009},
doi = {10.4230/LIPIcs.ESA.2021.19},
annote = {Keywords: Planarity Testing, Constrained Planarity, Cluster Planarity, Atomic Embeddability}
}
Document
Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry
Abstract
Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved running time. For the vertex cover problem, there is a gap between theory and practice when it comes to understanding this tradeoff. On the one hand, it is known that it is NP-hard to approximate a minimum vertex cover within a factor of √2. On the other hand, a simple greedy algorithm yields close to optimal approximations in practice.
A promising approach towards understanding this discrepancy is to recognize the differences between theoretical worst-case instances and real-world networks. Following this direction, we close the gap between theory and practice by providing an algorithm that efficiently computes nearly optimal vertex cover approximations on hyperbolic random graphs; a network model that closely resembles real-world networks in terms of degree distribution, clustering, and the small-world property. More precisely, our algorithm computes a (1 + o(1))-approximation, asymptotically almost surely, and has a running time of 𝒪(m log(n)).
The proposed algorithm is an adaption of the successful greedy approach, enhanced with a procedure that improves on parts of the graph where greedy is not optimal. This makes it possible to introduce a parameter that can be used to tune the tradeoff between approximation performance and running time. Our empirical evaluation on real-world networks shows that this allows for improving over the near-optimal results of the greedy approach.
Cite as
Thomas Bläsius, Tobias Friedrich, and Maximilian Katzmann. Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.20,
author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Katzmann, Maximilian},
title = {{Efficiently Approximating Vertex Cover on Scale-Free Networks with Underlying Hyperbolic Geometry}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {20:1--20:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.20},
URN = {urn:nbn:de:0030-drops-146012},
doi = {10.4230/LIPIcs.ESA.2021.20},
annote = {Keywords: vertex cover, approximation, random graphs, hyperbolic geometry, efficient algorithm}
}
Document
Efficiently Computing Maximum Flows in Scale-Free Networks
Abstract
We study the maximum-flow/minimum-cut problem on scale-free networks, i.e., graphs whose degree distribution follows a power-law. We propose a simple algorithm that capitalizes on the fact that often only a small fraction of such a network is relevant for the flow. At its core, our algorithm augments Dinitz’s algorithm with a balanced bidirectional search. Our experiments on a scale-free random network model indicate sublinear run time. On scale-free real-world networks, we outperform the commonly used highest-label Push-Relabel implementation by up to two orders of magnitude. Compared to Dinitz’s original algorithm, our modifications reduce the search space, e.g., by a factor of 275 on an autonomous systems graph.
Beyond these good run times, our algorithm has an additional advantage compared to Push-Relabel. The latter computes a preflow, which makes the extraction of a minimum cut potentially more difficult. This is relevant, for example, for the computation of Gomory-Hu trees. On a social network with 70000 nodes, our algorithm computes the Gomory-Hu tree in 3 seconds compared to 12 minutes when using Push-Relabel.
Cite as
Thomas Bläsius, Tobias Friedrich, and Christopher Weyand. Efficiently Computing Maximum Flows in Scale-Free Networks. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{blasius_et_al:LIPIcs.ESA.2021.21,
author = {Bl\"{a}sius, Thomas and Friedrich, Tobias and Weyand, Christopher},
title = {{Efficiently Computing Maximum Flows in Scale-Free Networks}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {21:1--21:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.21},
URN = {urn:nbn:de:0030-drops-146029},
doi = {10.4230/LIPIcs.ESA.2021.21},
annote = {Keywords: graphs, flow, network, scale-free}
}
Document
Asymptotically Optimal Welfare of Posted Pricing for Multiple Items with MHR Distributions
Abstract
We consider the problem of posting prices for unit-demand buyers if all n buyers have identically distributed valuations drawn from a distribution with monotone hazard rate. We show that even with multiple items asymptotically optimal welfare can be guaranteed.
Our main results apply to the case that either a buyer’s value for different items are independent or that they are perfectly correlated. We give mechanisms using dynamic prices that obtain a 1 - Θ (1/(log n))-fraction of the optimal social welfare in expectation. Furthermore, we devise mechanisms that only use static item prices and are 1 - Θ ((log log log n)/(log n))-competitive compared to the optimal social welfare. As we show, both guarantees are asymptotically optimal, even for a single item and exponential distributions.
Cite as
Alexander Braun, Matthias Buttkus, and Thomas Kesselheim. Asymptotically Optimal Welfare of Posted Pricing for Multiple Items with MHR Distributions. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{braun_et_al:LIPIcs.ESA.2021.22,
author = {Braun, Alexander and Buttkus, Matthias and Kesselheim, Thomas},
title = {{Asymptotically Optimal Welfare of Posted Pricing for Multiple Items with MHR Distributions}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {22:1--22:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.22},
URN = {urn:nbn:de:0030-drops-146038},
doi = {10.4230/LIPIcs.ESA.2021.22},
annote = {Keywords: Prophet Inequalities, Monotone Hazard Rate, Competitive Analysis, Posted Prices, Combinatorial Auctions, Matching}
}
Document
Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete
Abstract
Staged self-assembly has proven to be a powerful abstract model of self-assembly by modeling laboratory techniques where several nanoscale systems are allowed to assemble separately and then be mixed at a later stage. A fundamental problem in self-assembly is Unique Assembly Verification (UAV), which asks whether a single final assembly is uniquely constructed. This has previously been shown to be Π^{p}₂-hard in staged self-assembly with a constant number of stages, but a more precise complexity classification was left open related to the polynomial hierarchy.
Covert Computation was recently introduced as a way to compute a function while hiding the input to that function for self-assembly systems. These Tile Assembly Computers (TACs), in a growth only negative aTAM system, can compute arbitrary circuits, which proves UAV is coNP-hard in that model. Here, we show that the staged assembly model is capable of covert computation using only 3 stages. We then utilize this construction to show UAV with only 3 stages is Π^{p}₂-hard. We then extend this technique to open problems and prove that general staged UAV is PSPACE-complete. Measuring the complexity of n stage UAV, we show Π^{p}_{n - 1}-hardness. We finish by showing a Π^{p}_{n + 1} algorithm to solve n stage UAV leaving only a constant gap between membership and hardness.
Cite as
David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{caballero_et_al:LIPIcs.ESA.2021.23,
author = {Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim},
title = {{Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {23:1--23:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.23},
URN = {urn:nbn:de:0030-drops-146047},
doi = {10.4230/LIPIcs.ESA.2021.23},
annote = {Keywords: self-assembly, covert computation, staged self-assembly, assembly verification}
}
Document
An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility
Abstract
Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the order-oblivious setting. An algorithm A is instance-optimal in the order-oblivious setting for a certain class of algorithms 𝒜 if the following hold:
- A takes as input a sequence of objects from some domain;
- for any instance σ and any algorithm A' ∈ 𝒜, the runtime of A on σ is at most a constant factor removed from the runtime of A' on the worst possible permutation of σ. If we identify permutations of a sequence as representing the same instance, this essentially states that A is optimal on every possible input (and not only in the worst case).
We design instance-optimal algorithms for the problem of reporting, given a bichromatic set of points in the plane S, all pairs consisting of points of different color which span an empty axis-aligned rectangle (or reporting all points which appear in such a pair). This problem has applications for training-set reduction in nearest-neighbour classifiers. It is also related to the problem consisting of finding the decision boundaries of a euclidean nearest-neighbour classifier, for which Bremner et al. (2005) gave an optimal output-sensitive algorithm.
By showing the existence of an instance-optimal algorithm in the order-oblivious setting for this problem we push the methods of Afshani et al. closer to their limits by adapting and extending them to a setting which exhibits highly non-local features. Previous problems for which instance-optimal algorithms were proven to exist were based solely on local relationships between points in a set.
Cite as
Jean Cardinal, Justin Dallant, and John Iacono. An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cardinal_et_al:LIPIcs.ESA.2021.24,
author = {Cardinal, Jean and Dallant, Justin and Iacono, John},
title = {{An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {24:1--24:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.24},
URN = {urn:nbn:de:0030-drops-146057},
doi = {10.4230/LIPIcs.ESA.2021.24},
annote = {Keywords: computational geometry, instance-optimality, colored point sets, empty rectangles, visibility}
}
Document
Worst-Case Efficient Dynamic Geometric Independent Set
Abstract
We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present a data structure that maintains a constant-factor approximate maximum independent set for broad classes of fat objects in d dimensions, where d is assumed to be a constant, in sublinear worst-case update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic (4+ε)-approximation for squares, with O(log⁴ n) worst-case update time.
Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with amortized update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem.
Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.
Cite as
Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-Case Efficient Dynamic Geometric Independent Set. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cardinal_et_al:LIPIcs.ESA.2021.25,
author = {Cardinal, Jean and Iacono, John and Koumoutsos, Grigorios},
title = {{Worst-Case Efficient Dynamic Geometric Independent Set}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {25:1--25:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.25},
URN = {urn:nbn:de:0030-drops-146061},
doi = {10.4230/LIPIcs.ESA.2021.25},
annote = {Keywords: Maximum independent set, deamortization, approximation}
}
Document
Balanced Crown Decomposition for Connectivity Constraints
Abstract
We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems.
In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component.
Cite as
Katrin Casel, Tobias Friedrich, Davis Issac, Aikaterini Niklanovits, and Ziena Zeif. Balanced Crown Decomposition for Connectivity Constraints. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{casel_et_al:LIPIcs.ESA.2021.26,
author = {Casel, Katrin and Friedrich, Tobias and Issac, Davis and Niklanovits, Aikaterini and Zeif, Ziena},
title = {{Balanced Crown Decomposition for Connectivity Constraints}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {26:1--26:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.26},
URN = {urn:nbn:de:0030-drops-146076},
doi = {10.4230/LIPIcs.ESA.2021.26},
annote = {Keywords: crown decomposition, connected partition, balanced partition, approximation algorithms}
}
Document
All-Pairs Shortest Paths for Real-Weighted Undirected Graphs with Small Additive Error
Abstract
Given a graph with n vertices and real edge weights in [0,1], we investigate an approximate version of the standard all-pairs shortest paths (APSP) problem where distances are estimated with additive error at most ε. Yuster (2012) introduced this natural variant of approximate APSP, and presented an algorithm for directed graphs running in Õ(n^{(3+ω)/2}) ≤ O(n^{2.687}) time for an arbitrarily small constant ε > 0, where ω denotes the matrix multiplication exponent. We give a faster algorithm for undirected graphs running in Õ(n^{(3+ω²)/(ω+1)}) ≤ O(n^{2.559}) time for any constant ε > 0. If ω = 2, the time bound is Õ(n^{7/3}), matching a previous result for undirected graphs by Dor, Halperin, and Zwick (2000) which only guaranteed additive error at most 2.
Cite as
Timothy M. Chan. All-Pairs Shortest Paths for Real-Weighted Undirected Graphs with Small Additive Error. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 27:1-27:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chan:LIPIcs.ESA.2021.27,
author = {Chan, Timothy M.},
title = {{All-Pairs Shortest Paths for Real-Weighted Undirected Graphs with Small Additive Error}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {27:1--27:9},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.27},
URN = {urn:nbn:de:0030-drops-146086},
doi = {10.4230/LIPIcs.ESA.2021.27},
annote = {Keywords: Shortest paths, approximation, matrix multiplication}
}
Document
Dynamic Colored Orthogonal Range Searching
Abstract
In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)} n + klog^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + klog^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.
Cite as
Timothy M. Chan and Zhengcheng Huang. Dynamic Colored Orthogonal Range Searching. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chan_et_al:LIPIcs.ESA.2021.28,
author = {Chan, Timothy M. and Huang, Zhengcheng},
title = {{Dynamic Colored Orthogonal Range Searching}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {28:1--28:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.28},
URN = {urn:nbn:de:0030-drops-146090},
doi = {10.4230/LIPIcs.ESA.2021.28},
annote = {Keywords: Range searching, dynamic data structures, word RAM}
}
Document
𝓁_p-Norm Multiway Cut
Abstract
We introduce and study 𝓁_p-norm-multiway-cut: the input here is an undirected graph with non-negative edge weights along with k terminals and the goal is to find a partition of the vertex set into k parts each containing exactly one terminal so as to minimize the 𝓁_p-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when p = 1) and min-max multiway cut (when p = ∞), both of which are well-studied classic problems in the graph partitioning literature. We show that 𝓁_p-norm-multiway-cut is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an O(log² n)-approximation for all p ≥ 1. We also show an integrality gap of Ω(k^{1-1/p}) for a natural convex program and an O(k^{1-1/p-ε})-inapproximability for any constant ε > 0 assuming the small set expansion hypothesis.
Cite as
Karthekeyan Chandrasekaran and Weihang Wang. 𝓁_p-Norm Multiway Cut. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chandrasekaran_et_al:LIPIcs.ESA.2021.29,
author = {Chandrasekaran, Karthekeyan and Wang, Weihang},
title = {{𝓁\underlinep-Norm Multiway Cut}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {29:1--29:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.29},
URN = {urn:nbn:de:0030-drops-146103},
doi = {10.4230/LIPIcs.ESA.2021.29},
annote = {Keywords: multiway cut, approximation algorithms}
}
Document
Faster Algorithms for Longest Common Substring
Abstract
In the classic longest common substring (LCS) problem, we are given two strings S and T, each of length at most n, over an alphabet of size σ, and we are asked to find a longest string occurring as a fragment of both S and T. Weiner, in his seminal paper that introduced the suffix tree, presented an 𝒪(n log σ)-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an 𝒪(n)-time algorithm for the LCS problem [FOCS 1997]. However, for small alphabets, this is not necessarily optimal for the LCS problem in the word RAM model of computation, in which the strings can be stored in 𝒪(n log σ/log n) space and read in 𝒪(n log σ/log n) time. We show that, in this model, we can compute an LCS in time 𝒪(n log σ / √{log n}), which is sublinear in n if σ = 2^{o(√{log n})} (in particular, if σ = 𝒪(1)), using optimal space 𝒪(n log σ/log n).
We then lift our ideas to the problem of computing a k-mismatch LCS, which has received considerable attention in recent years. In this problem, the aim is to compute a longest substring of S that occurs in T with at most k mismatches. Flouri et al. showed how to compute a 1-mismatch LCS in 𝒪(n log n) time [IPL 2015]. Thankachan et al. extended this result to computing a k-mismatch LCS in 𝒪(n log^k n) time for k = 𝒪(1) [J. Comput. Biol. 2016]. We show an 𝒪(n log^{k-1/2} n)-time algorithm, for any constant integer k > 0 and irrespective of the alphabet size, using 𝒪(n) space as the previous approaches. We thus notably break through the well-known n log^k n barrier, which stems from a recursive heavy-path decomposition technique that was first introduced in the seminal paper of Cole et al. [STOC 2004] for string indexing with k errors.
Cite as
Panagiotis Charalampopoulos, Tomasz Kociumaka, Solon P. Pissis, and Jakub Radoszewski. Faster Algorithms for Longest Common Substring. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 30:1-30:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{charalampopoulos_et_al:LIPIcs.ESA.2021.30,
author = {Charalampopoulos, Panagiotis and Kociumaka, Tomasz and Pissis, Solon P. and Radoszewski, Jakub},
title = {{Faster Algorithms for Longest Common Substring}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {30:1--30:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.30},
URN = {urn:nbn:de:0030-drops-146114},
doi = {10.4230/LIPIcs.ESA.2021.30},
annote = {Keywords: longest common substring, k mismatches, wavelet tree}
}
Document
Feature Cross Search via Submodular Optimization
Abstract
In this paper, we study feature cross search as a fundamental primitive in feature engineering. The importance of feature cross search especially for the linear model has been known for a while, with well-known textbook examples. In this problem, the goal is to select a small subset of features, combine them to form a new feature (called the crossed feature) by considering their Cartesian product, and find feature crosses to learn an accurate model. In particular, we study the problem of maximizing a normalized Area Under the Curve (AUC) of the linear model trained on the crossed feature column.
First, we show that it is not possible to provide an n^{1/log log n}-approximation algorithm for this problem unless the exponential time hypothesis fails. This result also rules out the possibility of solving this problem in polynomial time unless 𝖯 = NP. On the positive side, by assuming the naïve Bayes assumption, we show that there exists a simple greedy (1-1/e)-approximation algorithm for this problem. This result is established by relating the AUC to the total variation of the commutator of two probability measures and showing that the total variation of the commutator is monotone and submodular. To show this, we relate the submodularity of this function to the positive semi-definiteness of a corresponding kernel matrix. Then, we use Bochner’s theorem to prove the positive semi-definiteness by showing that its inverse Fourier transform is non-negative everywhere. Our techniques and structural results might be of independent interest.
Cite as
Lin Chen, Hossein Esfandiari, Gang Fu, Vahab S. Mirrokni, and Qian Yu. Feature Cross Search via Submodular Optimization. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{chen_et_al:LIPIcs.ESA.2021.31,
author = {Chen, Lin and Esfandiari, Hossein and Fu, Gang and Mirrokni, Vahab S. and Yu, Qian},
title = {{Feature Cross Search via Submodular Optimization}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {31:1--31:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.31},
URN = {urn:nbn:de:0030-drops-146124},
doi = {10.4230/LIPIcs.ESA.2021.31},
annote = {Keywords: Feature engineering, feature cross, submodularity}
}
Document
An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes
Abstract
We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is homeomorphic to a surface.
It is known that the problem admits an algorithm with running time f(c)n^{O(c)}, where n is the size of the graph G and c is the size of the two-dimensional complex C. In other words, that algorithm is polynomial when C is fixed, but the degree of the polynomial depends on C. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing a deterministic f(c)n³-time algorithm. We also provide a randomized algorithm with expected running time 2^{c^{O(1)}}n^{O(1)}.
Our approach is to reduce to the case where G has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface; the only elaborated tool that we use is an algorithm to compute grid minors.
Cite as
Éric Colin de Verdière and Thomas Magnard. An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{colindeverdiere_et_al:LIPIcs.ESA.2021.32,
author = {Colin de Verdi\`{e}re, \'{E}ric and Magnard, Thomas},
title = {{An FPT Algorithm for the Embeddability of Graphs into Two-Dimensional Simplicial Complexes}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {32:1--32:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.32},
URN = {urn:nbn:de:0030-drops-146139},
doi = {10.4230/LIPIcs.ESA.2021.32},
annote = {Keywords: computational topology, embedding, simplicial complex, graph, surface, fixed-parameter tractability}
}
Document
Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity
Abstract
We consider the problem of solving integer programs of the form min {c^⊺ x : Ax = b, x ∈ ℤ_{⩾ 0}}, where A is a multistage stochastic matrix in the following sense: the primal treedepth of A is bounded by a parameter d, which means that the columns of A can be organized into a rooted forest of depth at most d so that columns not bound by the ancestor/descendant relation do not have non-zero entries in the same row. We give an algorithm that solves this problem in fixed-parameter time f(d,‖A‖_{∞})⋅ nlog^{𝒪(2^d)} n, where f is a computable function and n is the number of rows of A. The algorithm works in the strong model, where the running time only measures unit arithmetic operations on the input numbers and does not depend on their bitlength. This is the first fpt algorithm for multistage stochastic integer programming to achieve almost linear running time in the strong sense. For two-stage stochastic integer programs, our algorithm works in time 2^{((r+s)‖A‖_∞)^{𝒪(r(r+s))}}⋅ nlog^{𝒪(rs)} n, which improves over previous methods both in terms of the polynomial factor and in terms of the dependence on r and s. In fact, for r = 1 the dependence on s is asymptotically almost tight assuming the Exponential Time Hypothesis. Our algorithm can be also parallelized: we give an implementation in the PRAM model that achieves running time f(d,‖A‖_{∞})⋅ log^{𝒪(2^d)} n using n processors.
The main conceptual ingredient in our algorithms is a new proximity result for multistage stochastic integer programs. We prove that if we consider an integer program P, say with a constraint matrix A, then for every optimum solution to the linear relaxation of P there exists an optimum (integral) solution to P that lies, in the 𝓁_{∞}-norm, within distance bounded by a function of ‖A‖_{∞} and the primal treedepth of A. On the way to achieve this result, we prove a generalization and considerable improvement of a structural result of Klein for multistage stochastic integer programs. Once the proximity results are established, this allows us to apply a treedepth-based branching strategy guided by an optimum solution to the linear relaxation.
Cite as
Jana Cslovjecsek, Friedrich Eisenbrand, Michał Pilipczuk, Moritz Venzin, and Robert Weismantel. Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cslovjecsek_et_al:LIPIcs.ESA.2021.33,
author = {Cslovjecsek, Jana and Eisenbrand, Friedrich and Pilipczuk, Micha{\l} and Venzin, Moritz and Weismantel, Robert},
title = {{Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {33:1--33:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.33},
URN = {urn:nbn:de:0030-drops-146146},
doi = {10.4230/LIPIcs.ESA.2021.33},
annote = {Keywords: parameterized algorithm, multistage stochastic programming, proximity}
}
Document
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
Abstract
We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of k-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an n^{f(t,s)}-time algorithm to compute modulo 2^t the number of subgraph occurrences of patterns that are s vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo 2^t. Complementing our algorithm, we also give a simple and self-contained proof that counting k-matchings modulo odd integers q is {Mod}_q W[1]-complete and prove that counting k-paths modulo 2 is ⊕W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).
Cite as
Radu Curticapean, Holger Dell, and Thore Husfeldt. Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{curticapean_et_al:LIPIcs.ESA.2021.34,
author = {Curticapean, Radu and Dell, Holger and Husfeldt, Thore},
title = {{Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {34:1--34:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.34},
URN = {urn:nbn:de:0030-drops-146154},
doi = {10.4230/LIPIcs.ESA.2021.34},
annote = {Keywords: Counting complexity, matchings, paths, subgraphs, parameterized complexity}
}
Document
Minimum Common String Partition: Exact Algorithms
Abstract
In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2ⁿ upper bound (where n is the length of input strings) to ϕⁿ where ϕ ≈ 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis.
Cite as
Marek Cygan, Alexander S. Kulikov, Ivan Mihajlin, Maksim Nikolaev, and Grigory Reznikov. Minimum Common String Partition: Exact Algorithms. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 35:1-35:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{cygan_et_al:LIPIcs.ESA.2021.35,
author = {Cygan, Marek and Kulikov, Alexander S. and Mihajlin, Ivan and Nikolaev, Maksim and Reznikov, Grigory},
title = {{Minimum Common String Partition: Exact Algorithms}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {35:1--35:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.35},
URN = {urn:nbn:de:0030-drops-146167},
doi = {10.4230/LIPIcs.ESA.2021.35},
annote = {Keywords: similarity measure, string distance, exact algorithms, upper bounds, lower bounds}
}
Document
An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems
Abstract
We present an accelerated, or "look-ahead" version of the Newton-Dinkelbach method, a well-known technique for solving fractional and parametric optimization problems. This acceleration halves the Bregman divergence between the current iterate and the optimal solution within every two iterations. Using the Bregman divergence as a potential in conjunction with combinatorial arguments, we obtain strongly polynomial algorithms in three applications domains: (i) For linear fractional combinatorial optimization, we show a convergence bound of O(mlog m) iterations; the previous best bound was O(m²log m) by Wang et al. (2006). (ii) We obtain a strongly polynomial label-correcting algorithm for solving linear feasibility systems with two variables per inequality (2VPI). For a 2VPI system with n variables and m constraints, our algorithm runs in O(mn) iterations. Every iteration takes O(mn) time for general 2VPI systems, and O(m + nlog n) time for the special case of deterministic Markov Decision Processes (DMDPs). This extends and strengthens a previous result by Madani (2002) that showed a weakly polynomial bound for a variant of the Newton–Dinkelbach method for solving DMDPs. (iii) We give a simplified variant of the parametric submodular function minimization result by Goemans et al. (2017).
Cite as
Daniel Dadush, Zhuan Khye Koh, Bento Natura, and László A. Végh. An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dadush_et_al:LIPIcs.ESA.2021.36,
author = {Dadush, Daniel and Koh, Zhuan Khye and Natura, Bento and V\'{e}gh, L\'{a}szl\'{o} A.},
title = {{An Accelerated Newton-Dinkelbach Method and Its Application to Two Variables per Inequality Systems}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {36:1--36:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.36},
URN = {urn:nbn:de:0030-drops-146172},
doi = {10.4230/LIPIcs.ESA.2021.36},
annote = {Keywords: Newton-Dinkelbach method, fractional optimization, parametric optimization, strongly polynomial algorithms, two variables per inequality systems, Markov decision processes, submodular function minimization}
}
Document
Faster 3-Coloring of Small-Diameter Graphs
Abstract
We study the 3-Coloring problem in graphs with small diameter. In 2013, Mertzios and Spirakis showed that for n-vertex diameter-2 graphs this problem can be solved in subexponential time 2^{𝒪(√{n log n})}. Whether the problem can be solved in polynomial time remains a well-known open question in the area of algorithmic graphs theory.
In this paper we present an algorithm that solves 3-Coloring in n-vertex diameter-2 graphs in time 2^{𝒪(n^{1/3} log² n)}. This is the first improvement upon the algorithm of Mertzios and Spirakis in the general case, i.e., without putting any further restrictions on the instance graph.
In addition to standard branchings and reducing the problem to an instance of 2-Sat, the crucial building block of our algorithm is a combinatorial observation about 3-colorable diameter-2 graphs, which is proven using a probabilistic argument.
As a side result, we show that 3-Coloring can be solved in time 2^{𝒪((n log n)^{2/3})} in n-vertex diameter-3 graphs. We also generalize our algorithms to the problem of finding a list homomorphism from a small-diameter graph to a cycle.
Cite as
Michał Dębski, Marta Piecyk, and Paweł Rzążewski. Faster 3-Coloring of Small-Diameter Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{debski_et_al:LIPIcs.ESA.2021.37,
author = {D\k{e}bski, Micha{\l} and Piecyk, Marta and Rz\k{a}\.{z}ewski, Pawe{\l}},
title = {{Faster 3-Coloring of Small-Diameter Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {37:1--37:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.37},
URN = {urn:nbn:de:0030-drops-146181},
doi = {10.4230/LIPIcs.ESA.2021.37},
annote = {Keywords: 3-coloring, fine-grained complexity, subexponential-time algorithm, diameter}
}
Document
Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning
Abstract
In this paper, we study several important geometric optimization problems arising in machine learning. First, we revisit the Minimum Enclosing Ball (MEB) problem in Euclidean space ℝ^d. The problem has been extensively studied before, but real-world machine learning tasks often need to handle large-scale datasets so that we cannot even afford linear time algorithms. Motivated by the recent developments on beyond worst-case analysis, we introduce the notion of stability for MEB, which is natural and easy to understand. Roughly speaking, an instance of MEB is stable, if the radius of the resulting ball cannot be significantly reduced by removing a small fraction of the input points. Under the stability assumption, we present two sampling algorithms for computing radius-approximate MEB with sample complexities independent of the number of input points n. In particular, the second algorithm has the sample complexity even independent of the dimensionality d. We also consider the general case without the stability assumption. We present a hybrid algorithm that can output either a radius-approximate MEB or a covering-approximate MEB, which improves the running time and the number of passes for the previous sublinear MEB algorithms. Further, we extend our proposed notion of stability and design sublinear time algorithms for other geometric optimization problems including MEB with outliers, polytope distance, one-class and two-class linear SVMs (without or with outliers). Our proposed algorithms also work fine for kernels.
Cite as
Hu Ding. Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ding:LIPIcs.ESA.2021.38,
author = {Ding, Hu},
title = {{Stability Yields Sublinear Time Algorithms for Geometric Optimization in Machine Learning}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {38:1--38:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.38},
URN = {urn:nbn:de:0030-drops-146194},
doi = {10.4230/LIPIcs.ESA.2021.38},
annote = {Keywords: stability, sublinear time, geometric optimization, machine learning}
}
Document
Data Structures Lower Bounds and Popular Conjectures
Abstract
In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.
Cite as
Pavel Dvořák, Michal Koucký, Karel Král, and Veronika Slívová. Data Structures Lower Bounds and Popular Conjectures. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.39,
author = {Dvo\v{r}\'{a}k, Pavel and Kouck\'{y}, Michal and Kr\'{a}l, Karel and Sl{\'\i}vov\'{a}, Veronika},
title = {{Data Structures Lower Bounds and Popular Conjectures}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {39:1--39:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.39},
URN = {urn:nbn:de:0030-drops-146207},
doi = {10.4230/LIPIcs.ESA.2021.39},
annote = {Keywords: Data structures, Circuits, Lower bounds, Network Coding Conjecture}
}
Document
Approximation Schemes for Bounded Distance Problems on Fractionally Treewidth-Fragile Graphs
Abstract
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable on graphs of bounded treewidth. These schemes apply in all fractionally treewidth-fragile graph classes, a property which is true for many natural graph classes with sublinear separators. We also provide quasipolynomial-time approximation schemes for these problems in all classes with sublinear separators.
Cite as
Zdeněk Dvořák and Abhiruk Lahiri. Approximation Schemes for Bounded Distance Problems on Fractionally Treewidth-Fragile Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 40:1-40:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{dvorak_et_al:LIPIcs.ESA.2021.40,
author = {Dvo\v{r}\'{a}k, Zden\v{e}k and Lahiri, Abhiruk},
title = {{Approximation Schemes for Bounded Distance Problems on Fractionally Treewidth-Fragile Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {40:1--40:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.40},
URN = {urn:nbn:de:0030-drops-146216},
doi = {10.4230/LIPIcs.ESA.2021.40},
annote = {Keywords: approximation, sublinear separators}
}
Document
Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping
Abstract
A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of 2. Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a (0.385-ε)-approximation, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest.
Cite as
Yaron Fairstein, Ariel Kulik, and Hadas Shachnai. Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{fairstein_et_al:LIPIcs.ESA.2021.41,
author = {Fairstein, Yaron and Kulik, Ariel and Shachnai, Hadas},
title = {{Modular and Submodular Optimization with Multiple Knapsack Constraints via Fractional Grouping}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {41:1--41:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.41},
URN = {urn:nbn:de:0030-drops-146229},
doi = {10.4230/LIPIcs.ESA.2021.41},
annote = {Keywords: Sumodular Optimization, Multiple Knapsack, Randomized Rounding, Linear Grouping, Multiple Choice Multiple Knapsack}
}
Document
Differentially Private Algorithms for Graphs Under Continual Observation
Abstract
Differentially private algorithms protect individuals in data analysis scenarios by ensuring that there is only a weak correlation between the existence of the user in the data and the result of the analysis. Dynamic graph algorithms maintain the solution to a problem (e.g., a matching) on an evolving input, i.e., a graph where nodes or edges are inserted or deleted over time. They output the value of the solution after each update operation, i.e., continuously. We study (event-level and user-level) differentially private algorithms for graph problems under continual observation, i.e., differentially private dynamic graph algorithms. We present event-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor (in the length T of the update sequence) on the state of the art, resulting in the first algorithms with additive error polylogarithmic in T.
We also give ε-differentially private and partially dynamic algorithms for minimum spanning tree, minimum cut, densest subgraph, and maximum matching. The additive error of our improved MST algorithm is O(W log^{3/2}T / ε), where W is the maximum weight of any edge, which, as we show, is tight up to a (√{log T} / ε)-factor. For the other problems, we present a partially-dynamic algorithm with multiplicative error (1+β) for any constant β > 0 and additive error O(W log(nW) log(T) / (ε β)). Finally, we show that the additive error for a broad class of dynamic graph algorithms with user-level privacy must be linear in the value of the output solution’s range.
Cite as
Hendrik Fichtenberger, Monika Henzinger, and Lara Ost. Differentially Private Algorithms for Graphs Under Continual Observation. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 42:1-42:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{fichtenberger_et_al:LIPIcs.ESA.2021.42,
author = {Fichtenberger, Hendrik and Henzinger, Monika and Ost, Lara},
title = {{Differentially Private Algorithms for Graphs Under Continual Observation}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {42:1--42:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.42},
URN = {urn:nbn:de:0030-drops-146230},
doi = {10.4230/LIPIcs.ESA.2021.42},
annote = {Keywords: differential privacy, continual observation, dynamic graph algorithms}
}
Document
Experimental Comparison of PC-Trees and PQ-Trees
Abstract
PQ-trees and PC-trees are data structures that represent sets of linear and circular orders, respectively, subject to constraints that specific subsets of elements have to be consecutive. While equivalent to each other, PC-trees are conceptually much simpler than PQ-trees; updating a PC-tree so that a set of elements becomes consecutive requires only a single operation, whereas PQ-trees use an update procedure that is described in terms of nine transformation templates that have to be recursively matched and applied.
Despite these theoretical advantages, to date no practical PC-tree implementation is available. This might be due to the original description by Hsu and McConnell [Hsu et al., 2003] in some places only sketching the details of the implementation. In this paper, we describe two alternative implementations of PC-trees. For the first one, we follow the approach by Hsu and McConnell, filling in the necessary details and also proposing improvements on the original algorithm. For the second one, we use a different technique for efficiently representing the tree using a Union-Find data structure. In an extensive experimental evaluation we compare our implementations to a variety of other implementations of PQ-trees that are available on the web as part of academic and other software libraries. Our results show that both PC-tree implementations beat their closest fully correct competitor, the PQ-tree implementation from the OGDF library [Markus Chimani et al., 2014; Leipert, 1997], by a factor of 2 to 4, showing that PC-trees are not only conceptually simpler but also fast in practice. Moreover, we find the Union-Find-based implementation, while having a slightly worse asymptotic runtime, to be twice as fast as the one based on the description by Hsu and McConnell.
Cite as
Simon D. Fink, Matthias Pfretzschner, and Ignaz Rutter. Experimental Comparison of PC-Trees and PQ-Trees. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{fink_et_al:LIPIcs.ESA.2021.43,
author = {Fink, Simon D. and Pfretzschner, Matthias and Rutter, Ignaz},
title = {{Experimental Comparison of PC-Trees and PQ-Trees}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {43:1--43:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.43},
URN = {urn:nbn:de:0030-drops-146245},
doi = {10.4230/LIPIcs.ESA.2021.43},
annote = {Keywords: PQ-Tree, PC-Tree, circular consecutive ones, implementation, experimental evaluation}
}
Document
Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification
Abstract
The problem of nearest-neighbor classification is a fundamental technique in machine-learning. Given a training set P of n labeled points in ℝ^d, and an approximation parameter 0 < ε ≤ 1/2, any unlabeled query point should be classified with the class of any of its ε-approximate nearest-neighbors in P. Answering these queries efficiently has been the focus of extensive research, proposing techniques that are mainly tailored towards resolving the more general problem of ε-approximate nearest-neighbor search. While the latest can only hope to provide query time and space complexities dependent on n, the problem of nearest-neighbor classification accepts other parameters more suitable to its analysis. Such is the number k_ε of ε-border points, which describes the complexity of boundaries between sets of points of different classes.
This paper presents a new data structure called Chromatic AVD. This is the first approach for ε-approximate nearest-neighbor classification whose space and query time complexities are only dependent on ε, k_ε and d, while being independent on both n and Δ, the spread of P.
Cite as
Alejandro Flores-Velazco and David M. Mount. Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{floresvelazco_et_al:LIPIcs.ESA.2021.44,
author = {Flores-Velazco, Alejandro and Mount, David M.},
title = {{Boundary-Sensitive Approach for Approximate Nearest-Neighbor Classification}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {44:1--44:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.44},
URN = {urn:nbn:de:0030-drops-146252},
doi = {10.4230/LIPIcs.ESA.2021.44},
annote = {Keywords: approximate nearest-neighbor searching, nearest-neighbor classification, geometric data structures, space-time tradeoffs}
}
Document
Compression by Contracting Straight-Line Programs
Abstract
In grammar-based compression a string is represented by a context-free grammar, also called a straight-line program (SLP), that generates only that string. We refine a recent balancing result stating that one can transform an SLP of size g in linear time into an equivalent SLP of size 𝒪(g) so that the height of the unique derivation tree is 𝒪(log N) where N is the length of the represented string (FOCS 2019). We introduce a new class of balanced SLPs, called contracting SLPs, where for every rule A → β₁ … β_k the string length of every variable β_i on the right-hand side is smaller by a constant factor than the string length of A. In particular, the derivation tree of a contracting SLP has the property that every subtree has logarithmic height in its leaf size. We show that a given SLP of size g can be transformed in linear time into an equivalent contracting SLP of size 𝒪(g) with rules of constant length. This result is complemented by a lower bound, proving that converting SLPs into so called α-balanced SLPs or AVL-grammars can incur an increase by a factor of Ω(log N).
We present an application to the navigation problem in compressed unranked trees, represented by forest straight-line programs (FSLPs). A linear space data structure by Reh and Sieber (2020) supports navigation steps such as going to the parent, left/right sibling, or to the first/last child in constant time. We extend their solution by the operation of moving to the i-th child in time 𝒪(log d) where d is the degree of the current node.
Contracting SLPs are also applied to the finger search problem over SLP-compressed strings where one wants to access positions near to a pre-specified finger position, ideally in 𝒪(log d) time where d is the distance between the accessed position and the finger. We give a linear space solution for the dynamic variant where one can set the finger in 𝒪(log N) time, and then access symbols or move the finger in time 𝒪(log d + log^(t) N) for any constant t where log^(t) N is the t-fold logarithm of N. This improves a previous solution by Bille, Christiansen, Cording, and Gørtz (2018) with access/move time 𝒪(log d + log log N).
Cite as
Moses Ganardi. Compression by Contracting Straight-Line Programs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ganardi:LIPIcs.ESA.2021.45,
author = {Ganardi, Moses},
title = {{Compression by Contracting Straight-Line Programs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {45:1--45:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.45},
URN = {urn:nbn:de:0030-drops-146263},
doi = {10.4230/LIPIcs.ESA.2021.45},
annote = {Keywords: grammar-based compression, balancing, finger search}
}
Document
Space Efficient Two-Dimensional Orthogonal Colored Range Counting
Abstract
In the two-dimensional orthogonal colored range counting problem, we preprocess a set, P, of n colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space costs are in words and ε is an arbitrary constant in (0,1)):
- O(nlg³ n) space and O(√nlg^{5/2} n lg lg n) query time;
- O(nlg² n) space and O(√nlg^{4+ε} n) query time;
- O(n (lg² n)/(lg lg n)) space and O(√nlg^{5+ε} n) query time;
- O(nlg n) space and O(n^{1/2+ε}) query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than √n by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses O(nlg⁴ n) space to answer queries in O(√nlg⁸ n) time (SIAM. J. Comp. 2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.
Cite as
Younan Gao and Meng He. Space Efficient Two-Dimensional Orthogonal Colored Range Counting. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gao_et_al:LIPIcs.ESA.2021.46,
author = {Gao, Younan and He, Meng},
title = {{Space Efficient Two-Dimensional Orthogonal Colored Range Counting}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {46:1--46:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.46},
URN = {urn:nbn:de:0030-drops-146277},
doi = {10.4230/LIPIcs.ESA.2021.46},
annote = {Keywords: 2D Colored orthogonal range counting, stabbing queries, geometric data structures}
}
Document
Computing the 4-Edge-Connected Components of a Graph in Linear Time
Abstract
We present the first linear-time algorithm that computes the 4-edge-connected components of an undirected graph. Hence, we also obtain the first linear-time algorithm for testing 4-edge connectivity. Our results are based on a linear-time algorithm that computes the 3-edge cuts of a 3-edge-connected graph G, and a linear-time procedure that, given the collection of all 3-edge cuts, partitions the vertices of G into the 4-edge-connected components.
Cite as
Loukas Georgiadis, Giuseppe F. Italiano, and Evangelos Kosinas. Computing the 4-Edge-Connected Components of a Graph in Linear Time. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{georgiadis_et_al:LIPIcs.ESA.2021.47,
author = {Georgiadis, Loukas and Italiano, Giuseppe F. and Kosinas, Evangelos},
title = {{Computing the 4-Edge-Connected Components of a Graph in Linear Time}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {47:1--47:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.47},
URN = {urn:nbn:de:0030-drops-146286},
doi = {10.4230/LIPIcs.ESA.2021.47},
annote = {Keywords: Cuts, Edge Connectivity, Graph Algorithms}
}
Document
Deep Multilevel Graph Partitioning
Abstract
Partitioning a graph into blocks of "roughly equal" weight while cutting only few edges is a fundamental problem in computer science with a wide range of applications. In particular, the problem is a building block in applications that require parallel processing. While the amount of available cores in parallel architectures has significantly increased in recent years, state-of-the-art graph partitioning algorithms do not work well if the input needs to be partitioned into a large number of blocks. Often currently available algorithms compute highly imbalanced solutions, solutions of low quality, or have excessive running time for this case. This is due to the fact that most high-quality general-purpose graph partitioners are multilevel algorithms which perform graph coarsening to build a hierarchy of graphs, initial partitioning to compute an initial solution, and local improvement to improve the solution throughout the hierarchy. However, for large number of blocks, the smallest graph in the hierarchy that is used for initial partitioning still has to be large.
In this work, we substantially mitigate these problems by introducing deep multilevel graph partitioning and a shared-memory implementation thereof. Our scheme continues the multilevel approach deep into initial partitioning - integrating it into a framework where recursive bipartitioning and direct k-way partitioning are combined such that they can operate with high performance and quality. Our integrated approach is stronger, more flexible, arguably more elegant, and reduces bottlenecks for parallelization compared to existing multilevel approaches. For example, for large number of blocks our algorithm is on average at least an order of magnitude faster than competing algorithms while computing partitions with comparable solution quality. At the same time, our algorithm consistently produces balanced solutions. Moreover, for small number of blocks, our algorithms are the fastest among competing systems with comparable quality.
Cite as
Lars Gottesbüren, Tobias Heuer, Peter Sanders, Christian Schulz, and Daniel Seemaier. Deep Multilevel Graph Partitioning. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 48:1-48:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{gottesburen_et_al:LIPIcs.ESA.2021.48,
author = {Gottesb\"{u}ren, Lars and Heuer, Tobias and Sanders, Peter and Schulz, Christian and Seemaier, Daniel},
title = {{Deep Multilevel Graph Partitioning}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {48:1--48:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.48},
URN = {urn:nbn:de:0030-drops-146298},
doi = {10.4230/LIPIcs.ESA.2021.48},
annote = {Keywords: graph partitioning, graph algorithms, multilevel, shared-memory, parallel}
}
Document
Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back
Abstract
Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss.
Cite as
Fabrizio Grandoni, Tobias Mömke, and Andreas Wiese. Faster (1+ε)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{grandoni_et_al:LIPIcs.ESA.2021.49,
author = {Grandoni, Fabrizio and M\"{o}mke, Tobias and Wiese, Andreas},
title = {{Faster (1+\epsilon)-Approximation for Unsplittable Flow on a Path via Resource Augmentation and Back}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {49:1--49:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.49},
URN = {urn:nbn:de:0030-drops-146301},
doi = {10.4230/LIPIcs.ESA.2021.49},
annote = {Keywords: Approximation Algorithms, Unsplittable Flow, Dynamic Programming}
}
Document
Quantum Sub-Gaussian Mean Estimator
Abstract
We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup over the number of classical i.i.d. samples needed to estimate the mean of a heavy-tailed distribution with a sub-Gaussian error rate. This result subsumes (up to logarithmic factors) earlier works on the mean estimation problem that were not optimal for heavy-tailed distributions [Brassard et al., 2002; Brassard et al., 2011], or that require prior information on the variance [Heinrich, 2002; Montanaro, 2015; Hamoudi and Magniez, 2019]. As an application, we obtain new quantum algorithms for the (ε,δ)-approximation problem with an optimal dependence on the coefficient of variation of the input random variable.
Cite as
Yassine Hamoudi. Quantum Sub-Gaussian Mean Estimator. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 50:1-50:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{hamoudi:LIPIcs.ESA.2021.50,
author = {Hamoudi, Yassine},
title = {{Quantum Sub-Gaussian Mean Estimator}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {50:1--50:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.50},
URN = {urn:nbn:de:0030-drops-146318},
doi = {10.4230/LIPIcs.ESA.2021.50},
annote = {Keywords: Quantum algorithm, statistical analysis, mean estimator, sub-Gaussian estimator, (\epsilon,\delta)-approximation, lower bound}
}
Document
Improved Approximation Algorithms for Tverberg Partitions
Abstract
Tverberg’s theorem states that a set of n points in ℝ^d can be partitioned into ⌈n/(d+1)⌉ sets whose convex hulls all intersect. A point in the intersection (aka Tverberg point) is a centerpoint, or high-dimensional median, of the input point set. While randomized algorithms exist to find centerpoints with some failure probability, a partition for a Tverberg point provides a certificate of its correctness.
Unfortunately, known algorithms for computing exact Tverberg points take n^{O(d²)} time. We provide several new approximation algorithms for this problem, which improve running time or approximation quality over previous work. In particular, we provide the first strongly polynomial (in both n and d) approximation algorithm for finding a Tverberg point.
Cite as
Sariel Har-Peled and Timothy Zhou. Improved Approximation Algorithms for Tverberg Partitions. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{harpeled_et_al:LIPIcs.ESA.2021.51,
author = {Har-Peled, Sariel and Zhou, Timothy},
title = {{Improved Approximation Algorithms for Tverberg Partitions}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {51:1--51:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.51},
URN = {urn:nbn:de:0030-drops-146323},
doi = {10.4230/LIPIcs.ESA.2021.51},
annote = {Keywords: Geometric spanners, vertex failures, robustness}
}
Document
Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs
Abstract
Let G be a graph and S, T ⊆ V(G) be (possibly overlapping) sets of terminals, |S| = |T| = k. We are interested in computing a vertex sparsifier for terminal cuts in G, i.e., a graph H on a smallest possible number of vertices, where S ∪ T ⊆ V(H) and such that for every A ⊆ S and B ⊆ T the size of a minimum (A,B)-vertex cut is the same in G as in H. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlström (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier H with O(k³) vertices can be computed in randomized polynomial time, even for arbitrary digraphs G. However, since then, no improvements on the size O(k³) have been shown.
In this paper, we draw inspiration from the renowned Bollobás’s Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlström’s methods. This new perspective allows us to construct a sparsifier H of Θ(k²) vertices for the case that G is a DAG. We also show how to compute H in time near-linear in the size of G, improving on the previous O(n^{ω+1}). Furthermore, H recovers the closest min-cut in G for every partition (A,B), which was not previously known. Finally, we show that a sparsifier of size Ω(k²) is required, both for DAGs and for undirected edge cuts.
Cite as
Zhiyang He, Jason Li, and Magnus Wahlström. Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{he_et_al:LIPIcs.ESA.2021.52,
author = {He, Zhiyang and Li, Jason and Wahlstr\"{o}m, Magnus},
title = {{Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.52},
URN = {urn:nbn:de:0030-drops-146331},
doi = {10.4230/LIPIcs.ESA.2021.52},
annote = {Keywords: graph theory, vertex sparsifier, representative family, matroid}
}
Document
Closing the Gap for Single Resource Constraint Scheduling
Abstract
In the problem called single resource constraint scheduling, we are given m identical machines and a set of jobs, each needing one machine to be processed as well as a share of a limited renewable resource R. A schedule of these jobs is feasible if, at each point in the schedule, the number of machines and resources required by jobs processed at this time is not exceeded. It is NP-hard to approximate this problem with a ratio better than 3/2. On the other hand, the best algorithm so far has an absolute approximation ratio of 2+ε. In this paper, we present an algorithm with absolute approximation ratio (3/2+ε), which closes the gap between inapproximability and best algorithm with exception of a negligible small ε.
Cite as
Klaus Jansen and Malin Rau. Closing the Gap for Single Resource Constraint Scheduling. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{jansen_et_al:LIPIcs.ESA.2021.53,
author = {Jansen, Klaus and Rau, Malin},
title = {{Closing the Gap for Single Resource Constraint Scheduling}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {53:1--53:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.53},
URN = {urn:nbn:de:0030-drops-146344},
doi = {10.4230/LIPIcs.ESA.2021.53},
annote = {Keywords: resource constraint scheduling, approximation algorithm}
}
Document
Certified Approximation Algorithms for the Fermat Point and n-Ellipses
Abstract
Given a set A of n points in ℝ^d with weight function w: A→ℝ_{> 0}, the Fermat distance function is φ(x): = ∑_{a∈A}w(a)‖x-a‖. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function φ. We consider the problem of computing a point x̃* that is an ε-approximation of x* in the sense that ‖x̃*-x*‖<ε. The algorithmic literature has so far used a different notion based on ε-approximation of the value φ(x*). We devise a certified subdivision algorithm for computing x̃*, enhanced by Newton operator techniques. We also revisit the classic Weiszfeld-Kuhn iteration scheme for x*, turning it into an ε-approximate Fermat point algorithm. Our second problem is the certified construction of ε-isotopic approximations of n-ellipses. These are the level sets φ^{-1}(r) for r > φ(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques.
Cite as
Kolja Junginger, Ioannis Mantas, Evanthia Papadopoulou, Martin Suderland, and Chee Yap. Certified Approximation Algorithms for the Fermat Point and n-Ellipses. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 54:1-54:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{junginger_et_al:LIPIcs.ESA.2021.54,
author = {Junginger, Kolja and Mantas, Ioannis and Papadopoulou, Evanthia and Suderland, Martin and Yap, Chee},
title = {{Certified Approximation Algorithms for the Fermat Point and n-Ellipses}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {54:1--54:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.54},
URN = {urn:nbn:de:0030-drops-146359},
doi = {10.4230/LIPIcs.ESA.2021.54},
annote = {Keywords: Fermat point, n-ellipse, subdivision, approximation, certified, algorithms}
}
Document
Parameterized Algorithms for Diverse Multistage Problems
Abstract
The world is rarely static - many problems need not only be solved once but repeatedly, under changing conditions. This setting is addressed by the multistage view on computational problems. We study the diverse multistage variant, where consecutive solutions of large variety are preferable to similar ones, e.g. for reasons of fairness or wear minimization. While some aspects of this model have been tackled before, we introduce a framework allowing us to prove that a number of diverse multistage problems are fixed-parameter tractable by diversity, namely Perfect Matching, s-t Path, Matroid Independent Set, and Plurality Voting. This is achieved by first solving special, colored variants of these problems, which might also be of independent interest.
Cite as
Leon Kellerhals, Malte Renken, and Philipp Zschoche. Parameterized Algorithms for Diverse Multistage Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 55:1-55:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kellerhals_et_al:LIPIcs.ESA.2021.55,
author = {Kellerhals, Leon and Renken, Malte and Zschoche, Philipp},
title = {{Parameterized Algorithms for Diverse Multistage Problems}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {55:1--55:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.55},
URN = {urn:nbn:de:0030-drops-146360},
doi = {10.4230/LIPIcs.ESA.2021.55},
annote = {Keywords: Temporal graphs, dissimilar solutions, fixed-parameter tractability, perfect matchings, s-t paths, committee election, spanning forests, matroids}
}
Document
Fast and Space-Efficient Construction of AVL Grammars from the LZ77 Parsing
Abstract
Grammar compression is, next to Lempel-Ziv (LZ77) and run-length Burrows-Wheeler transform (RLBWT), one of the most flexible approaches to representing and processing highly compressible strings. The main idea is to represent a text as a context-free grammar whose language is precisely the input string. This is called a straight-line grammar (SLG). An AVL grammar, proposed by Rytter [Theor. Comput. Sci., 2003] is a type of SLG that additionally satisfies the AVL property: the heights of parse trees for children of every nonterminal differ by at most one. In contrast to other SLG constructions, AVL grammars can be constructed from the LZ77 parsing in compressed time: 𝒪(z log n) where z is the size of the LZ77 parsing and n is the length of the input text. Despite these advantages, AVL grammars are thought to be too large to be practical.
We present a new technique for rapidly constructing a small AVL grammar from an LZ77 or LZ77-like parse. Our algorithm produces grammars that are always at least five times smaller than those produced by the original algorithm, and usually not more than double the size of grammars produced by the practical Re-Pair compressor [Larsson and Moffat, Proc. IEEE, 2000]. Our algorithm also achieves low peak RAM usage. By combining this algorithm with recent advances in approximating the LZ77 parsing, we show that our method has the potential to construct a run-length BWT in about one third of the time and peak RAM required by other approaches. Overall, we show that AVL grammars are surprisingly practical, opening the door to much faster construction of key compressed data structures.
Cite as
Dominik Kempa and Ben Langmead. Fast and Space-Efficient Construction of AVL Grammars from the LZ77 Parsing. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{kempa_et_al:LIPIcs.ESA.2021.56,
author = {Kempa, Dominik and Langmead, Ben},
title = {{Fast and Space-Efficient Construction of AVL Grammars from the LZ77 Parsing}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {56:1--56:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.56},
URN = {urn:nbn:de:0030-drops-146373},
doi = {10.4230/LIPIcs.ESA.2021.56},
annote = {Keywords: grammar compression, straight-line program, SLP, AVL grammar, Lempel-Ziv compression, LZ77, dictionary compression}
}
Document
Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing
Abstract
A hierarchical plane st-graph H can be thought of as a combinatorial description of a planar drawing Γ of a 2-connected graph G in which each edge is a y-monotone curve and each face encloses a y-monotone region (that is, a region whose intersection with any horizontal line is a line segment, a point, or empty). A drawing Γ' of H is a drawing of G such that each horizontal line intersects the same left-to-right order of edges and vertices in Γ and Γ', that is, the underlying hierarchical plane st-graph of both drawings is H. A straight-line planar drawing of a graph is convex if the boundary of each face is realized as a convex polygon.
We study the computation of convex drawings of hierarchical plane st-graphs such that the outer face is realized as a prescribed polygon. Chrobak, Goodrich, and Tamassia [SoCG'96] and, independently, Kleist et al. [CGTA'19] described an idea to solve this problem in O(n^{1.1865}) time, where n is the number of vertices of the graph. Also independently, Hong and Nagamochi [J. Discrete Algorithms'10] described a completely different approach, which can be executed in O(n²) time.
In this paper, we present an optimal O(n) time algorithm to solve the above problem, thereby improving the previous results by Chrobak, Goodrich, and Tamassia, Kleist et al., and by Hong and Nagamochi. Our result has applications in graph morphing. A planar morph is a continuous deformation of a graph drawing that preserves straight-line planarity. We show that our algorithm can be used as a drop-in replacement to speed up a procedure by Alamdari et al. [SICOMP'17] to morph between any two given straight-line planar drawings of the same plane graph. The running time improves from O(n^{2.1865}) to O(n²log n). To obtain our results, we devise a new strategy for computing so-called archfree paths in hierarchical plane st-graphs, which might be of independent interest.
Cite as
Boris Klemz. Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{klemz:LIPIcs.ESA.2021.57,
author = {Klemz, Boris},
title = {{Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {57:1--57:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.57},
URN = {urn:nbn:de:0030-drops-146381},
doi = {10.4230/LIPIcs.ESA.2021.57},
annote = {Keywords: convex drawing, hierarchical graph, graph drawing, computational geometry, planarity, planar graph, morphing, convexity}
}
Document
QCSP on Reflexive Tournaments
Abstract
We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.
Cite as
Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58,
author = {Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav},
title = {{QCSP on Reflexive Tournaments}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {58:1--58:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.58},
URN = {urn:nbn:de:0030-drops-146392},
doi = {10.4230/LIPIcs.ESA.2021.58},
annote = {Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction}
}
Document
Learnable and Instance-Robust Predictions for Online Matching, Flows and Load Balancing
Abstract
We propose a new model for augmenting algorithms with predictions by requiring that they are formally learnable and instance robust. Learnability ensures that predictions can be efficiently constructed from a reasonable amount of past data. Instance robustness ensures that the prediction is robust to modest changes in the problem input, where the measure of the change may be problem specific. Instance robustness insists on a smooth degradation in performance as a function of the change. Ideally, the performance is never worse than worst-case bounds. This also allows predictions to be objectively compared.
We design online algorithms with predictions for a network flow allocation problem and restricted assignment makespan minimization. For both problems, two key properties are established: high quality predictions can be learned from a small sample of prior instances and these predictions are robust to errors that smoothly degrade as the underlying problem instance changes.
Cite as
Thomas Lavastida, Benjamin Moseley, R. Ravi, and Chenyang Xu. Learnable and Instance-Robust Predictions for Online Matching, Flows and Load Balancing. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lavastida_et_al:LIPIcs.ESA.2021.59,
author = {Lavastida, Thomas and Moseley, Benjamin and Ravi, R. and Xu, Chenyang},
title = {{Learnable and Instance-Robust Predictions for Online Matching, Flows and Load Balancing}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {59:1--59:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.59},
URN = {urn:nbn:de:0030-drops-146405},
doi = {10.4230/LIPIcs.ESA.2021.59},
annote = {Keywords: Learning-augmented algorithms, Online algorithms, Flow allocation}
}
Document
Telescoping Filter: A Practical Adaptive Filter
Abstract
Filters are small, fast, and approximate set membership data structures. They are often used to filter out expensive accesses to a remote set S for negative queries (that is, filtering out queries x ∉ S). Filters have one-sided errors: on a negative query, a filter may say "present" with a tunable false-positive probability of ε. Correctness is traded for space: filters only use log (1/ε) + O(1) bits per element.
The false-positive guarantees of most filters, however, hold only for a single query. In particular, if x is a false positive, a subsequent query to x is a false positive with probability 1, not ε. With this in mind, recent work has introduced the notion of an adaptive filter. A filter is adaptive if each query is a false positive with probability ε, regardless of answers to previous queries. This requires "fixing" false positives as they occur.
Adaptive filters not only provide strong false positive guarantees in adversarial environments but also improve query performance on practical workloads by eliminating repeated false positives.
Existing work on adaptive filters falls into two categories. On the one hand, there are practical filters, based on the cuckoo filter, that attempt to fix false positives heuristically without meeting the adaptivity guarantee. On the other hand, the broom filter is a very complex adaptive filter that meets the optimal theoretical bounds.
In this paper, we bridge this gap by designing the telescoping adaptive filter (TAF), a practical, provably adaptive filter. We provide theoretical false-positive and space guarantees for our filter, along with empirical results where we compare its performance against state-of-the-art filters. We also implement the broom filter and compare it to the TAF. Our experiments show that theoretical adaptivity can lead to improved false-positive performance on practical inputs, and can be achieved while maintaining throughput that is similar to non-adaptive filters.
Cite as
David J. Lee, Samuel McCauley, Shikha Singh, and Max Stein. Telescoping Filter: A Practical Adaptive Filter. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 60:1-60:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lee_et_al:LIPIcs.ESA.2021.60,
author = {Lee, David J. and McCauley, Samuel and Singh, Shikha and Stein, Max},
title = {{Telescoping Filter: A Practical Adaptive Filter}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {60:1--60:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.60},
URN = {urn:nbn:de:0030-drops-146410},
doi = {10.4230/LIPIcs.ESA.2021.60},
annote = {Keywords: Filters, approximate-membership query data structures (AMQs), Bloom filters, quotient filters, cuckoo filters, adaptivity, succinct data structures}
}
Document
Finding an Approximate Mode of a Kernel Density Estimate
Abstract
Given points P = {p₁,...,p_n} subset of ℝ^d, how do we find a point x which approximately maximizes the function 1/n ∑_{p_i ∈ P} e^{-‖p_i-x‖²}? In other words, how do we find an approximate mode of a Gaussian kernel density estimate (KDE) of P? Given the power of KDEs in representing probability distributions and other continuous functions, the basic mode finding problem is widely applicable. However, it is poorly understood algorithmically. We provide fast and provably accurate approximation algorithms for mode finding in both the low and high dimensional settings. For low (constant) dimension, our main contribution is a reduction to solving systems of polynomial inequalities. For high dimension, we prove the first dimensionality reduction result for KDE mode finding. The latter result leverages Johnson-Lindenstrauss projection, Kirszbraun’s classic extension theorem, and perhaps surprisingly, the mean-shift heuristic for mode finding. For constant approximation factor these algorithms run in O(n (log n)^{O(d)}) and O(nd + (log n)^{O(log³ n)}), respectively; these are proven more precisely as a (1+ε)-approximation guarantee. Furthermore, for the special case of d = 2, we give a combinatorial algorithm running in O(n log² n) time. We empirically demonstrate that the random projection approach and the 2-dimensional algorithm improves over the state-of-the-art mode-finding heuristics.
Cite as
Jasper C.H. Lee, Jerry Li, Christopher Musco, Jeff M. Phillips, and Wai Ming Tai. Finding an Approximate Mode of a Kernel Density Estimate. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lee_et_al:LIPIcs.ESA.2021.61,
author = {Lee, Jasper C.H. and Li, Jerry and Musco, Christopher and Phillips, Jeff M. and Tai, Wai Ming},
title = {{Finding an Approximate Mode of a Kernel Density Estimate}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {61:1--61:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.61},
URN = {urn:nbn:de:0030-drops-146428},
doi = {10.4230/LIPIcs.ESA.2021.61},
annote = {Keywords: Kernel density estimation, Dimensionality reduction, Coresets, Means-shift}
}
Document
An Efficient Reduction of a Gammoid to a Partition Matroid
Abstract
Our main contribution is a polynomial-time algorithm to reduce a k-colorable gammoid to a (2k-2)-colorable partition matroid. It is known that there are gammoids that can not be reduced to any (2k-3)-colorable partition matroid, so this result is tight. We then discuss how such a reduction can be used to obtain polynomial-time algorithms with better approximation ratios for various natural problems related to coloring and list coloring the intersection of matroids.
Cite as
Marilena Leichter, Benjamin Moseley, and Kirk Pruhs. An Efficient Reduction of a Gammoid to a Partition Matroid. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{leichter_et_al:LIPIcs.ESA.2021.62,
author = {Leichter, Marilena and Moseley, Benjamin and Pruhs, Kirk},
title = {{An Efficient Reduction of a Gammoid to a Partition Matroid}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {62:1--62:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.62},
URN = {urn:nbn:de:0030-drops-146432},
doi = {10.4230/LIPIcs.ESA.2021.62},
annote = {Keywords: Matroid, Gammoid, Reduction, Algorithms}
}
Document
Efficient Algorithms for Least Square Piecewise Polynomial Regression
Abstract
We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (𝐱₁, y₁),… , (𝐱_n, y_n) ∈ ℝ^d × ℝ where d ∈ {1,2}, the goal is to segment 𝐱_i’s into some (arbitrary) number of disjoint pieces P₁, … , P_k, where each piece P_j is associated with a fixed-degree polynomial f_j: ℝ^d → ℝ, to minimize the total loss function λ k + ∑_{i = 1}ⁿ (y_i - f(𝐱_i))², where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f: ⨆_{j = 1}^k P_j → ℝ is the piecewise polynomial function defined as f|_{P_j} = f_j. The pieces P₁, … , P_k are disjoint intervals of ℝ in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions.
Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(n/(ε) log 1/(ε)), assuming that data is presented in sorted order of x_i’s. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(√n)}; a polynomial-time constant-approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness.
Cite as
Daniel Lokshtanov, Subhash Suri, and Jie Xue. Efficient Algorithms for Least Square Piecewise Polynomial Regression. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lokshtanov_et_al:LIPIcs.ESA.2021.63,
author = {Lokshtanov, Daniel and Suri, Subhash and Xue, Jie},
title = {{Efficient Algorithms for Least Square Piecewise Polynomial Regression}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {63:1--63:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.63},
URN = {urn:nbn:de:0030-drops-146443},
doi = {10.4230/LIPIcs.ESA.2021.63},
annote = {Keywords: regression analysis, piecewise polynomial, least square error}
}
Document
Bidirectional String Anchors: A New String Sampling Mechanism
Abstract
The minimizers sampling mechanism is a popular mechanism for string sampling introduced independently by Schleimer et al. [SIGMOD 2003] and by Roberts et al. [Bioinf. 2004]. Given two positive integers w and k, it selects the lexicographically smallest length-k substring in every fragment of w consecutive length-k substrings (in every sliding window of length w+k-1). Minimizers samples are approximately uniform, locally consistent, and computable in linear time. Although they do not have good worst-case guarantees on their size, they are often small in practice. They thus have been successfully employed in several string processing applications. Two main disadvantages of minimizers sampling mechanisms are: first, they also do not have good guarantees on the expected size of their samples for every combination of w and k; and, second, indexes that are constructed over their samples do not have good worst-case guarantees for on-line pattern searches.
To alleviate these disadvantages, we introduce bidirectional string anchors (bd-anchors), a new string sampling mechanism. Given a positive integer 𝓁, our mechanism selects the lexicographically smallest rotation in every length-𝓁 fragment (in every sliding window of length 𝓁). We show that bd-anchors samples are also approximately uniform, locally consistent, and computable in linear time. In addition, our experiments using several datasets demonstrate that the bd-anchors sample sizes decrease proportionally to 𝓁; and that these sizes are competitive to or smaller than the minimizers sample sizes using the analogous sampling parameters. We provide theoretical justification for these results by analyzing the expected size of bd-anchors samples.
We also show that by using any bd-anchors sample, we can construct, in near-linear time, an index which requires linear (extra) space in the size of the sample and answers on-line pattern searches in near-optimal time. We further show, using several datasets, that a simple implementation of our index is consistently faster for on-line pattern searches than an analogous implementation of a minimizers-based index [Grabowski and Raniszewski, Softw. Pract. Exp. 2017].
Finally, we highlight the applicability of bd-anchors by developing an efficient and effective heuristic for top-K similarity search under edit distance. We show, using synthetic datasets, that our heuristic is more accurate and more than one order of magnitude faster in top-K similarity searches than the state-of-the-art tool for the same purpose [Zhang and Zhang, KDD 2020].
Cite as
Grigorios Loukides and Solon P. Pissis. Bidirectional String Anchors: A New String Sampling Mechanism. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 64:1-64:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{loukides_et_al:LIPIcs.ESA.2021.64,
author = {Loukides, Grigorios and Pissis, Solon P.},
title = {{Bidirectional String Anchors: A New String Sampling Mechanism}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {64:1--64:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.64},
URN = {urn:nbn:de:0030-drops-146456},
doi = {10.4230/LIPIcs.ESA.2021.64},
annote = {Keywords: string algorithms, string sampling, text indexing, top-K similarity search}
}
Document
The Visibility Center of a Simple Polygon
Abstract
We introduce the visibility center of a set of points inside a polygon - a point c_V such that the maximum geodesic distance from c_V to see any point in the set is minimized. For a simple polygon of n vertices and a set of m points inside it, we give an O((n+m) log (n+m)) time algorithm to find the visibility center. We find the visibility center of all points in a simple polygon in O(n log n) time.
Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an O((n+k) log (n+k)) time algorithm for this problem, where k is the number of half-polygons.
Cite as
Anna Lubiw and Anurag Murty Naredla. The Visibility Center of a Simple Polygon. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{lubiw_et_al:LIPIcs.ESA.2021.65,
author = {Lubiw, Anna and Naredla, Anurag Murty},
title = {{The Visibility Center of a Simple Polygon}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {65:1--65:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.65},
URN = {urn:nbn:de:0030-drops-146466},
doi = {10.4230/LIPIcs.ESA.2021.65},
annote = {Keywords: Visibility, Shortest Paths, Simple Polygons, Facility Location}
}
Document
Extension of Additive Valuations to General Valuations on the Existence of EFX
Abstract
Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when n = 2 or (ii) when all agents have identical valuations, where n is the number of agents. it is also known that an EFX allocation always exists when one can leave at most n-1 items unallocated.
We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n+3. We also show that an EFX allocation always exists when one can leave at most n-2 items unallocated. In addition to the positive results, we construct an instance with n = 3 in which an existing approach does not work as it is.
Cite as
Ryoga Mahara. Extension of Additive Valuations to General Valuations on the Existence of EFX. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{mahara:LIPIcs.ESA.2021.66,
author = {Mahara, Ryoga},
title = {{Extension of Additive Valuations to General Valuations on the Existence of EFX}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {66:1--66:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.66},
URN = {urn:nbn:de:0030-drops-146478},
doi = {10.4230/LIPIcs.ESA.2021.66},
annote = {Keywords: Discrete Fair Division, EFX allocations, General Valuations}
}
Document
FPT and FPT-Approximation Algorithms for Unsplittable Flow on Trees
Abstract
We study the unsplittable flow on trees (UFT) problem in which we are given a tree with capacities on its edges and a set of tasks, where each task is described by a path and a demand. Our goal is to select a subset of the given tasks of maximum size such that the demands of the selected tasks respect the edge capacities. The problem models throughput maximization in tree networks. The best known approximation ratio for (unweighted) UFT is O(log n). We study the problem under the angle of FPT and FPT-approximation algorithms. We prove that
- UFT is FPT if the parameters are the cardinality k of the desired solution and the number of different task demands in the input,
- UFT is FPT under (1+δ)-resource augmentation of the edge capacities for parameters k and 1/δ, and
- UFT admits an FPT-5-approximation algorithm for parameter k. One key to our results is to compute structured hitting sets of the input edges which partition the given tree into O(k) clean components. This allows us to guess important properties of the optimal solution. Also, in some settings we can compute core sets of subsets of tasks out of which at least one task i is contained in the optimal solution. These sets have bounded size, and hence we can guess this task i easily.
A consequence of our results is that the integral multicommodity flow problem on trees is FPT if the parameter is the desired amount of sent flow. Also, even under (1+δ)-resource augmentation UFT is APX-hard, and hence our FPT-approximation algorithm for this setting breaks this boundary.
Cite as
Tomás Martínez-Muñoz and Andreas Wiese. FPT and FPT-Approximation Algorithms for Unsplittable Flow on Trees. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 67:1-67:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{martinezmunoz_et_al:LIPIcs.ESA.2021.67,
author = {Mart{\'\i}nez-Mu\~{n}oz, Tom\'{a}s and Wiese, Andreas},
title = {{FPT and FPT-Approximation Algorithms for Unsplittable Flow on Trees}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {67:1--67:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.67},
URN = {urn:nbn:de:0030-drops-146486},
doi = {10.4230/LIPIcs.ESA.2021.67},
annote = {Keywords: FPT algorithms, FPT-approximation algorithms, packing problems, unsplittable flow, trees}
}
Document
A Simple Algorithm for Graph Reconstruction
Abstract
How efficiently can we find an unknown graph using distance queries between its vertices? We assume that the unknown graph is connected, unweighted, and has bounded degree. The goal is to find every edge in the graph. This problem admits a reconstruction algorithm based on multi-phase Voronoi-cell decomposition and using Õ(n^{3/2}) distance queries [Kannan et al., 2018].
In our work, we analyze a simple reconstruction algorithm. We show that, on random Δ-regular graphs, our algorithm uses Õ(n) distance queries. As by-products, we can reconstruct those graphs using O(log² n) queries to an all-distances oracle or Õ(n) queries to a betweenness oracle, and we bound the metric dimension of those graphs by log² n.
Our reconstruction algorithm has a very simple structure, and is highly parallelizable. On general graphs of bounded degree, our reconstruction algorithm has subquadratic query complexity.
Cite as
Claire Mathieu and Hang Zhou. A Simple Algorithm for Graph Reconstruction. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 68:1-68:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{mathieu_et_al:LIPIcs.ESA.2021.68,
author = {Mathieu, Claire and Zhou, Hang},
title = {{A Simple Algorithm for Graph Reconstruction}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {68:1--68:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.68},
URN = {urn:nbn:de:0030-drops-146496},
doi = {10.4230/LIPIcs.ESA.2021.68},
annote = {Keywords: reconstruction, network topology, random regular graphs, metric dimension}
}
Document
Generalized Max-Flows and Min-Cuts in Simplicial Complexes
Abstract
We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For d-dimensional simplicial complexes embedded into ℝ^{d+1} we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.
Cite as
William Maxwell and Amir Nayyeri. Generalized Max-Flows and Min-Cuts in Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{maxwell_et_al:LIPIcs.ESA.2021.69,
author = {Maxwell, William and Nayyeri, Amir},
title = {{Generalized Max-Flows and Min-Cuts in Simplicial Complexes}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {69:1--69:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.69},
URN = {urn:nbn:de:0030-drops-146509},
doi = {10.4230/LIPIcs.ESA.2021.69},
annote = {Keywords: Max-flow min-cut, simplicial complexes, algebraic topology}
}
Document
Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima
Abstract
We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures.
We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of 2n+o(n) bits, but the space drops to 1.736n + o(n) bits on average for random permutations of n elements, and 2lg binom{n}{r} + o(n) for arrays with r increasing runs, respectively. Both results are optimal; the former answers an open problem of Davoodi et al. (2014) and Golin et al. (2016).
Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that they simultaneously achieve the optimal space usage to within lower order terms for a wide range of distributions over tree shapes, including: binary search trees (BSTs) generated by insertions in random order / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees.
Cite as
J. Ian Munro, Patrick K. Nicholson, Louisa Seelbach Benkner, and Sebastian Wild. Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 70:1-70:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{munro_et_al:LIPIcs.ESA.2021.70,
author = {Munro, J. Ian and Nicholson, Patrick K. and Benkner, Louisa Seelbach and Wild, Sebastian},
title = {{Hypersuccinct Trees - New Universal Tree Source Codes for Optimal Compressed Tree Data Structures and Range Minima}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {70:1--70:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.70},
URN = {urn:nbn:de:0030-drops-146512},
doi = {10.4230/LIPIcs.ESA.2021.70},
annote = {Keywords: analysis of algorithms, universal source code, compressed trees, succinct data structure, succinct trees, tree covering, random binary search trees, range-minimum queries}
}
Document
Determining 4-Edge-Connected Components in Linear Time
Abstract
In this work, we present the first linear time deterministic algorithm computing the 4-edge-connected components of an undirected graph. First, we show an algorithm listing all 3-edge-cuts in a given 3-edge-connected graph, and then we use the output of this algorithm in order to determine the 4-edge-connected components of the graph.
Cite as
Wojciech Nadara, Mateusz Radecki, Marcin Smulewicz, and Marek Sokołowski. Determining 4-Edge-Connected Components in Linear Time. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{nadara_et_al:LIPIcs.ESA.2021.71,
author = {Nadara, Wojciech and Radecki, Mateusz and Smulewicz, Marcin and Soko{\l}owski, Marek},
title = {{Determining 4-Edge-Connected Components in Linear Time}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {71:1--71:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.71},
URN = {urn:nbn:de:0030-drops-146523},
doi = {10.4230/LIPIcs.ESA.2021.71},
annote = {Keywords: graphs, connectivity, cuts}
}
Document
Isomorphism Testing Parameterized by Genus and Beyond
Abstract
We present an isomorphism test for graphs of Euler genus g running in time 2^{{O}(g⁴ log g)}n^{{O}(1)}. Our algorithm provides the first explicit upper bound on the dependence on g for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time f(g)n for some function f (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude K_{3,h} as a minor. For such graphs, no fpt isomorphism test was known before.
The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce (t,k)-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm. This concept may be of independent interest.
Cite as
Daniel Neuen. Isomorphism Testing Parameterized by Genus and Beyond. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 72:1-72:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{neuen:LIPIcs.ESA.2021.72,
author = {Neuen, Daniel},
title = {{Isomorphism Testing Parameterized by Genus and Beyond}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {72:1--72:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.72},
URN = {urn:nbn:de:0030-drops-146533},
doi = {10.4230/LIPIcs.ESA.2021.72},
annote = {Keywords: graph isomorphism, fixed-parameter tractability, Euler genus, Weisfeiler-Leman algorithm}
}
Document
Restricted t-Matchings via Half-Edges
Abstract
For a bipartite graph G we consider the problem of finding a maximum size/weight square-free 2-matching and its generalization - the problem of finding a maximum size/weight K_{t,t}-free t-matching, where t is an integer greater than two and K_{t,t} denotes a bipartite clique with t vertices on each of the two sides. Since the weighted versions of these problems are NP-hard in general, we assume that the weights are vertex-induced on any subgraph isomorphic to K_{t,t}. We present simple combinatorial algorithms for these problems. Our algorithms are significantly simpler and faster than those previously known. We dispense with the need to shrink squares and, more generally subgraphs isomorphic to K_{t,t}, the operation which occurred in all previous algorithms for such t-matchings and instead use so-called half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints.
Additionally, we consider another problem concerning restricted matchings. Given a (not necessarily bipartite) graph G = (V,E), a set of k subsets of edges E₁, E₂, …, E_k and k natural numbers r₁, r₂, …, r_k, the Restricted Matching Problem asks to find a maximum size matching of G among such ones that for each 1 ≤ i ≤ k, M contains at most r_i edges of E_i. This problem is NP-hard even when G is bipartite. We show that it is solvable in polynomial time if (i) for each i the graph G contains a clique or a bipartite clique on all endpoints of E_i; in the case of a bipartite clique it is required to contain E_i and (ii) the sets E₁, …, E_k are almost vertex-disjoint - the endpoints of any two different sets have at most one vertex in common.
Cite as
Katarzyna Paluch and Mateusz Wasylkiewicz. Restricted t-Matchings via Half-Edges. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 73:1-73:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{paluch_et_al:LIPIcs.ESA.2021.73,
author = {Paluch, Katarzyna and Wasylkiewicz, Mateusz},
title = {{Restricted t-Matchings via Half-Edges}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {73:1--73:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.73},
URN = {urn:nbn:de:0030-drops-146541},
doi = {10.4230/LIPIcs.ESA.2021.73},
annote = {Keywords: restricted 2-matchings}
}
Document
Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems
Abstract
The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists.
Cite as
Ojas Parekh and Kevin Thompson. Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{parekh_et_al:LIPIcs.ESA.2021.74,
author = {Parekh, Ojas and Thompson, Kevin},
title = {{Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {74:1--74:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.74},
URN = {urn:nbn:de:0030-drops-146554},
doi = {10.4230/LIPIcs.ESA.2021.74},
annote = {Keywords: Quantum Approximation Algorithms, Local Hamiltonian}
}
Document
Additive Sparsification of CSPs
Abstract
Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20].
Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs.
As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{0,1}^k → {0,1} of a fixed arity k, we show that CSP(P)} admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P)} admits an additive sparsifier for any predicate P:D^k → {0,1} of a fixed arity k on an arbitrary finite domain D.
Cite as
Eden Pelleg and Stanislav Živný. Additive Sparsification of CSPs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{pelleg_et_al:LIPIcs.ESA.2021.75,
author = {Pelleg, Eden and \v{Z}ivn\'{y}, Stanislav},
title = {{Additive Sparsification of CSPs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {75:1--75:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.75},
URN = {urn:nbn:de:0030-drops-146562},
doi = {10.4230/LIPIcs.ESA.2021.75},
annote = {Keywords: additive sparsification, graphs, hypergraphs, minimum cuts}
}
Document
Faster Deterministic Modular Subset Sum
Abstract
We consider the Modular Subset Sum problem: given a multiset X of integers from ℤ_m and a target integer t, decide if there exists a subset of X with a sum equal to t (mod m). Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in 𝒪(m log m) time, while Axiotis et al. gave a deterministic algorithm that runs in 𝒪(m polylog m) time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure.
In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees.
We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the 𝒪(m log m) runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in 𝒪(m log m ⋅ α(m)) time, where α is the inverse Ackermann function.
Cite as
Krzysztof Potępa. Faster Deterministic Modular Subset Sum. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 76:1-76:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{potepa:LIPIcs.ESA.2021.76,
author = {Pot\k{e}pa, Krzysztof},
title = {{Faster Deterministic Modular Subset Sum}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {76:1--76:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.76},
URN = {urn:nbn:de:0030-drops-146574},
doi = {10.4230/LIPIcs.ESA.2021.76},
annote = {Keywords: Modular Subset Sum, String Problem, Segment Tree, Data Structure}
}
Document
Hardness of Detecting Abelian and Additive Square Factors in Strings
Abstract
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality of the state-of-the-art algorithms for finding such factors.
Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of an odd half-length, (b) computing centers of all Abelian square factors, (c) detecting an additive square factor in a length-n string of integers of magnitude n^{𝒪(1)}, and (d) a problem of computing a double 3-term arithmetic progression (i.e., finding indices i ≠ j such that (x_i+x_j)/2 = x_{(i+j)/2}) in a sequence of integers x₁,… ,x_n of magnitude n^{𝒪(1)}.
Problem (d) is essentially a convolution version of the AVERAGE problem that was proposed in a manuscript of Erickson. We obtain a conditional lower bound for it with the aid of techniques recently developed by Dudek et al. [STOC 2020]. Problem (d) immediately reduces to problem (c) and is a step in reductions to problems (a) and (b). In conditional lower bounds for problems (a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it using several string gadgets that include arbitrarily long Abelian-square-free strings.
Our reductions also imply conditional lower bounds for detecting Abelian squares in strings over a constant-sized alphabet. We also show a subquadratic upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].
Cite as
Jakub Radoszewski, Wojciech Rytter, Juliusz Straszyński, Tomasz Waleń, and Wiktor Zuba. Hardness of Detecting Abelian and Additive Square Factors in Strings. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 77:1-77:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{radoszewski_et_al:LIPIcs.ESA.2021.77,
author = {Radoszewski, Jakub and Rytter, Wojciech and Straszy\'{n}ski, Juliusz and Wale\'{n}, Tomasz and Zuba, Wiktor},
title = {{Hardness of Detecting Abelian and Additive Square Factors in Strings}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {77:1--77:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.77},
URN = {urn:nbn:de:0030-drops-146581},
doi = {10.4230/LIPIcs.ESA.2021.77},
annote = {Keywords: Abelian square, additive square, 3SUM problem}
}
Document
On Approximate Compressions for Connected Minor-Hitting Sets
Abstract
In the Connected ℱ-Deletion problem, ℱ is a fixed finite family of graphs and the objective is to compute a minimum set of vertices (or a vertex set of size at most k for some given k) such that (a) this set induces a connected subgraph of the given graph and (b) deleting this set results in a graph which excludes every F ∈ ℱ as a minor. In the area of kernelization, this problem is well known to exclude a polynomial kernel subject to standard complexity hypotheses even in very special cases such as ℱ = K₂, i.e., Connected Vertex Cover.
In this work, we give a (2+ε)-approximate polynomial compression for the Connected ℱ-Deletion problem when ℱ contains at least one planar graph. This is the first approximate polynomial compression result for this generic problem. As a corollary, we obtain the first approximate polynomial compression result for the special case of Connected η-Treewidth Deletion.
Cite as
M. S. Ramanujan. On Approximate Compressions for Connected Minor-Hitting Sets. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 78:1-78:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{ramanujan:LIPIcs.ESA.2021.78,
author = {Ramanujan, M. S.},
title = {{On Approximate Compressions for Connected Minor-Hitting Sets}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {78:1--78:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.78},
URN = {urn:nbn:de:0030-drops-146590},
doi = {10.4230/LIPIcs.ESA.2021.78},
annote = {Keywords: Parameterized Complexity, Kernelization, Approximation Algorithms}
}
Document
Restricted Adaptivity in Stochastic Scheduling
Abstract
We consider the stochastic scheduling problem of minimizing the expected makespan on m parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of 2, any (non-adaptive) fixed assignment policy has performance guarantee Ω((log m)/(log log m)). Although the performance of the latter class of policies are worse, there are applications in which non-adaptive policies are desired. In this work, we introduce the two classes of δ-delay and τ-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an 𝒪(log log m)-approximation for reasonably bounded parameters. In other words, an exponential improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any δ-delay and τ-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy.
Cite as
Guillaume Sagnol and Daniel Schmidt genannt Waldschmidt. Restricted Adaptivity in Stochastic Scheduling. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{sagnol_et_al:LIPIcs.ESA.2021.79,
author = {Sagnol, Guillaume and Schmidt genannt Waldschmidt, Daniel},
title = {{Restricted Adaptivity in Stochastic Scheduling}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {79:1--79:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.79},
URN = {urn:nbn:de:0030-drops-146603},
doi = {10.4230/LIPIcs.ESA.2021.79},
annote = {Keywords: stochastic scheduling, makespan minimzation, approximation algorithm, fixed assignment policy, non-anticipatory policy}
}
Document
A General Framework for Enumerating Equivalence Classes of Solutions
Abstract
When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e., to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. Furthermore, depending on the application, many of these solutions can be considered equivalent. The problem of an efficient enumeration of the equivalence classes or of one representative per class (without generating all the solutions), although identified as a need in many areas, has been addressed only for very few specific cases. In this paper, we provide a general framework that solves this problem in polynomial delay for a wide variety of contexts, including optimization ones that can be addressed by dynamic programming algorithms, and for certain types of equivalence relations between solutions.
Cite as
Yishu Wang, Arnaud Mary, Marie-France Sagot, and Blerina Sinaimeri. A General Framework for Enumerating Equivalence Classes of Solutions. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{wang_et_al:LIPIcs.ESA.2021.80,
author = {Wang, Yishu and Mary, Arnaud and Sagot, Marie-France and Sinaimeri, Blerina},
title = {{A General Framework for Enumerating Equivalence Classes of Solutions}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {80:1--80:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.80},
URN = {urn:nbn:de:0030-drops-146614},
doi = {10.4230/LIPIcs.ESA.2021.80},
annote = {Keywords: Enumeration algorithms, Equivalence relation, Dynamic programming}
}
Document
Engineering MultiQueues: Fast Relaxed Concurrent Priority Queues
Abstract
Priority queues with parallel access are an attractive data structure for applications like prioritized online scheduling, discrete event simulation, or greedy algorithms. However, a classical priority queue constitutes a severe bottleneck in this context, leading to very small throughput. Hence, there has been significant interest in concurrent priority queues with relaxed semantics. We investigate the complementary quality criteria rank error (how close are deleted elements to the global minimum) and delay (for each element x, how many elements with lower priority are deleted before x). In this paper, we introduce MultiQueues as a natural approach to relaxed priority queues based on multiple sequential priority queues. Their naturally high theoretical scalability is further enhanced by using three orthogonal ways of batching operations on the sequential queues. Experiments indicate that MultiQueues present a very good performance-quality tradeoff and considerably outperform competing approaches in at least one of these aspects.
We employ a seemingly paradoxical technique of "wait-free locking" that might be of more general interest to convert sequential data structures to relaxed concurrent data structures.
Cite as
Marvin Williams, Peter Sanders, and Roman Dementiev. Engineering MultiQueues: Fast Relaxed Concurrent Priority Queues. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 81:1-81:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{williams_et_al:LIPIcs.ESA.2021.81,
author = {Williams, Marvin and Sanders, Peter and Dementiev, Roman},
title = {{Engineering MultiQueues: Fast Relaxed Concurrent Priority Queues}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {81:1--81:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.81},
URN = {urn:nbn:de:0030-drops-146627},
doi = {10.4230/LIPIcs.ESA.2021.81},
annote = {Keywords: concurrent data structure, priority queues, randomized algorithms, wait-free locking}
}
Document
Evidence for Long-Tails in SLS Algorithms
Abstract
Stochastic local search (SLS) is a successful paradigm for solving the satisfiability problem of propositional logic. A recent development in this area involves solving not the original instance, but a modified, yet logically equivalent one [Jan{-}Hendrik Lorenz and Florian Wörz, 2020]. Empirically, this technique was found to be promising as it improves the performance of state-of-the-art SLS solvers.
Currently, there is only a shallow understanding of how this modification technique affects the runtimes of SLS solvers. Thus, we model this modification process and conduct an empirical analysis of the hardness of logically equivalent formulas. Our results are twofold. First, if the modification process is treated as a random process, a lognormal distribution perfectly characterizes the hardness; implying that the hardness is long-tailed. This means that the modification technique can be further improved by implementing an additional restart mechanism. Thus, as a second contribution, we theoretically prove that all algorithms exhibiting this long-tail property can be further improved by restarts. Consequently, all SAT solvers employing this modification technique can be enhanced.
Cite as
Florian Wörz and Jan-Hendrik Lorenz. Evidence for Long-Tails in SLS Algorithms. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 82:1-82:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
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@InProceedings{worz_et_al:LIPIcs.ESA.2021.82,
author = {W\"{o}rz, Florian and Lorenz, Jan-Hendrik},
title = {{Evidence for Long-Tails in SLS Algorithms}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {82:1--82:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus 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.2021.82},
URN = {urn:nbn:de:0030-drops-146634},
doi = {10.4230/LIPIcs.ESA.2021.82},
annote = {Keywords: Stochastic Local Search, Runtime Distribution, Statistical Analysis, Lognormal Distribution, Long-Tailed Distribution, SAT Solving}
}