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

We study the following KS₂(c) problem: let c ∈ ℝ^+ be some constant, and v₁,…, v_m ∈ ℝ^d be vectors such that ‖v_i‖² ≤ α for any i ∈ [m] and ∑_{i=1}^m ⟨v_i, x⟩² = 1 for any x ∈ ℝ^d with ‖x‖ = 1. The KS₂(c) problem asks to find some S ⊂ [m], such that it holds for all x ∈ ℝ^d with ‖x‖ = 1 that |∑_{i∈S} ⟨v_i, x⟩² - 1/2| ≤ c⋅√α, or report no if such S doesn't exist. Based on the work of Marcus et al. [Adam Marcus et al., 2013] and Weaver [Nicholas Weaver, 2004], the KS₂(c) problem can be seen as the algorithmic Kadison-Singer problem with parameter c ∈ ℝ^+.
Our first result is a randomised algorithm with one-sided error for the KS₂(c) problem such that (1) our algorithm finds a valid set S ⊂ [m] with probability at least 1-2/d, if such S exists, or (2) reports no with probability 1, if no valid sets exist. The algorithm has running time O(binom(m,n)⋅poly(m, d)) for n = O(d/ε² log(d) log(1/(c√α))), where ε is a parameter which controls the error of the algorithm. This presents the first algorithm for the Kadison-Singer problem whose running time is quasi-polynomial in m in a certain regime, although having exponential dependency on d. Moreover, it shows that the algorithmic Kadison-Singer problem is easier to solve in low dimensions. Our second result is on the computational complexity of the KS₂(c) problem. We show that the KS₂(1/(4√2)) problem is FNP-hard for general values of d, and solving the KS₂(1/(4√2)) problem is as hard as solving the NAE-3SAT problem.

Ben Jourdan, Peter Macgregor, and He Sun. Is the Algorithmic Kadison-Singer Problem Hard?. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 43:1-43:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jourdan_et_al:LIPIcs.ISAAC.2023.43, author = {Jourdan, Ben and Macgregor, Peter and Sun, He}, title = {{Is the Algorithmic Kadison-Singer Problem Hard?}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {43:1--43:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.43}, URN = {urn:nbn:de:0030-drops-193457}, doi = {10.4230/LIPIcs.ISAAC.2023.43}, annote = {Keywords: Kadison-Singer problem, spectral sparsification} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

A closed random walk of length 𝓁 on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step 𝓁, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics '21) and spectral graph theory (McKenzie et al., STOC '21). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length 𝓁 ⩾ 1 is Ω(𝓁^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of Ω(𝓁^{1/2}) holds for any regular graph.
First, we show that the support of a closed random walk of length 𝓁 is at least Ω(𝓁^{1/2} / √{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every 𝓁 ⩾ 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(𝓁^{1/2}/√{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be Ω(𝓁^{1/2}) for all 𝓁 ⩾ 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log 𝓁). This rules out a general polynomial lower bound in 𝓁 for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs.

Thomas Sauerwald, He Sun, and Danny Vagnozzi. The Support of Open Versus Closed Random Walks. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 103:1-103:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{sauerwald_et_al:LIPIcs.ICALP.2023.103, author = {Sauerwald, Thomas and Sun, He and Vagnozzi, Danny}, title = {{The Support of Open Versus Closed Random Walks}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {103:1--103:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.103}, URN = {urn:nbn:de:0030-drops-181556}, doi = {10.4230/LIPIcs.ICALP.2023.103}, annote = {Keywords: support of random walks, eigenvalue multiplicity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Designing efficient dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms and has witnessed many exciting recent developments in, e.g., dynamic matching (Wajc STOC'20) and decremental shortest paths (Chuzhoy and Khanna STOC'19). Compared to other graph primitives (e.g. spanning trees and matchings), designing such algorithms for graph spanners and (more broadly) graph sparsifiers poses a unique challenge since there is no fast deterministic algorithm known for static computation and the lack of a way to adjust the output slowly (known as "small recourse/replacements").
This paper presents the first non-trivial efficient adaptive algorithms for maintaining many sparsifiers against an adaptive adversary. Specifically, we present algorithms that maintain
1) a polylog(n)-spanner of size Õ(n) in polylog(n) amortized update time,
2) an O(k)-approximate cut sparsifier of size Õ(n) in Õ(n^{1/k}) amortized update time, and
3) a polylog(n)-approximate spectral sparsifier in polylog(n) amortized update time. Our bounds are the first non-trivial ones even when only the recourse is concerned. Our results hold even against a stronger adversary, who can access the random bits previously used by the algorithms and the amortized update time of all algorithms can be made worst-case by paying sub-polynomial factors. Our spanner result resolves an open question by Ahmed et al. (2019) and our results and techniques imply additional improvements over existing results, including (i) answering open questions about decremental single-source shortest paths by Chuzhoy and Khanna (STOC'19) and Gutenberg and Wulff-Nilsen (SODA'20), implying a nearly-quadratic time algorithm for approximating minimum-cost unit-capacity flow and (ii) de-amortizing a result of Abraham et al. (FOCS'16) for dynamic spectral sparsifiers.
Our results are based on two novel techniques. The first technique is a generic black-box reduction that allows us to assume that the graph is initially an expander with almost uniform-degree and, more importantly, stays as an almost uniform-degree expander while undergoing only edge deletions. The second technique is called proactive resampling: here we constantly re-sample parts of the input graph so that, independent of an adversary’s computational power, a desired structure of the underlying graph can be always maintained. Despite its simplicity, the analysis of this sampling scheme is far from trivial, because the adversary can potentially create dependencies between the random choices used by the algorithm. We believe these two techniques could be useful for developing other adaptive algorithms.

Aaron Bernstein, Jan van den Brand, Maximilian Probst Gutenberg, Danupon Nanongkai, Thatchaphol Saranurak, Aaron Sidford, and He Sun. Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bernstein_et_al:LIPIcs.ICALP.2022.20, author = {Bernstein, Aaron and van den Brand, Jan and Probst Gutenberg, Maximilian and Nanongkai, Danupon and Saranurak, Thatchaphol and Sidford, Aaron and Sun, He}, title = {{Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {20:1--20:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.20}, URN = {urn:nbn:de:0030-drops-163611}, doi = {10.4230/LIPIcs.ICALP.2022.20}, annote = {Keywords: dynamic graph algorithm, adaptive adversary, spanner, sparsifier} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

For any undirected graph G = (V,E) and a set E_W of candidate edges with E ∩ E_W = ∅, the (k,γ)-spectral augmentability problem is to find a set F of k edges from E_W with appropriate weighting, such that the algebraic connectivity of the resulting graph H = (V, E ∪ F) is least γ. Because of a tight connection between the algebraic connectivity and many other graph parameters, including the graph’s conductance and the mixing time of random walks in a graph, maximising the resulting graph’s algebraic connectivity by adding a small number of edges has been studied over the past 15 years, and has many practical applications in network optimisation.
In this work we present an approximate and efficient algorithm for the (k,γ)-spectral augmentability problem, and our algorithm runs in almost-linear time under a wide regime of parameters. Our main algorithm is based on the following two novel techniques developed in the paper, which might have applications beyond the (k,γ)-spectral augmentability problem:
- We present a fast algorithm for solving a feasibility version of an SDP for the algebraic connectivity maximisation problem from [Ghosh and Boyd, 2006]. Our algorithm is based on the classic primal-dual framework for solving SDP, which in turn uses the multiplicative weight update algorithm. We present a novel approach of unifying SDP constraints of different matrix and vector variables and give a good separation oracle accordingly.
- We present an efficient algorithm for the subgraph sparsification problem, and for a wide range of parameters our algorithm runs in almost-linear time, in contrast to the previously best known algorithm running in at least Ω(n²mk) time [Kolla et al., 2010]. Our analysis shows how the randomised BSS framework can be generalised in the setting of subgraph sparsification, and how the potential functions can be applied to approximately keep track of different subspaces.

Bogdan-Adrian Manghiuc, Pan Peng, and He Sun. Augmenting the Algebraic Connectivity of Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 70:1-70:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{manghiuc_et_al:LIPIcs.ESA.2020.70, author = {Manghiuc, Bogdan-Adrian and Peng, Pan and Sun, He}, title = {{Augmenting the Algebraic Connectivity of Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {70:1--70:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.70}, URN = {urn:nbn:de:0030-drops-129367}, doi = {10.4230/LIPIcs.ESA.2020.70}, annote = {Keywords: Graph sparsification, Algebraic connectivity, Semidefinite programming} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We study spectral approaches for the MAX-2-LIN(k) problem, in which we are given a system of m linear equations of the form x_i - x_j is equivalent to c_{ij} mod k, and required to find an assignment to the n variables {x_i} that maximises the total number of satisfied equations.
We consider Hermitian Laplacians related to this problem, and prove a Cheeger inequality that relates the smallest eigenvalue of a Hermitian Laplacian to the maximum number of satisfied equations of a MAX-2-LIN(k) instance I. We develop an O~(kn^2) time algorithm that, for any (1-epsilon)-satisfiable instance, produces an assignment satisfying a (1 - O(k)sqrt{epsilon})-fraction of equations. We also present a subquadratic-time algorithm that, when the graph associated with I is an expander, produces an assignment satisfying a (1- O(k^2)epsilon)-fraction of the equations. Our Cheeger inequality and first algorithm can be seen as generalisations of the Cheeger inequality and algorithm for MAX-CUT developed by Trevisan.

Huan Li, He Sun, and Luca Zanetti. Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{li_et_al:LIPIcs.ESA.2019.71, author = {Li, Huan and Sun, He and Zanetti, Luca}, title = {{Hermitian Laplacians and a Cheeger Inequality for the Max-2-Lin Problem}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {71:1--71:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.71}, URN = {urn:nbn:de:0030-drops-111926}, doi = {10.4230/LIPIcs.ESA.2019.71}, annote = {Keywords: Spectral methods, Hermitian Laplacians, the Max-2-Lin problem, Unique Games} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m=n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m>=n.

Karl Bringmann, Thomas Sauerwald, Alexandre Stauffer, and He Sun. Balls into bins via local search: cover time and maximum load. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 187-198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bringmann_et_al:LIPIcs.STACS.2014.187, author = {Bringmann, Karl and Sauerwald, Thomas and Stauffer, Alexandre and Sun, He}, title = {{Balls into bins via local search: cover time and maximum load}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {187--198}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.187}, URN = {urn:nbn:de:0030-drops-44570}, doi = {10.4230/LIPIcs.STACS.2014.187}, annote = {Keywords: Balls and Bins, Stochastic Process, Randomized Algorithm} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We consider the classical rumor spreading problem, where a piece of information must be disseminated from a single node to all n nodes of a given network. We devise two simple push-based protocols, in which nodes choose the neighbor they send the information to in each round using pairwise independent hash functions, or a pseudo-random generator, respectively. For several well-studied topologies our algorithms use exponentially fewer random bits than previous protocols. For example, in complete graphs, expanders, and random graphs only a polylogarithmic number of random bits are needed in total to spread the rumor in O(log n) rounds with high probability.
Previous explicit algorithms require Omega(n) random bits to achieve the same round complexity. For complete graphs, the amount of randomness used by our hashing-based algorithm is within an O(log n)-factor of the theoretical minimum determined by [Giakkoupis and Woelfel, 2011].

George Giakkoupis, Thomas Sauerwald, He Sun, and Philipp Woelfel. Low Randomness Rumor Spreading via Hashing. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 314-325, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{giakkoupis_et_al:LIPIcs.STACS.2012.314, author = {Giakkoupis, George and Sauerwald, Thomas and Sun, He and Woelfel, Philipp}, title = {{Low Randomness Rumor Spreading via Hashing}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {314--325}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.314}, URN = {urn:nbn:de:0030-drops-34417}, doi = {10.4230/LIPIcs.STACS.2012.314}, annote = {Keywords: Parallel and Distributed Computing, Randomness, Rumor Spreading} }

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**Published in:** LIPIcs, Volume 172, 20th International Workshop on Algorithms in Bioinformatics (WABI 2020)

Considering a set of intervals on the real line, an interval graph records these intervals as nodes and their intersections as edges. Identifying (i.e. merging) pairs of nodes in an interval graph results in a multiple-interval graph. Given only the nodes and the edges of the multiple-interval graph without knowing the underlying intervals, we are interested in the following questions. Can one determine how many intervals correspond to each node? Can one compute a walk over the multiple-interval graph nodes that reflects the ordering of the original intervals? These questions are closely related to linked-read DNA sequencing, where barcodes are assigned to long molecules whose intersection graph forms an interval graph. Each barcode may correspond to multiple molecules, which complicates downstream analysis, and corresponds to the identification of nodes of the corresponding interval graph. Resolving the above graph-theoretic problems would facilitate analyses of linked-reads sequencing data, through enabling the conceptual separation of barcodes into molecules and providing, through the molecules order, a skeleton for accurately assembling the genome. Here, we propose a framework that takes as input an arbitrary intersection graph (such as an overlap graph of barcodes) and constructs a heuristic approximation of the ordering of the original intervals.

Yoann Dufresne, Chen Sun, Pierre Marijon, Dominique Lavenier, Cedric Chauve, and Rayan Chikhi. A Graph-Theoretic Barcode Ordering Model for Linked-Reads. In 20th International Workshop on Algorithms in Bioinformatics (WABI 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 172, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dufresne_et_al:LIPIcs.WABI.2020.11, author = {Dufresne, Yoann and Sun, Chen and Marijon, Pierre and Lavenier, Dominique and Chauve, Cedric and Chikhi, Rayan}, title = {{A Graph-Theoretic Barcode Ordering Model for Linked-Reads}}, booktitle = {20th International Workshop on Algorithms in Bioinformatics (WABI 2020)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-161-0}, ISSN = {1868-8969}, year = {2020}, volume = {172}, editor = {Kingsford, Carl and Pisanti, Nadia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2020.11}, URN = {urn:nbn:de:0030-drops-128001}, doi = {10.4230/LIPIcs.WABI.2020.11}, annote = {Keywords: DNA sequencing, graph algorithms, linked-reads, interval graphs, cliques} }

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