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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We revisit a natural variant of the geometric set cover problem, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set S of points and a set ℛ of geometric objects, and the goal is to find a subset ℛ^* ⊆ ℛ to cover all points in S such that the membership of S with respect to ℛ^*, denoted by memb(S,ℛ^*), is minimized, where memb(S,ℛ^*) = max_{p ∈ S} |{R ∈ ℛ^*: p ∈ R}|. We give the first polynomial-time approximation algorithms for MMGSC in ℝ². Specifically, we achieve the following two main results.
- We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time n^{O(opt)} where opt is the optimum of the problem (i.e., the minimum membership).
- We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of o(log n) in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find ℛ^* ⊆ ℛ to cover S such that the ply of ℛ^* is minimized, where the ply is defined as the maximum number of objects in ℛ^* which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in O(n^{12}) time. We give a significantly simpler constant-approximation algorithm with near-linear running time.

Sayan Bandyapadhyay, William Lochet, Saket Saurabh, and Jie Xue. Minimum-Membership Geometric Set Cover, Revisited. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 11:1-11:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.11, author = {Bandyapadhyay, Sayan and Lochet, William and Saurabh, Saket and Xue, Jie}, title = {{Minimum-Membership Geometric Set Cover, Revisited}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.11}, URN = {urn:nbn:de:0030-drops-178610}, doi = {10.4230/LIPIcs.SoCG.2023.11}, annote = {Keywords: geometric set cover, geometric optimization, approximation algorithms} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into k clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints (Capacitated Sum of Radii ). In particular, we give a (15+ε)-approximation algorithm that runs in 2^𝒪(k²log k) ⋅ n³ time.
When capacities are uniform, we obtain the following improved approximation bounds.
- A (4 + ε)-approximation with running time 2^𝒪(klog(k/ε)) n³, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020].
- A (2 + ε)-approximation with running time 2^𝒪(k/ε² ⋅log(k/ε)) dn³ and a (1+ε)-approxim- ation with running time 2^𝒪(kdlog ((k/ε))) n³ in the Euclidean space. Here d is the dimension.
- A (1 + ε)-approximation in the Euclidean space with running time 2^𝒪(k/ε² ⋅log(k/ε)) dn³ if we are allowed to violate the capacities by (1 + ε)-factor. We complement this result by showing that there is no (1 + ε)-approximation algorithm running in time f(k)⋅ n^𝒪(1), if any capacity violation is not allowed.

Sayan Bandyapadhyay, William Lochet, and Saket Saurabh. FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 12:1-12:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.12, author = {Bandyapadhyay, Sayan and Lochet, William and Saurabh, Saket}, title = {{FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.12}, URN = {urn:nbn:de:0030-drops-178628}, doi = {10.4230/LIPIcs.SoCG.2023.12}, annote = {Keywords: Clustering, FPT-approximation} }

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

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set 𝒟 of n unit disks inducing a unit-disk graph G_𝒟 and a number p ∈ [n], one can partition 𝒟 into p subsets 𝒟₁,… ,𝒟_p such that for every i ∈ [p] and every 𝒟' ⊆ 𝒟_i, the graph obtained from G_𝒟 by contracting all edges between the vertices in 𝒟_i $1𝒟' admits a tree decomposition in which each bag consists of O(p+|𝒟'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22].
By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} ⋅ n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} ⋅ n^{O(1)} time assuming the ETH.

Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, and Jie Xue. True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 11:1-11:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2022.11, author = {Bandyapadhyay, Sayan and Lochet, William and Lokshtanov, Daniel and Saurabh, Saket and Xue, Jie}, title = {{True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {11:1--11:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.11}, URN = {urn:nbn:de:0030-drops-160190}, doi = {10.4230/LIPIcs.SoCG.2022.11}, annote = {Keywords: unit-disk graphs, tree decomposition, contraction decomposition, bipartization} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. [Ivona Bezáková et al., 2019] proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O(k)}⋅ n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O (k)}· n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs.
In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k(diam(G)denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path above Diameter is solvable in polynomial time for each k ∈ {1,..., 4} and is NP-complete for every k ≥ 5. The parameterized complexity of Longest Detour on general directed graphs remains an interesting open problem.

Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov, Kirill Simonov, and Saket Saurabh. Detours in Directed Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 29:1-29:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2022.29, author = {Fomin, Fedor V. and Golovach, Petr A. and Lochet, William and Sagunov, Danil and Simonov, Kirill and Saurabh, Saket}, title = {{Detours in Directed Graphs}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {29:1--29:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.29}, URN = {urn:nbn:de:0030-drops-158390}, doi = {10.4230/LIPIcs.STACS.2022.29}, annote = {Keywords: longest path, longest detour, diameter, directed graphs, parameterized complexity} }

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**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most Ω(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized.

William Lochet, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Exploiting Dense Structures in Parameterized Complexity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 50:1-50:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lochet_et_al:LIPIcs.STACS.2021.50, author = {Lochet, William and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav}, title = {{Exploiting Dense Structures in Parameterized Complexity}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {50:1--50:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.50}, URN = {urn:nbn:de:0030-drops-136950}, doi = {10.4230/LIPIcs.STACS.2021.50}, annote = {Keywords: Dense graphs, disjoint paths, odd cycle transversal, kernels} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

For a fixed graph H, the H-free Edge Editing problem asks whether we can modify a given graph G by adding or deleting at most k edges such that the resulting graph does not contain H as an induced subgraph. The problem is known to be NP-complete for all fixed H with at least 3 vertices and it admits a 2^O(k)n^O(1) algorithm. Cai and Cai [Algorithmica (2015) 71:731–757] showed that, assuming coNP ⊈ NP/poly, H-free Edge Editing does not admit a polynomial kernel whenever H or its complement is a path or a cycle with at least 4 edges or a 3-connected graph with at least one edge missing. Based on their result, very recently Marx and Sandeep [ESA 2020] conjectured that if H is a graph with at least 5 vertices, then H-free Edge Editing has a polynomial kernel if and only if H is a complete or empty graph, unless coNP ⊆ NP/poly. Furthermore they gave a list of 9 graphs, each with five vertices, such that if H-free Edge Editing for these graphs does not admit a polynomial kernel, then the conjecture is true. Therefore, resolving the kernelization of H-free Edge Editing for graphs H with 4 and 5 vertices plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs H on 4 vertices. Namely, we give the first polynomial kernel for Paw-free Edge Editing with O(k⁶) vertices.

Eduard Eiben, William Lochet, and Saket Saurabh. A Polynomial Kernel for Paw-Free Editing. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 10:1-10:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2020.10, author = {Eiben, Eduard and Lochet, William and Saurabh, Saket}, title = {{A Polynomial Kernel for Paw-Free Editing}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.10}, URN = {urn:nbn:de:0030-drops-133136}, doi = {10.4230/LIPIcs.IPEC.2020.10}, annote = {Keywords: Kernelization, Paw-free graph, H-free editing, graph modification problem} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m-k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m-k arcs. We show that
- Directed Multiplicative Spanner admits a polynomial kernel of size 𝒪(k⁴t⁵) and can be solved in randomized (4t)^k⋅ n^𝒪(1) time,
- Directed Additive Spanner is W[1]-hard when parameterized by k even if t = 1 and the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is FPT when parameterized by t and k.

Fedor V. Fomin, Petr A. Golovach, William Lochet, Pranabendu Misra, Saket Saurabh, and Roohani Sharma. Parameterized Complexity of Directed Spanner Problems. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 12:1-12:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.IPEC.2020.12, author = {Fomin, Fedor V. and Golovach, Petr A. and Lochet, William and Misra, Pranabendu and Saurabh, Saket and Sharma, Roohani}, title = {{Parameterized Complexity of Directed Spanner Problems}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {12:1--12:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.12}, URN = {urn:nbn:de:0030-drops-133156}, doi = {10.4230/LIPIcs.IPEC.2020.12}, annote = {Keywords: Graph spanners, directed graphs, parameterized complexity, kernelization} }

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

The line graph of a graph G is the graph L(G) whose vertex set is the edge set of G and there is an edge between e,f ∈ E(G) if e and f share an endpoint in G. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most k edges from the input graph G such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with O(k⁵) vertices. This answers an open question posed by Falk Hüffner at Workshop on Kernels (WorKer) in 2013.

Eduard Eiben and William Lochet. A Polynomial Kernel for Line Graph Deletion. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 42:1-42:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2020.42, author = {Eiben, Eduard and Lochet, William}, title = {{A Polynomial Kernel for Line Graph Deletion}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {42:1--42:15}, 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.2020.42}, URN = {urn:nbn:de:0030-drops-129088}, doi = {10.4230/LIPIcs.ESA.2020.42}, annote = {Keywords: Kernelization, line graphs, H-free editing, graph modification problem} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

In the past decade, the design of fault tolerant data structures for networks has become a central topic of research. Particular attention has been given to the construction of a subgraph H of a given digraph D with as fewest arcs/vertices as possible such that, after the failure of any set F of at most k ≥ 1 arcs, testing whether D-F has a certain property P is equivalent to testing whether H-F has that property. Here, reachability (or, more generally, distance preservation) is the most basic requirement to maintain to ensure that the network functions properly. Given a vertex s ∈ V(D), Baswana et al. [STOC'16] presented a construction of H with O(2^kn) arcs in time O(2^{k}nm) where n=|V(D)| and m= |E(D)| such that for any vertex v ∈ V(D): if there exists a path from s to v in D-F, then there also exists a path from s to v in H-F. Additionally, they gave a tight matching lower bound. While the question of the improvement of the dependency on k arises for special classes of digraphs, an arguably more basic research direction concerns the dependency on n (for reachability between a pair of vertices s,t ∈ V(D)) - which are the largest classes of digraphs where the dependency on n can be made sublinear, logarithmic or even constant? Already for the simple classes of directed paths and tournaments, Ω(n) arcs are mandatory. Nevertheless, we prove that "almost acyclicity" suffices to eliminate the dependency on n entirely for a broad class of dense digraphs called bounded independence digraphs. Also, the dependence in k is only a polynomial factor for this class of digraphs. In fact, our sparsification procedure extends to preserve parity-based reachability. Additionally, it finds notable applications in Kernelization: we prove that the classic Directed Feedback Arc Set (DFAS) problem as well as Directed Edge Odd Cycle Transversal (DEOCT) (which, in sharp contrast to DFAS, is W[1]-hard on general digraphs) admit polynomial kernels on bounded independence digraphs. In fact, for any p ∈ N, we can design a polynomial kernel for the problem of hitting all cycles of length ℓ where (ℓ mod p = 1). As a complementary result, we prove that DEOCT is NP-hard on tournaments by establishing a combinatorial identity between the minimum size of a feedback arc set and the minimum size of an edge odd cycle transversal. In passing, we also improve upon the running time of the sub-exponential FPT algorithm for DFAS in digraphs of bounded independence number given by Misra et at. [FSTTCS 2018], and give the first sub-exponential FPT algorithm for DEOCT in digraphs of bounded independence number.

William Lochet, Daniel Lokshtanov, Pranabendu Misra, Saket Saurabh, Roohani Sharma, and Meirav Zehavi. Fault Tolerant Subgraphs with Applications in Kernelization. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 47:1-47:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lochet_et_al:LIPIcs.ITCS.2020.47, author = {Lochet, William and Lokshtanov, Daniel and Misra, Pranabendu and Saurabh, Saket and Sharma, Roohani and Zehavi, Meirav}, title = {{Fault Tolerant Subgraphs with Applications in Kernelization}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {47:1--47:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.47}, URN = {urn:nbn:de:0030-drops-117326}, doi = {10.4230/LIPIcs.ITCS.2020.47}, annote = {Keywords: sparsification, kernelization, fault tolerant subgraphs, directed feedback arc set, directed edge odd cycle transversal, bounded independence number digraphs} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively.
- Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree.
- Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor.
- Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n).
- Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n).

Pallavi Jain, Lawqueen Kanesh, William Lochet, Saket Saurabh, and Roohani Sharma. Exact and Approximate Digraph Bandwidth. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 18:1-18:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jain_et_al:LIPIcs.FSTTCS.2019.18, author = {Jain, Pallavi and Kanesh, Lawqueen and Lochet, William and Saurabh, Saket and Sharma, Roohani}, title = {{Exact and Approximate Digraph Bandwidth}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.18}, URN = {urn:nbn:de:0030-drops-115802}, doi = {10.4230/LIPIcs.FSTTCS.2019.18}, annote = {Keywords: directed bandwidth, digraph bandwidth, approximation scheme, exact exponential algorithms} }

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