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

A framework consists of an undirected graph G and a matroid M whose elements correspond to the vertices of G. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank k in frameworks. More precisely, for vertices s and t of G, and an integer k, they gave FPT algorithms parameterized by k deciding whether there is an (s,t)-path in G whose vertex set contains a subset of elements of M of rank k. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open.
We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph G is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid M is represented over rationals.
Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, and Giannos Stamoulis. Computing Paths of Large Rank in Planar Frameworks Deterministically. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 32:1-32:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ISAAC.2023.32, author = {Fomin, Fedor V. and Golovach, Petr A. and Korhonen, Tuukka and Stamoulis, Giannos}, title = {{Computing Paths of Large Rank in Planar Frameworks Deterministically}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {32:1--32:15}, 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.2023.32}, URN = {urn:nbn:de:0030-drops-193341}, doi = {10.4230/LIPIcs.ISAAC.2023.32}, annote = {Keywords: Planar graph, longest path, linear matroid, irrelevant vertex} }

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

We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is "close" to each other. More precisely, for a family of n points, an integer k, and a real number d > 0, we ask whether at most k points could be relocated, each point at distance at most d from its original location, such that the distance between each pair of points is at least a fixed constant, say 1. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with 𝒪(d²k³) points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by k and d. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in k alone, unless NP ⊆ coNP/poly.

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Kernelization for Spreading Points. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 48:1-48:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2023.48, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav}, title = {{Kernelization for Spreading Points}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {48:1--48:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.48}, URN = {urn:nbn:de:0030-drops-187017}, doi = {10.4230/LIPIcs.ESA.2023.48}, annote = {Keywords: parameterized algorithms, kernelization, spreading points, distant representatives, unit disk packing} }

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

We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with 𝒪(k^d) bits (𝒪(k^d) sets and 𝒪(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization.
In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (𝒪(k^{1+ε})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+ε)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments.

Fedor V. Fomin, Tien-Nam Le, Daniel Lokshtanov, Saket Saurabh, Stéphan Thomassé, and Meirav Zehavi. Lossy Kernelization for (Implicit) Hitting Set Problems. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 49:1-49:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2023.49, author = {Fomin, Fedor V. and Le, Tien-Nam and Lokshtanov, Daniel and Saurabh, Saket and Thomass\'{e}, St\'{e}phan and Zehavi, Meirav}, title = {{Lossy Kernelization for (Implicit) Hitting Set Problems}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {49:1--49:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.49}, URN = {urn:nbn:de:0030-drops-187020}, doi = {10.4230/LIPIcs.ESA.2023.49}, annote = {Keywords: Hitting Set, Lossy Kernelization} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0 ≤ α ≤ 1, the question is whether there is a set S ⊆ V of size k with a specified coverage function cov_α(S) at least p (or at most p for the minimization version). The coverage function cov_α(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1 - α. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and Max (k,n-k)-Cut.
A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Max k-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α > 0 and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α > 1/3 and minimization with α < 1/3.

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Tomohiro Koana. FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 46:1-46:8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.MFCS.2023.46, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Koana, Tomohiro}, title = {{FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {46:1--46:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.46}, URN = {urn:nbn:de:0030-drops-185806}, doi = {10.4230/LIPIcs.MFCS.2023.46}, annote = {Keywords: Partial Vertex Cover, Approximation Algorithms, Max Cut} }

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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)

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m/2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n/2 + k for some k ≥ 1 and how difficult is it to find such a vertex cover?
The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac’s theorem from 1952. Dirac’s theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ(G) ≤ n/2, the length of the longest cycle L is at least 2δ(G). Thus the "essential" part of finding the longest cycle is in approximating the "offset" k = L - 2δ(G). The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2δ(G)+Ω(f(k)).

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Approximating Long Cycle Above Dirac’s Guarantee. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 60:1-60:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2023.60, author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill}, title = {{Approximating Long Cycle Above Dirac’s Guarantee}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {60:1--60:18}, 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.2023.60}, URN = {urn:nbn:de:0030-drops-181128}, doi = {10.4230/LIPIcs.ICALP.2023.60}, annote = {Keywords: Longest path, longest cycle, approximation algorithms, above guarantee parameterization, minimum degree, Dirac theorem} }

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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)

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or G-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle’s theorem does not apply.

Fedor V. Fomin, Petr A. Golovach, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Compound Logics for Modification Problems. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 61:1-61:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2023.61, author = {Fomin, Fedor V. and Golovach, Petr A. and Sau, Ignasi and Stamoulis, Giannos and Thilikos, Dimitrios M.}, title = {{Compound Logics for Modification Problems}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {61:1--61:21}, 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.2023.61}, URN = {urn:nbn:de:0030-drops-181137}, doi = {10.4230/LIPIcs.ICALP.2023.61}, annote = {Keywords: Algorithmic meta-theorems, Graph modification problems, Model-checking, Graph minors, First-order logic, Monadic second-order logic, Flat Wall theorem, Irrelevant vertex technique} }

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

Designing coresets - small-space sketches of the data preserving cost of the solutions within (1± ε)-approximate factor - is an important research direction in the study of center-based k-clustering problems, such as k-means or k-median. Feldman and Langberg [STOC'11] have shown that for k-clustering of n points in general metrics, it is possible to obtain coresets whose size depends logarithmically in n. Moreover, such a dependency in n is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independent of n for special metrics, like d-dimensional Euclidean space [Huang, Vishnoi, STOC'20], doubling metrics [Huang, Jiang, Li, Wu, FOCS'18], metrics of graphs of bounded treewidth [Baker, Braverman, Huang, Jiang, Krauthgamer, Wu, ICML’20], or graphs excluding a fixed minor [Braverman, Jiang, Krauthgamer, Wu, SODA’21].
In this paper, we provide the first constructions of coresets whose size does not depend on n for k-clustering in the metrics induced by geometric intersection graphs. For example, we obtain (k log²k)/ε^𝒪(1) size coresets for k-clustering in Euclidean-weighted unit-disk graphs (UDGs) and unit-square graphs (USGs). These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining coresets whose size is independent of n. The proof of our theorem builds on the recent work of Cohen-Addad, Saulpic, and Schwiegelshohn [STOC '21], which ensures small-sized coresets conditioned on the existence of an interesting set of centers, called centroid set. The main technical contribution of our work is the proof of the existence of such a small-sized centroid set for graphs that satisfy the two canonical properties. Loosely speaking, the metrics of geometric intersection graphs are "similar" to the Euclidean metrics for points that are close, and to the shortest path metrics of planar graphs for points that are far apart. The main technical challenge in constructing centroid sets of small sizes is in combining these two very different metrics.
The new coreset construction helps to design the first (1+ε)-approximation for center-based clustering problems in UDGs and USGs, that is fixed-parameter tractable in k and ε (FPT-AS).

Sayan Bandyapadhyay, Fedor V. Fomin, and Tanmay Inamdar. Coresets for Clustering in Geometric Intersection Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.SoCG.2023.10, author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Inamdar, Tanmay}, title = {{Coresets for Clustering in Geometric Intersection Graphs}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {10:1--10:16}, 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.2023.10}, URN = {urn:nbn:de:0030-drops-178605}, doi = {10.4230/LIPIcs.SoCG.2023.10}, annote = {Keywords: k-median, k-means, clustering, coresets, geometric graphs} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In this paper, we consider the Minimum-Load k-Clustering/Facility Location (MLkC) problem where we are given a set P of n points in a metric space that we have to cluster and an integer k > 0 that denotes the number of clusters. Additionally, we are given a set F of cluster centers in the same metric space. The goal is to select a set C ⊆ F of k centers and assign each point in P to a center in C, such that the maximum load over all centers is minimized. Here the load of a center is the sum of the distances between it and the points assigned to it.
Although clustering/facility location problems have rich literature, the minimum-load objective has not been studied substantially, and hence MLkC has remained a poorly understood problem. More interestingly, the problem is notoriously hard even in some special cases including the one in line metrics as shown by Ahmadian et al. [APPROX 2014, ACM Trans. Algorithms 2018]. They also show APX-hardness of the problem in the plane. On the other hand, the best-known approximation factor for MLkC is O(k), even in the plane.
In this work, we study a fair version of MLkC inspired by the work of Chierichetti et al. [NeurIPS, 2017]. Here the input points are partitioned into 𝓁 protected groups, and only clusters that proportionally represent each group are allowed. MLkC is the special case with 𝓁 = 1. For the fair version, we are able to obtain a randomized 3-approximation algorithm in f(k,𝓁)⋅ n^O(1) time. Also, our scheme leads to an improved (1 + ε)-approximation in the case of Euclidean norm with the same running time (depending also linearly on the dimension d). Our results imply the same approximations for MLkC with running time f(k)⋅ n^O(1), achieving the first constant-factor FPT approximations for this problem in general and Euclidean metric spaces.

Sayan Bandyapadhyay, Fedor V. Fomin, Petr A. Golovach, Nidhi Purohit, and Kirill Simonov. FPT Approximation for Fair Minimum-Load Clustering. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.IPEC.2022.4, author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Golovach, Petr A. and Purohit, Nidhi and Simonov, Kirill}, title = {{FPT Approximation for Fair Minimum-Load Clustering}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.4}, URN = {urn:nbn:de:0030-drops-173600}, doi = {10.4230/LIPIcs.IPEC.2022.4}, annote = {Keywords: fair clustering, load balancing, parameterized approximation} }

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**Published in:** LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

In this paper we initiate a systematic study of exact algorithms for some of the well known clustering problems, namely k-MEDIAN and k-MEANS. In k-MEDIAN, the input consists of a set X of n points belonging to a metric space, and the task is to select a subset C ⊆ X of k points as centers, such that the sum of the distances of every point to its nearest center is minimized. In k-MEANS, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time max_{k ≤ n} {n choose k} n^𝒪(1) = 𝒪^*(2ⁿ) (here, 𝒪^*(⋅) notation hides polynomial factors in n). In this paper we design first non-trivial exact algorithms for these problems. In particular, we obtain an 𝒪^*((1.89)ⁿ) time exact algorithm for k-MEDIAN that works for any value of k. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space - it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for k-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless the Exponential Time Hypothesis fails, there is no algorithm for these problems running in time 2^o(n)⋅n^𝒪(1).
Finally, we consider the "facility location" or "supplier" versions of these clustering problems, where, in addition to the set X we are additionally given a set of m candidate centers (or facilities) F, and objective is to find a subset of k centers from F. The goal is still to minimize the k-Median/k-Means/k-Center objective. For these versions we give a 𝒪(2ⁿ (mn)^𝒪(1)) time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the "supplier" versions of these problems do not admit an exact algorithm running in time 2^{(1-ε) n} (mn)^𝒪(1).

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Nidhi Purohit, and Saket Saurabh. Exact Exponential Algorithms for Clustering Problems. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 13:1-13:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fomin_et_al:LIPIcs.IPEC.2022.13, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Purohit, Nidhi and Saurabh, Saket}, title = {{Exact Exponential Algorithms for Clustering Problems}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.13}, URN = {urn:nbn:de:0030-drops-173691}, doi = {10.4230/LIPIcs.IPEC.2022.13}, annote = {Keywords: clustering, k-median, k-means, exact algorithms} }

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

In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2^𝒪(k)⋅n^𝒪(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Longest Cycle Above Erdős-Gallai Bound. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 55:1-55:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2022.55, author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill}, title = {{Longest Cycle Above Erd\H{o}s-Gallai Bound}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {55:1--55:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.55}, URN = {urn:nbn:de:0030-drops-169935}, doi = {10.4230/LIPIcs.ESA.2022.55}, annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization, average degree, Erd\H{o}s and Gallai theorem} }

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Invited Talk

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We discuss recent algorithmic extensions of two classic results of extremal combinatorics about long paths in graphs. First, the theorem of Dirac from 1952 asserts that a 2-connected graph G with the minimum vertex degree d > 1, is either Hamiltonian or contains a cycle of length at least 2d. Second, the theorem of Erdős-Gallai from 1959, states that a graph G with the average vertex degree D > 1, contains a cycle of length at least D. The proofs of these theorems are constructive, they provide polynomial-time algorithms constructing cycles of lengths 2d and D. We extend these algorithmic results by showing that each of the problems, to decide whether a 2-connected graph contains a cycle of length at least 2d+k or of a cycle of length at least D+k, is fixed-parameter tractable parameterized by k.

Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms (Invited Talk). In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 1:1-1:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fomin_et_al:LIPIcs.MFCS.2022.1, author = {Fomin, Fedor V. and Golovach, Petr A. and Sagunov, Danil and Simonov, Kirill}, title = {{Long Cycles in Graphs: Extremal Combinatorics Meets Parameterized Algorithms}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {1:1--1:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.1}, URN = {urn:nbn:de:0030-drops-167999}, doi = {10.4230/LIPIcs.MFCS.2022.1}, annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization, average degree, dense graph, Dirac theorem, Erd\H{o}s-Gallai theorem} }

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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)

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of n equal disks packed into a rectangle and integers k and h, we ask whether it is possible by changing positions of at most h disks to pack n+k disks. Thus the problem of packing equal disks is the special case of our problem with n = h = 0.
While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for h = 0. Our main algorithmic contribution is an algorithm that solves the repacking problem in time (h+k)^𝒪(h+k)⋅|I|^𝒪(1), where |I| is the input size. That is, the problem is fixed-parameter tractable parameterized by k and h.

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Meirav Zehavi. (Re)packing Equal Disks into Rectangle. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 60:1-60:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2022.60, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Zehavi, Meirav}, title = {{(Re)packing Equal Disks into Rectangle}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {60:1--60:17}, 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.2022.60}, URN = {urn:nbn:de:0030-drops-164011}, doi = {10.4230/LIPIcs.ICALP.2022.60}, annote = {Keywords: circle packing, unit disks, parameterized complexity, fixed-parameter tractability} }

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

De Berg et al. in [SICOMP 2020] gave an algorithmic framework for subexponential algorithms on geometric graphs with tight (up to ETH) running times. This framework is based on dynamic programming on graphs of weighted treewidth resulting in algorithms that use super-polynomial space. We introduce the notion of weighted treedepth and use it to refine the framework of de Berg et al. for obtaining polynomial space (with tight running times) on geometric graphs. As a result, we prove that for any fixed dimension d ≥ 2 on intersection graphs of similarly-sized fat objects many well-known graph problems including Independent Set, r-Dominating Set for constant r, Cycle Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable in time 2^𝒪(n^{1-1/d}) and within polynomial space.

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, and Saket Saurabh. ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 21:1-21:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2021.21, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket}, title = {{ETH Tight Algorithms for Geometric Intersection Graphs: Now in Polynomial Space}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.21}, URN = {urn:nbn:de:0030-drops-155323}, doi = {10.4230/LIPIcs.FSTTCS.2021.21}, annote = {Keywords: Subexponential Algorithms, Geometric Intersection Graphs, Treedepth, Treewidth} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We develop new algorithmic methods with provable guarantees for feature selection in regard to categorical data clustering. While feature selection is one of the most common approaches to reduce dimensionality in practice, most of the known feature selection methods are heuristics. We study the following mathematical model. We assume that there are some inadvertent (or undesirable) features of the input data that unnecessarily increase the cost of clustering. Consequently, we want to select a subset of the original features from the data such that there is a small-cost clustering on the selected features. More precisely, for given integers l (the number of irrelevant features) and k (the number of clusters), budget B, and a set of n categorical data points (represented by m-dimensional vectors whose elements belong to a finite set of values Σ), we want to select m-l relevant features such that the cost of any optimal k-clustering on these features does not exceed B. Here the cost of a cluster is the sum of Hamming distances (l0-distances) between the selected features of the elements of the cluster and its center. The clustering cost is the total sum of the costs of the clusters.
We use the framework of parameterized complexity to identify how the complexity of the problem depends on parameters k, B, and |Σ|. Our main result is an algorithm that solves the Feature Selection problem in time f(k,B,|Σ|)⋅m^{g(k,|Σ|)}⋅n² for some functions f and g. In other words, the problem is fixed-parameter tractable parameterized by B when |Σ| and k are constants. Our algorithm for Feature Selection is based on a solution to a more general problem, Constrained Clustering with Outliers. In this problem, we want to delete a certain number of outliers such that the remaining points could be clustered around centers satisfying specific constraints. One interesting fact about Constrained Clustering with Outliers is that besides Feature Selection, it encompasses many other fundamental problems regarding categorical data such as Robust Clustering, Binary and Boolean Low-rank Matrix Approximation with Outliers, and Binary Robust Projective Clustering. Thus as a byproduct of our theorem, we obtain algorithms for all these problems. We also complement our algorithmic findings with complexity lower bounds.

Sayan Bandyapadhyay, Fedor V. Fomin, Petr A. Golovach, and Kirill Simonov. Parameterized Complexity of Feature Selection for Categorical Data Clustering. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 14:1-14:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.MFCS.2021.14, author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Golovach, Petr A. and Simonov, Kirill}, title = {{Parameterized Complexity of Feature Selection for Categorical Data Clustering}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.14}, URN = {urn:nbn:de:0030-drops-144544}, doi = {10.4230/LIPIcs.MFCS.2021.14}, annote = {Keywords: Robust clustering, PCA, Low rank approximation, Hypergraph enumeration} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Fair clustering is a variant of constrained clustering where the goal is to partition a set of colored points. The fraction of points of each color in every cluster should be more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS 2017] and became widely popular. This paper proposes a new construction of coresets for fair k-means and k-median clustering for Euclidean and general metrics based on random sampling. For the Euclidean space ℝ^d, we provide the first coresets whose size does not depend exponentially on the dimension d. The question of whether such constructions exist was asked by Schmidt, Schwiegelshohn, and Sohler [WAOA 2019] and Huang, Jiang, and Vishnoi [NeurIPS 2019]. For general metric, our construction provides the first coreset for fair k-means and k-median.
New coresets appear to be a handy tool for designing better approximation and streaming algorithms for fair and other constrained clustering variants. In particular, we obtain
- the first fixed-parameter tractable (FPT) PTAS for fair k-means and k-median clustering in ℝ^d. The near-linear time of our PTAS improves over the previous scheme of Böhm, Fazzone, Leonardi, and Schwiegelshohn [ArXiv 2020] with running time n^{poly(k/ε)};
- FPT "true" constant-approximation for metric fair clustering. All previous algorithms for fair k-means and k-median in general metric are bicriteria and violate the fairness constraints;
- FPT 3-approximation for lower-bounded k-median improving the best-known 3.736 factor of Bera, Chakrabarty, and Negahbani [ArXiv 2019];
- the first FPT constant-approximations for metric chromatic clustering and 𝓁-Diversity clustering;
- near linear-time (in n) PTAS for capacitated and lower-bounded clustering improving over PTAS of Bhattacharya, Jaiswal, and Kumar [TOCS 2018] with super-quadratic running time;
- a streaming (1+ε)-approximation for fair k-means and k-median of space complexity polynomial in k, d, ε and log{n} (the previous algorithms have exponential space complexity on either d or k).

Sayan Bandyapadhyay, Fedor V. Fomin, and Kirill Simonov. On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bandyapadhyay_et_al:LIPIcs.ICALP.2021.23, author = {Bandyapadhyay, Sayan and Fomin, Fedor V. and Simonov, Kirill}, title = {{On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {23:1--23:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.23}, URN = {urn:nbn:de:0030-drops-140923}, doi = {10.4230/LIPIcs.ICALP.2021.23}, annote = {Keywords: fair clustering, coresets, approximation algorithms} }

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

We investigate the parameterized complexity of finding diverse sets of solutions to three fundamental combinatorial problems, two from the theory of matroids and the third from graph theory. The input to the Weighted Diverse Bases problem consists of a matroid M, a weight function ω:E(M)→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k bases B_1, ..., B_k of M such that the weight of the symmetric difference of any pair of these bases is at least d. This is a diverse variant of the classical matroid base packing problem. The input to the Weighted Diverse Common Independent Sets problem consists of two matroids M₁,M₂ defined on the same ground set E, a weight function ω:E→N, and integers k ≥ 1, d ≥ 0. The task is to decide if there is a collection of k common independent sets I_1, ..., I_k of M₁ and M₂ such that the weight of the symmetric difference of any pair of these sets is at least d. This is motivated by the classical weighted matroid intersection problem. The input to the Diverse Perfect Matchings problem consists of a graph G and integers k ≥ 1, d ≥ 0. The task is to decide if G contains k perfect matchings M_1, ..., M_k such that the symmetric difference of any two of these matchings is at least d.
The underlying problem of finding one solution (basis, common independent set, or perfect matching) is known to be doable in polynomial time for each of these problems, and Diverse Perfect Matchings is known to be NP-hard for k = 2. We show that Weighted Diverse Bases and Weighted Diverse Common Independent Sets are both NP-hard. We show also that Diverse Perfect Matchings cannot be solved in polynomial time (unless P=NP) even for the case d = 1. We derive fixed-parameter tractable (FPT) algorithms for all three problems with (k,d) as the parameter.
The above results on matroids are derived under the assumption that the input matroids are given as independence oracles. For Weighted Diverse Bases we present a polynomial-time algorithm that takes a representation of the input matroid over a finite field and computes a poly(k,d)-sized kernel for the problem.

Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, Geevarghese Philip, and Saket Saurabh. Diverse Collections in Matroids and Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2021.31, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket}, title = {{Diverse Collections in Matroids and Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {31:1--31:14}, 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.2021.31}, URN = {urn:nbn:de:0030-drops-136769}, doi = {10.4230/LIPIcs.STACS.2021.31}, annote = {Keywords: Matroids, Diverse solutions, Fixed-parameter tractable algorithms} }

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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 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is NP-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is FPT parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on 𝒪(k²) vertices.

Fedor V. Fomin, Petr A. Golovach, Lars Jaffke, Geevarghese Philip, and Danil Sagunov. Diverse Pairs of Matchings. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 26:1-26:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ISAAC.2020.26, author = {Fomin, Fedor V. and Golovach, Petr A. and Jaffke, Lars and Philip, Geevarghese and Sagunov, Danil}, title = {{Diverse Pairs of Matchings}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {26:1--26:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.26}, URN = {urn:nbn:de:0030-drops-133706}, doi = {10.4230/LIPIcs.ISAAC.2020.26}, annote = {Keywords: Matching, Solution Diversity, Fixed-Parameter Tractability} }

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

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs G and H are 2-isomorphic, or equivalently, their cycle matroids are isomorphic, if and only if G can be transformed into H by a series of operations called Whitney switches. In this paper we consider the quantitative question arising from Whitney’s theorem: Given 2-isomorphic graphs, can we transform one into another by applying at most k Whitney switches? This problem is already NP-complete for cycles, and we investigate its parameterized complexity. We show that the problem admits a kernel of size 𝒪(k), and thus, is fixed-parameter tractable when parameterized by k.

Fedor V. Fomin and Petr A. Golovach. Kernelization of Whitney Switches. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 48:1-48:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2020.48, author = {Fomin, Fedor V. and Golovach, Petr A.}, title = {{Kernelization of Whitney Switches}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {48:1--48:19}, 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.2020.48}, URN = {urn:nbn:de:0030-drops-129144}, doi = {10.4230/LIPIcs.ESA.2020.48}, annote = {Keywords: Whitney switch, 2-isomorphism, Parameterized Complexity, kernelization} }

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

We study algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. While various parameterized algorithms on graphs for many structural parameters like vertex cover or treewidth can be found in the literature, up to the Exponential Time Hypothesis (ETH), the existence of subexponential parameterized algorithms for most of the structural parameters and optimization problems is highly unlikely. This is why we find the algorithmic behavior of the "fill-in parameterization" very unusual.
Being intrigued by this behaviour, we identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^𝒪(√k log k) ⋅ n^𝒪(1). Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails.
Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.

Fedor V. Fomin and Petr A. Golovach. Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 49:1-49:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2020.49, author = {Fomin, Fedor V. and Golovach, Petr A.}, title = {{Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {49:1--49:17}, 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.2020.49}, URN = {urn:nbn:de:0030-drops-129157}, doi = {10.4230/LIPIcs.ESA.2020.49}, annote = {Keywords: Parameterized complexity, structural parameterization, subexponential algorithms, kernelization, chordal graphs, fill-in, independent set, clique, coloring} }

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

The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and co-graphic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition.
When L is of small rank, we show that the problem is NP-complete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upper-bound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs.

Fedor V. Fomin, Petr Golovach, Pranabendu Misra, and M. S. Ramanujan. On the Complexity of Recovering Incidence Matrices. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 50:1-50:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2020.50, author = {Fomin, Fedor V. and Golovach, Petr and Misra, Pranabendu and Ramanujan, M. S.}, title = {{On the Complexity of Recovering Incidence Matrices}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {50:1--50:13}, 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.2020.50}, URN = {urn:nbn:de:0030-drops-129164}, doi = {10.4230/LIPIcs.ESA.2020.50}, annote = {Keywords: Graph Incidence Matrix, Matrix Recovery, Enumeration Algorithm} }

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

In general, a graph modification problem is defined by a graph modification operation ⊠ and a target graph property 𝒫. Typically, the modification operation ⊠ may be vertex removal, edge removal, edge contraction, or edge addition and the question is, given a graph G and an integer k, whether it is possible to transform G to a graph in 𝒫 after applying k times the operation ⊠ on G. This problem has been extensively studied for particilar instantiations of ⊠ and 𝒫. In this paper we consider the general property 𝒫_ϕ of being planar and, moreover, being a model of some First-Order Logic sentence ϕ (an FOL-sentence). We call the corresponding meta-problem Graph ⊠-Modification to Planarity and ϕ and prove the following algorithmic meta-theorem: there exists a function f: ℕ² → ℕ such that, for every ⊠ and every FOL sentence ϕ, the Graph ⊠-Modification to Planarity and ϕ is solvable in f(k,|ϕ|)⋅n² time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman’s Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

Fedor V. Fomin, Petr A. Golovach, Giannos Stamoulis, and Dimitrios M. Thilikos. An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 51:1-51:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2020.51, author = {Fomin, Fedor V. and Golovach, Petr A. and Stamoulis, Giannos and Thilikos, Dimitrios M.}, title = {{An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {51:1--51:17}, 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.2020.51}, URN = {urn:nbn:de:0030-drops-129172}, doi = {10.4230/LIPIcs.ESA.2020.51}, annote = {Keywords: Graph modification Problems, Algorithmic meta-theorems, First Order Logic, Irrelevant vertex technique, Planar graphs, Surface embeddable graphs} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

A popular model to measure network stability is the k-core, that is the maximal induced subgraph in which every vertex has degree at least k. For example, k-cores are commonly used to model the unraveling phenomena in social networks. In this model, users having less than k connections within the network leave it, so the remaining users form exactly the k-core. In this paper we study the question of whether it is possible to make the network more robust by spending only a limited amount of resources on new connections. A mathematical model for the k-core construction problem is the following Edge k-Core optimization problem. We are given a graph G and integers k, b and p. The task is to ensure that the k-core of G has at least p vertices by adding at most b edges.
The previous studies on Edge k-Core demonstrate that the problem is computationally challenging. In particular, it is NP-hard when k = 3, W[1]-hard when parameterized by k+b+p (Chitnis and Talmon, 2018), and APX-hard (Zhou et al, 2019). Nevertheless, we show that there are efficient algorithms with provable guarantee when the k-core has to be constructed from a sparse graph with some additional structural properties. Our results are
- When the input graph is a forest, Edge k-Core is solvable in polynomial time;
- Edge k-Core is fixed-parameter tractable (FPT) when parameterized by the minimum size of a vertex cover in the input graph. On the other hand, with such parameterization, the problem does not admit a polynomial kernel subject to a widely-believed assumption from complexity theory;
- Edge k-Core is FPT parameterized by the treewidth of the graph plus k. This improves upon a result of Chitnis and Talmon by not requiring b to be small. Each of our algorithms is built upon a new graph-theoretical result interesting in its own.

Fedor V. Fomin, Danil Sagunov, and Kirill Simonov. Building Large k-Cores from Sparse Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 35:1-35:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.MFCS.2020.35, author = {Fomin, Fedor V. and Sagunov, Danil and Simonov, Kirill}, title = {{Building Large k-Cores from Sparse Graphs}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {35:1--35:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.35}, URN = {urn:nbn:de:0030-drops-127026}, doi = {10.4230/LIPIcs.MFCS.2020.35}, annote = {Keywords: parameterized complexity, k-core, vertex cover, treewidth} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We consider 𝓁₁-Rank-r Approximation over {GF}(2), where for a binary m× n matrix 𝐀 and a positive integer constant r, one seeks a binary matrix 𝐁 of rank at most r, minimizing the column-sum norm ‖ 𝐀 -𝐁‖₁. We show that for every ε ∈ (0, 1), there is a {randomized} (1+ε)-approximation algorithm for 𝓁₁-Rank-r Approximation over {GF}(2) of running time m^{O(1)}n^{O(2^{4r}⋅ ε^{-4})}. This is the first polynomial time approximation scheme (PTAS) for this problem.

Fedor V. Fomin, Petr A. Golovach, Fahad Panolan, and Kirill Simonov. Low-Rank Binary Matrix Approximation in Column-Sum Norm. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 32:1-32:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2020.32, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad and Simonov, Kirill}, title = {{Low-Rank Binary Matrix Approximation in Column-Sum Norm}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.32}, URN = {urn:nbn:de:0030-drops-126355}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.32}, annote = {Keywords: Binary Matrix Factorization, PTAS, Column-sum norm} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time n^o(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of n^o(n)-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.

Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, and Meirav Zehavi. Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 49:1-49:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2020.49, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Mihajlin, Ivan and Saurabh, Saket and Zehavi, Meirav}, title = {{Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {49:1--49:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.49}, URN = {urn:nbn:de:0030-drops-124568}, doi = {10.4230/LIPIcs.ICALP.2020.49}, annote = {Keywords: Hadwiger Number, Exponential-Time Hypothesis, Exact Algorithms, Edge Contraction Problems} }

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Invited Talk

**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We discuss some recent progress in the study of Principal Component Analysis (PCA) from the perspective of Parameterized Complexity.

Fedor V. Fomin, Petr A. Golovach, and Kirill Simonov. Parameterized Complexity of PCA (Invited Talk). In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 1:1-1:5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.SWAT.2020.1, author = {Fomin, Fedor V. and Golovach, Petr A. and Simonov, Kirill}, title = {{Parameterized Complexity of PCA}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {1:1--1:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.1}, URN = {urn:nbn:de:0030-drops-122487}, doi = {10.4230/LIPIcs.SWAT.2020.1}, annote = {Keywords: parameterized complexity, Robust PCA, outlier detection} }

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

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2^{𝒪(√k)}(n+m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2^{o(√k)}(n+m)^𝒪(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2^{𝒪(√k)}(n+m)^𝒪(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2^{𝒪(√klog k)}(n+m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width 𝒪(√k).

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 44:1-44:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2020.44, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {44:1--44:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.44}, URN = {urn:nbn:de:0030-drops-122024}, doi = {10.4230/LIPIcs.SoCG.2020.44}, annote = {Keywords: Optimality Program, ETH, Unit Disk Graphs, Parameterized Complexity, Long Path, Long Cycle} }

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

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem Π with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ⋅ g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε>0, multiplicative parameterization above g(I)^(1+ε) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Parameterization Above a Multiplicative Guarantee. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 39:1-39:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fomin_et_al:LIPIcs.ITCS.2020.39, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Parameterization Above a Multiplicative Guarantee}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {39:1--39:13}, 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.2020.39}, URN = {urn:nbn:de:0030-drops-117248}, doi = {10.4230/LIPIcs.ITCS.2020.39}, annote = {Keywords: Parameterized Complexity, Above-Guarantee Parameterization, Girth} }

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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 k-Clustering we are given a multiset of n vectors X subset Z^d and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost sum_{i=1}^k min_{c_i in R^d} sum_{x in C_i} |x-c_i|_p^p <= D, where |*|_p is the Minkowski (L_p) norm of order p. For p=1, k-Clustering is the well-known k-Median. For p=2, the case of the Euclidean distance, k-Clustering is k-Means. We study k-Clustering from the perspective of parameterized complexity. The problem is known to be NP-hard for k=2 and it is also NP-hard for d=2. It is a long-standing open question, whether the problem is fixed-parameter tractable (FPT) for the combined parameter d+k. In this paper, we focus on the parameterization by D. We complement the known negative results by showing that for p=0 and p=infty, k-Clustering is W1-hard when parameterized by D. Interestingly, the complexity landscape of the problem appears to be more intricate than expected. We discover a tractability island of k-Clustering: for every p in (0,1], k-Clustering is solvable in time 2^O(D log D) (nd)^O(1).

Fedor V. Fomin, Petr A. Golovach, and Kirill Simonov. Parameterized k-Clustering: Tractability Island. 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. 14:1-14:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2019.14, author = {Fomin, Fedor V. and Golovach, Petr A. and Simonov, Kirill}, title = {{Parameterized k-Clustering: Tractability Island}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {14:1--14:15}, 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.2019.14}, URN = {urn:nbn:de:0030-drops-115761}, doi = {10.4230/LIPIcs.FSTTCS.2019.14}, annote = {Keywords: clustering, parameterized complexity, k-means, k-median} }

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

An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^O(1). In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log{n} can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^O(1). We complement these results by showing that the choice of degeneracy as the "above guarantee parameterization" is optimal in the following sense: For any epsilon>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+epsilon)d.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Going Far From Degeneracy. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 47:1-47:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2019.47, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Going Far From Degeneracy}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {47:1--47:14}, 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.2019.47}, URN = {urn:nbn:de:0030-drops-111688}, doi = {10.4230/LIPIcs.ESA.2019.47}, annote = {Keywords: Longest path, longest cycle, fixed-parameter tractability, above guarantee parameterization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

A graph G is contractible to a graph H if there is a set X subseteq E(G), such that G/X is isomorphic to H. Here, G/X is the graph obtained from G by contracting all the edges in X. For a family of graphs F, the F-Contraction problem takes as input a graph G on n vertices, and the objective is to output the largest integer t, such that G is contractible to a graph H in F, where |V(H)|=t. When F is the family of paths, then the corresponding F-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time 2^n * n^{O(1)}. In spite of the deceptive simplicity of the problem, beating the 2^n * n^{O(1)} bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time 1.99987^n * n^{O(1)}. We also define a problem called 3-Disjoint Connected Subgraphs, and design an algorithm for it that runs in time 1.88^n * n^{O(1)}. The above algorithm is used as a sub-routine in our algorithm for Path Contraction.

Akanksha Agrawal, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Prafullkumar Tale. Path Contraction Faster Than 2^n. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{agrawal_et_al:LIPIcs.ICALP.2019.11, author = {Agrawal, Akanksha and Fomin, Fedor V. and Lokshtanov, Daniel and Saurabh, Saket and Tale, Prafullkumar}, title = {{Path Contraction Faster Than 2^n}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.11}, URN = {urn:nbn:de:0030-drops-105874}, doi = {10.4230/LIPIcs.ICALP.2019.11}, annote = {Keywords: path contraction, exact exponential time algorithms, graph algorithms, enumerating connected sets, 3-disjoint connected subgraphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, an r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids.
We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals T subseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of E \ T of size at most k.
We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for r <= 2 and |T|<= 2.
On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi. Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 59:1-59:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2019.59, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Saurabh, Saket and Zehavi, Meirav}, title = {{Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {59:1--59:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.59}, URN = {urn:nbn:de:0030-drops-106351}, doi = {10.4230/LIPIcs.ICALP.2019.59}, annote = {Keywords: Binary matroids, perturbed graphic matroids, spanning set, parameterized complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a sqrt{k} x sqrt{k}-grid as a minor, or its treewidth is O(sqrt{k}). However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs like unit disk or map graphs. This is mainly due to the presence of large cliques in these classes of graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs, the intersection graphs of finitely many simply-connected and interior-disjoint regions of the Euclidean plane. Informally, our lemma states the following. For any map graph G, there exists a collection (U_1,...,U_t) of cliques of G with the following property: G either contains a sqrt{k} x sqrt{k}-grid as a minor, or it admits a tree decomposition where every bag is the union of O(sqrt{k}) cliques in the above collection.
The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time 2^{O({sqrt{k}log{k}})} * n^{O(1)} for Connected Planar F-Deletion (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions.
For Longest Cycle/Path, these are the first subexponential-time parameterized algorithm on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known 2^{O({k^{0.75}log{k}})} * n^{O(1)}-time algorithms on map graphs.

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of Map Graphs with Applications. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 60:1-60:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2019.60, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Decomposition of Map Graphs with Applications}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {60:1--60:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.60}, URN = {urn:nbn:de:0030-drops-106366}, doi = {10.4230/LIPIcs.ICALP.2019.60}, annote = {Keywords: Longest Cycle, Cycle Packing, Feedback Vertex Set, Map Graphs, FPT} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 1 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 19041 "New Horizons in Parameterized Complexity".
Parameterized Complexity is celebrating its 30th birthday in 2019. In these three decades, there has been tremendous progress in developing the area. The central vision of Parameterized Complexity through all these years has been to provide the algorithmic and complexity-theoretic toolkit for studying multivariate algorithmics in different disciplines and subfields of Computer Science. These tools are universal as they did not only help in the development of the core of Parameterized Complexity, but also led to its success in other subfields of Computer Science such as Approximation Algorithms, Computational Social Choice, Computational Geometry, problems solvable in P (polynomial time), to name a few.
In the last few years, we have witnessed several exciting developments of new parameterized techniques and tools in the following subfields of Computer Science and Optimization: Mathematical Programming, Computational Linear Algebra, Computational Counting, Derandomization, and Approximation Algorithms.
The main objective of the seminar was to initiate the discussion on which of the recent
domain-specific algorithms and complexity advances can become useful in other domains.

Fedor V. Fomin, Dániel Marx, Saket Saurabh, and Meirav Zehavi. New Horizons in Parameterized Complexity (Dagstuhl Seminar 19041). In Dagstuhl Reports, Volume 9, Issue 1, pp. 67-87, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@Article{fomin_et_al:DagRep.9.1.67, author = {Fomin, Fedor V. and Marx, D\'{a}niel and Saurabh, Saket and Zehavi, Meirav}, title = {{New Horizons in Parameterized Complexity (Dagstuhl Seminar 19041)}}, pages = {67--87}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {1}, editor = {Fomin, Fedor V. and Marx, D\'{a}niel and Saurabh, Saket and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.1.67}, URN = {urn:nbn:de:0030-drops-105706}, doi = {10.4230/DagRep.9.1.67}, annote = {Keywords: Intractability, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems.

Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Modification to Planarity is Fixed Parameter Tractable. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 28:1-28:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2019.28, author = {Fomin, Fedor V. and Golovach, Petr A. and Thilikos, Dimitrios M.}, title = {{Modification to Planarity is Fixed Parameter Tractable}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.28}, URN = {urn:nbn:de:0030-drops-102677}, doi = {10.4230/LIPIcs.STACS.2019.28}, annote = {Keywords: Modification problems, Planar graphs, Irrelevant vertex technique} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. These graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This notion was introduced in the early 1990s by Biro, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Clique, Independent Set, or Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions.
First we show that for every fixed H, the class of H-graphs is of logarithmically-bounded boolean-width. We also prove that H-graphs are graphs with polynomially many minimal separators. Pipelined with the plethora of known algorithms on graphs of bounded boolean-width and graphs with polynomially many minimal separators, this describes a large class of optimization problems that are solvable in polynomial time on H-graphs.
The most fundamental optimization problems among those solvable in polynomial time on H-graphs are Clique, Independent Set, and Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Independent Set and Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, Dominating Set is fixed-parameter tractable (FPT) parameterized by the size of H. Besides, we show that Clique admits a polynomial kernel parameterized by H and the solution size.

Fedor V. Fomin, Petr A. Golovach, and Jean-Florent Raymond. On the Tractability of Optimization Problems on H-Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 30:1-30:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2018.30, author = {Fomin, Fedor V. and Golovach, Petr A. and Raymond, Jean-Florent}, title = {{On the Tractability of Optimization Problems on H-Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {30:1--30:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.30}, URN = {urn:nbn:de:0030-drops-94930}, doi = {10.4230/LIPIcs.ESA.2018.30}, annote = {Keywords: H-topological intersection graphs, parameterized complexity, minimal separators, boolean-width, mim-width} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input.
The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH.
This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.

Fedor V. Fomin, Fahad Panolan, M. S. Ramanujan, and Saket Saurabh. On the Optimality of Pseudo-polynomial Algorithms for Integer Programming. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 31:1-31:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ESA.2018.31, author = {Fomin, Fedor V. and Panolan, Fahad and Ramanujan, M. S. and Saurabh, Saket}, title = {{On the Optimality of Pseudo-polynomial Algorithms for Integer Programming}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {31:1--31:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.31}, URN = {urn:nbn:de:0030-drops-94949}, doi = {10.4230/LIPIcs.ESA.2018.31}, annote = {Keywords: Integer Programming, Strong Exponential Time Hypothesis, Branch-width of a matrix, Fine-grained Complexity} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We provide a number of algorithmic results for the following family of problems: For a given binary m x n matrix A and a nonnegative integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an integer r, the "simplicity" of B is characterized as follows.
- Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r=2. We show that the problem is solvable in time 2^{O(k log k)}*(nm)^O(1) and thus is fixed-parameter tractable parameterized by k. We also complement this result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}).
- Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r=1. It is also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2^{O(r^{3/2}* sqrt{k log k})}(nm)^O(1), which is subexponential in k for r in o((k/log k)^{1/3}).
- Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k=0 as well as for r=1. We show that it is solvable in subexponential in k time 2^{O(r2^r * sqrt{k log k})}(nm)^O(1).

Fedor V. Fomin, Petr A. Golovach, and Fahad Panolan. Parameterized Low-Rank Binary Matrix Approximation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 53:1-53:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2018.53, author = {Fomin, Fedor V. and Golovach, Petr A. and Panolan, Fahad}, title = {{Parameterized Low-Rank Binary Matrix Approximation}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {53:1--53:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.53}, URN = {urn:nbn:de:0030-drops-90571}, doi = {10.4230/LIPIcs.ICALP.2018.53}, annote = {Keywords: Binary matrices, clustering, low-rank approximation, fixed-parameter tractability} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric X. Computing such an embedding (exactly or approximately) is a non-trivial task even when X is the metric induced by a path, or, equivalently, the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph H, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs G, H and integer c, is it possible to embed G with distortion c into a graph homeomorphic to H? Then embedding into the line is the special case H=K_2, and embedding into the cycle is the case H=K_3, where K_k denotes the complete graph on k vertices. For this problem we give
- an approximation algorithm, which in time f(H)* poly (n), for some function f, either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion poly(c);
- an exact algorithm, which in time f'(H, c)* poly (n), for some function f', either correctly decides that there is no embedding of G with distortion c into any graph homeomorphic to H, or finds an embedding with distortion c. Prior to our work, poly(OPT)-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

Timothy Carpenter, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Anastasios Sidiropoulos. Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carpenter_et_al:LIPIcs.SoCG.2018.21, author = {Carpenter, Timothy and Fomin, Fedor V. and Lokshtanov, Daniel and Saurabh, Saket and Sidiropoulos, Anastasios}, title = {{Algorithms for Low-Distortion Embeddings into Arbitrary 1-Dimensional Spaces}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.21}, URN = {urn:nbn:de:0030-drops-87344}, doi = {10.4230/LIPIcs.SoCG.2018.21}, annote = {Keywords: Metric embeddings, minimum-distortion embeddings, 1-dimensional simplicial complex, Fixed-parameter tractable algorithms, Approximation algorithms} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

A partial complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G, is there a partial complement of G which is in G? We show that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of r-regular graphs.

Fedor V. Fomin, Petr A. Golovach, Torstein J. F. Strømme, and Dimitrios M. Thilikos. Partial Complementation of Graphs. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 21:1-21:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fomin_et_al:LIPIcs.SWAT.2018.21, author = {Fomin, Fedor V. and Golovach, Petr A. and Str{\o}mme, Torstein J. F. and Thilikos, Dimitrios M.}, title = {{Partial Complementation of Graphs}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.21}, URN = {urn:nbn:de:0030-drops-88476}, doi = {10.4230/LIPIcs.SWAT.2018.21}, annote = {Keywords: Partial complementation, graph editing, graph classes} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHPi_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHPi_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas. A Fixed-Parameter Perspective on #BIS. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 13:1-13:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{curticapean_et_al:LIPIcs.IPEC.2017.13, author = {Curticapean, Radu and Dell, Holger and Fomin, Fedor V. and Goldberg, Leslie Ann and Lapinskas, John}, title = {{A Fixed-Parameter Perspective on #BIS}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.13}, URN = {urn:nbn:de:0030-drops-85613}, doi = {10.4230/LIPIcs.IPEC.2017.13}, annote = {Keywords: Approximate counting, parameterised complexity, independent sets} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping \phi:V(H)->V(G) if every edge uv of F is either an edge of G, or \phi^{-1}(u)\phi^{-1}(v) is an edge of H. Thus F contains both G and H as subgraphs, and the edge set of F is the union of the edge sets of G and \phi(H). We consider the following optimization problem. Given graphs G, H, and a weight function \omega assigning non-negative weights to pairs of vertices of V(G), the task is to find \phi of minimum weight \omega(\phi)=\sum_{xy\in E(H)}\omega(\phi(x)\phi(y)) such that the edge connectivity of the superposition F of G and H with respect to \phi is higher than the edge connectivity of G. Our main result is the following ``dichotomy'' complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when H\in C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when H\in C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.

Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Structured Connectivity Augmentation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 29:1-29:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fomin_et_al:LIPIcs.MFCS.2017.29, author = {Fomin, Fedor V. and Golovach, Petr A. and Thilikos, Dimitrios M.}, title = {{Structured Connectivity Augmentation}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {29:1--29:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.29}, URN = {urn:nbn:de:0030-drops-80603}, doi = {10.4230/LIPIcs.MFCS.2017.29}, annote = {Keywords: connectivity augmentation, graph superposition, complexity} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We consider the following natural "above guarantee" parameterization of the classical longest path problem: For given vertices s and t of a graph G, and an integer k, the longest detour problem asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that the longest detour problem is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) * poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study a related problem, exact detour, that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k * poly(n), and a deterministic algorithm with running time about 6.745^k * poly(n), showing that this problem is FPT as well. Our algorithms for the exact detour problem apply to both undirected and directed graphs.

Ivona Bezáková, Radu Curticapean, Holger Dell, and Fedor V. Fomin. Finding Detours is Fixed-Parameter Tractable. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2017.54, author = {Bez\'{a}kov\'{a}, Ivona and Curticapean, Radu and Dell, Holger and Fomin, Fedor V.}, title = {{Finding Detours is Fixed-Parameter Tractable}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.54}, URN = {urn:nbn:de:0030-drops-74790}, doi = {10.4230/LIPIcs.ICALP.2017.54}, annote = {Keywords: longest path, fixed-parameter tractable algorithms, above-guarantee parameterization, graph minors} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, we study the Space Cover problem, where we are given a matrix M and a subset of its columns T; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains all vectors of T. This is a fundamental problem arising in different domains, such as coding theory, machine learning, and graph algorithms.
We give a parameterized algorithm with running time 2^{O(k)}||M|| ^{O(1)} solving this problem in the case when M is a totally unimodular matrix over rationals, where k is the size of F. In other words, we show that the problem is fixed-parameter tractable parameterized by the rank of the covering subspace. The algorithm is "asymptotically optimal" for the following reasons.
Choice of matrices: Vector matroids corresponding to totally unimodular matrices over rationals are exactly the regular matroids. It is known that for matrices corresponding to a more general class of matroids, namely, binary matroids, the problem becomes W[1]-hard being parameterized by k.
Choice of the parameter: The problem is NP-hard even if |T|=3 on matrix-representations of a subclass of regular matroids, namely cographic matroids. Thus for a stronger parameterization, like by the size of T, the problem becomes intractable.
Running Time: The exponential dependence in the running time of our algorithm cannot be asymptotically improved unless Exponential Time Hypothesis (ETH) fails.
Our algorithm exploits the classical decomposition theorem of Seymour for regular matroids.

Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, and Saket Saurabh. Covering Vectors by Spaces: Regular Matroids. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 56:1-56:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2017.56, author = {Fomin, Fedor V. and Golovach, Petr A. and Lokshtanov, Daniel and Saurabh, Saket}, title = {{Covering Vectors by Spaces: Regular Matroids}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {56:1--56:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.56}, URN = {urn:nbn:de:0030-drops-73865}, doi = {10.4230/LIPIcs.ICALP.2017.56}, annote = {Keywords: regular matroids, spanning set, parameterized complexity} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We give algorithms with running time 2^{O({\sqrt{k}\log{k}})} n^{O(1)} for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains (i) a path on exactly/at least k vertices, (ii) a cycle on exactly k vertices, (iii) a cycle on at least k vertices, (iv) a feedback vertex set of size at most k, and (v) a set of k pairwise vertex disjoint cycles.
For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time 2^{O(k^{0.75}\log{k})} n^{O(1)}. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to k^{O(1)} and there exists a solution that crosses every separator at most O(\sqrt{k}) times. The running times of our algorithms are optimal up to the log{k} factor in the exponent, assuming the Exponential Time Hypothesis.

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 65:1-65:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fomin_et_al:LIPIcs.ICALP.2017.65, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket and Zehavi, Meirav}, title = {{Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {65:1--65:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.65}, URN = {urn:nbn:de:0030-drops-73937}, doi = {10.4230/LIPIcs.ICALP.2017.65}, annote = {Keywords: Longest Cycle, Cycle Packing, Feedback Vertex Set, Unit Disk Graph, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size on T that guards all of the points on T. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm [SODA 2005]. However, only in 2010 King and Krohn [SODA 2010] finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this paper, we answer the first question affirmatively by developing an n^O(sqrt{k})-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.

Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, and Meirav Zehavi. Exact Algorithms for Terrain Guarding. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 11:1-11:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ashok_et_al:LIPIcs.SoCG.2017.11, author = {Ashok, Pradeesha and Fomin, Fedor V. and Kolay, Sudeshna and Saurabh, Saket and Zehavi, Meirav}, title = {{Exact Algorithms for Terrain Guarding}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.11}, URN = {urn:nbn:de:0030-drops-71975}, doi = {10.4230/LIPIcs.SoCG.2017.11}, annote = {Keywords: Terrain Guarding, Art Gallery, Exponential-Time Algorithms} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 1 (2017)

Dagstuhl Seminar 17041 "Randomization in Parameterized Complexity" took place from January 22nd to January 27th 2017 with the objective to bridge the gap between randomization and parameterized complexity theory. This report documents the talks held during the seminar as well as the open questions arised in the discussion sessions.

Marek Cygan, Fedor V. Fomin, Danny Hermelin, and Magnus Wahlström. Randomization in Parameterized Complexity (Dagstuhl Seminar 17041). In Dagstuhl Reports, Volume 7, Issue 1, pp. 103-128, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@Article{cygan_et_al:DagRep.7.1.103, author = {Cygan, Marek and Fomin, Fedor V. and Hermelin, Danny and Wahlstr\"{o}m, Magnus}, title = {{Randomization in Parameterized Complexity (Dagstuhl Seminar 17041)}}, pages = {103--128}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {7}, number = {1}, editor = {Cygan, Marek and Fomin, Fedor V. and Hermelin, Danny and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.1.103}, URN = {urn:nbn:de:0030-drops-72479}, doi = {10.4230/DagRep.7.1.103}, annote = {Keywords: fixed-parameter tractability, intractability, parameterized complexity, randomness} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

The rigidity of a matrix A for a target rank r over a field F is the minimum Hamming distance between A and a matrix of rank at most r. Rigidity is a classical concept in Computational Complexity Theory: constructions of rigid matrices are known to imply lower bounds of significant importance relating to arithmetic circuits. Yet, from the viewpoint of Parameterized Complexity, the study of central properties of matrices in general, and of the rigidity of a matrix in particular, has been neglected. In this paper, we conduct a comprehensive study of different aspects of the computation of the rigidity of general matrices in the framework of Parameterized Complexity. Naturally, given parameters r and k, the Matrix Rigidity problem asks whether the rigidity of A for the target rank r is at most k. We show that in case F equals the reals or F is any finite field, this problem is fixed-parameter tractable with respect to k+r. To this end, we present a dimension reduction procedure, which may be a valuable primitive in future studies of problems of this nature. We also employ central tools in Real Algebraic Geometry, which are not well known in Parameterized Complexity, as a black box. In particular, we view the output of our dimension reduction procedure as an algebraic variety. Our main results are complemented by a W[1]-hardness result and a subexponential-time parameterized algorithm for a special case of Matrix Rigidity, highlighting the different flavors of this problem.

Fedor V. Fomin, Daniel Lokshtanov, S. M. Meesum, Saket Saurabh, and Meirav Zehavi. Matrix Rigidity from the Viewpoint of Parameterized Complexity. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2017.32, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Meesum, S. M. and Saurabh, Saket and Zehavi, Meirav}, title = {{Matrix Rigidity from the Viewpoint of Parameterized Complexity}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.32}, URN = {urn:nbn:de:0030-drops-70019}, doi = {10.4230/LIPIcs.STACS.2017.32}, annote = {Keywords: Matrix Rigidity, Parameterized Complexity, Linear Algebra} }

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Invited Talk

**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

We overview the recent progress in solving intractable optimization problems on planar graphs as well as other classes of sparse graphs. In particular, we discuss how tools from Graph Minors theory can be used to obtain:
* subexponential parameterized algorithms
* approximation algorithms, and
* preprocessing and kernelization algorithms
on these classes of graphs.

Fedor V. Fomin. Graph Decompositions and Algorithms (Invited Talk). In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, p. 5:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin:LIPIcs.FSTTCS.2016.5, author = {Fomin, Fedor V.}, title = {{Graph Decompositions and Algorithms}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {5:1--5:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.5}, URN = {urn:nbn:de:0030-drops-68903}, doi = {10.4230/LIPIcs.FSTTCS.2016.5}, annote = {Keywords: Algorithms, logic, graph minor} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We prove that for every positive integer r and for every graph class G of bounded expansion, the r-DOMINATING SET problem admits a linear kernel on graphs from G. Moreover, in the more general case when G is only assumed to be nowhere dense, we give an almost linear kernel on G for the classic DOMINATING SET problem, i.e., for the case r=1. These results generalize a line of previous research on finding linear kernels for DOMINATING SET and r-DOMINATING SET (Alber et al., JACM 2004, Bodlaender et al., FOCS 2009, Fomin et al., SODA 2010, Fomin et al., SODA 2012, Fomin et al., STACS 2013). However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related CONNECTED DOMINATING SET problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on H-topological-minor-free graphs (Fomin et al., STACS 2013). Also, we prove that for any somewhere dense class G, there is some r for which r-DOMINATING SET is W[2]-hard on G. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of r-DOMINATING SET on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

Pål Grønås Drange, Markus Dregi, Fedor V. Fomin, Stephan Kreutzer, Daniel Lokshtanov, Marcin Pilipczuk, Michal Pilipczuk, Felix Reidl, Fernando Sánchez Villaamil, Saket Saurabh, Sebastian Siebertz, and Somnath Sikdar. Kernelization and Sparseness: the Case of Dominating Set. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 31:1-31:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{drange_et_al:LIPIcs.STACS.2016.31, author = {Drange, P\r{a}l Gr{\o}n\r{a}s and Dregi, Markus and Fomin, Fedor V. and Kreutzer, Stephan and Lokshtanov, Daniel and Pilipczuk, Marcin and Pilipczuk, Michal and Reidl, Felix and S\'{a}nchez Villaamil, Fernando and Saurabh, Saket and Siebertz, Sebastian and Sikdar, Somnath}, title = {{Kernelization and Sparseness: the Case of Dominating Set}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.31}, URN = {urn:nbn:de:0030-drops-57327}, doi = {10.4230/LIPIcs.STACS.2016.31}, annote = {Keywords: kernelization, dominating set, bounded expansion, nowhere dense} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In the EDGE EDITING TO CONNECTED f-DEGREE GRAPH problem we are given a graph G, an integer k and a function f assigning integers to vertices of G. The task is to decide whether there is a connected graph F on the same vertex set as G, such that for every vertex v, its degree in F is f(v) and the number of edges inthe symmetric difference of E(G) and E(F), is at most k. We show that EDGE EDITING TO CONNECTED f-DEGREE GRAPH is fixed-parameter tractable (FPT) by providing an algorithm solving the problem on an n-vertex graph in time 2^{O(k)}n^{O(1)}. Our FPT algorithm is based on a non-trivial combination of color-coding and fast computations of representative families over direct sum matroid of l-elongation of co-graphic matroid associated with G and uniform matroid over the set of non-edges of G. We believe that this combination could be useful in designing parameterized algorithms for other edge editing problems.

Fedor V. Fomin, Petr Golovach, Fahad Panolan, and Saket Saurabh. Editing to Connected f-Degree Graph. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 36:1-36:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2016.36, author = {Fomin, Fedor V. and Golovach, Petr and Panolan, Fahad and Saurabh, Saket}, title = {{Editing to Connected f-Degree Graph}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {36:1--36:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.36}, URN = {urn:nbn:de:0030-drops-57370}, doi = {10.4230/LIPIcs.STACS.2016.36}, annote = {Keywords: Connected f-factor, FPT, Representative Family, Color Coding} }

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**Published in:** LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)

The Secluded Path problem introduced by Chechik et al. in [ESA 2013] models a situation where a sensitive information has to be transmitted between a pair of nodes along a path in a network.
The measure of the quality of a selected path is its exposure, which is the total weight of vertices in its closed neighborhood. In order to minimize the risk of intercepting the information, we are interested in selecting a secluded path, i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure of the tree is minimized. In this work, we obtain the following results about parameterized complexity of secluded connectivity problems.
We start from an observation that being parameterized by the size of the exposure, the problem is fixed-parameter tractable (FPT). More precisely, we give an algorithm deciding if a graph G with a given cost function w:V(G)->N contains a secluded path of exposure at most k with the cost at most C in time O(3^{k/3}(n+m) log W), where W is the maximum value of w on an input graph G. Similarly,
Secluded Steiner Tree is solvable in time O(2^{k}k^2 (n+m) log W).
The main result of this paper is about "above guarantee" parameterizations for secluded problems. We show that Secluded Steiner Tree is FPT being parameterized by r+p, where p is the number of the terminals, l the size of an optimum Steiner tree, and r=k-l. We complement this result by showing that the problem is co-W[1]-hard when parameterized by r only.
We also investigate Secluded Steiner Tree from kernelization perspective and provide several lower and upper bounds when parameters are the treewidth, the size of a vertex cover, maximum vertex degree and the solution size. Finally, we refine the algorithmic result of Chechik et al. by improving the exponential dependence from the treewidth of the input graph.

Fedor V. Fomin, Petr A. Golovach, Nikolay Karpov, and Alexander S. Kulikov. Parameterized Complexity of Secluded Connectivity Problems. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 408-419, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2015.408, author = {Fomin, Fedor V. and Golovach, Petr A. and Karpov, Nikolay and Kulikov, Alexander S.}, title = {{Parameterized Complexity of Secluded Connectivity Problems}}, booktitle = {35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)}, pages = {408--419}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-97-2}, ISSN = {1868-8969}, year = {2015}, volume = {45}, editor = {Harsha, Prahladh and Ramalingam, G.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.408}, URN = {urn:nbn:de:0030-drops-56318}, doi = {10.4230/LIPIcs.FSTTCS.2015.408}, annote = {Keywords: Secluded path, Secluded Steiner tree, parameterized complexity} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We study the paramereteized complexity of the following connectivity problem. For a vertex subset U of a graph G, trees T_1,...,T_s of G are completely independent spanning trees of U if each of them contains U, and for every two distinct vertices u,v in U, the paths from u to v in T_1,...,T_s are pairwise vertex disjoint except for end-vertices u and v. Then for a given s >= 2 and a parameter k, the task is to decide if a given n-vertex graph G contains a set U of size at least k such that there are s completely independent spanning trees of U. The problem is known to be NP-complete already for s=2. We prove the following results: (*) For s=2 the problem is solvable in time 2^{O(k)}*n^{O(1)}. (*) For s=2 the problem does not admit a polynomial kernel unless NP subseteq coNP/poly. (*) For arbitrary s, we show that the problem is solvable in time f(s,k)n^{O(1)} for some function f of s and k only.

Manu Basavaraju, Fedor V. Fomin, Petr A. Golovach, and Saket Saurabh. Connecting Vertices by Independent Trees. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 73-84, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{basavaraju_et_al:LIPIcs.FSTTCS.2014.73, author = {Basavaraju, Manu and Fomin, Fedor V. and Golovach, Petr A. and Saurabh, Saket}, title = {{Connecting Vertices by Independent Trees}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {73--84}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.73}, URN = {urn:nbn:de:0030-drops-48340}, doi = {10.4230/LIPIcs.FSTTCS.2014.73}, annote = {Keywords: Parameterized complexity, FPT-algorithms, completely independent spanning trees} }

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

Let F be a family of graphs. In the F-Completion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It appeared recently that special cases of F-Completion, the problem of completing into a chordal graph known as "Minimum Fill-in", corresponding to the case of F={C_4,C_5,C_6,...}, and the problem of completing into a split graph, i.e., the case of F={C_4,2K_2,C_5}, are solvable in parameterized subexponential time. The exploration of this phenomenon is the main motivation for our research on F-Completion.
In this paper we prove that completions into several well studied classes of graphs without long induced cycles also admit parameterized subexponential time algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized subexponential time, that is F-Completion for F={C_4,P_4}, a cycle and a path on four vertices.
- The problems known in the literature as Pseudosplit Completion, the case where F={2K_2,C_4}, and Threshold Completion, where F={2K_2,P_4,C_4}, are also solvable in subexponential time.
We complement our algorithms for $F$-Completion with the following lower bounds:
- For F={2K_2}, F={C_4}, F={P_4}, and F={2K_2,P_4}, F-Completion cannot be solved in time 2^o(k).n^O(1) unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F-Completion problems for F contained inside {2K_2,C_4,P_4}.

Pal Gronas Drange, Fedor V. Fomin, Michal Pilipczuk, and Yngve Villanger. Exploring Subexponential Parameterized Complexity of Completion Problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 288-299, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{drange_et_al:LIPIcs.STACS.2014.288, author = {Drange, Pal Gronas and Fomin, Fedor V. and Pilipczuk, Michal and Villanger, Yngve}, title = {{Exploring Subexponential Parameterized Complexity of Completion Problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {288--299}, 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.2014.288}, URN = {urn:nbn:de:0030-drops-44659}, doi = {10.4230/LIPIcs.STACS.2014.288}, annote = {Keywords: edge completion, modification, subexponential parameterized complexity} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

We consider the Directed Anchored k-Core problem, where the task is for a given directed graph G and integers b, k and p, to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (the anchors) of H have in-degree at least k. For undirected graphs, this problem was introduced by Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012]. We undertake a
systematic analysis of the computational complexity of Directed Anchored k-Core and show that:
- The decision version of the problem is NP-complete for every k>=1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k+2.
- The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k=1, and W[1]-hard for k>=2.
- When the maximum degree of the graph is at most Delta, the problem is FPT parameterized by p+Delta if k>=Delta/2.

Rajesh Chitnis, Fedor V. Fomin, and Petr A. Golovach. Parameterized Complexity of the Anchored k-Core Problem for Directed Graphs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 79-90, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chitnis_et_al:LIPIcs.FSTTCS.2013.79, author = {Chitnis, Rajesh and Fomin, Fedor V. and Golovach, Petr A.}, title = {{Parameterized Complexity of the Anchored k-Core Problem for Directed Graphs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {79--90}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.79}, URN = {urn:nbn:de:0030-drops-43636}, doi = {10.4230/LIPIcs.FSTTCS.2013.79}, annote = {Keywords: Parameterized complexity, directed graphs, anchored \$k\$-core} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 3 (2013)

We provide a report on the Dagstuhl Seminar 13121: "Bidimensional Structures: Algorithms, Combinatorics and Logic" held at Schloss Dagstuhl in Wadern, Germany
between Monday 18 and Friday 22 of March 2013. The report contains the motivation of the seminar, the abstracts of the talks given during the seminar, and the list of open problems.

Erik D. Demaine, Fedor V. Fomin, MohammadTaghi Hajiaghayi, and Dimitrios M. Thilikos. Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121). In Dagstuhl Reports, Volume 3, Issue 3, pp. 51-74, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@Article{demaine_et_al:DagRep.3.3.51, author = {Demaine, Erik D. and Fomin, Fedor V. and Hajiaghayi, MohammadTaghi and Thilikos, Dimitrios M.}, title = {{Bidimensional Structures: Algorithms, Combinatorics and Logic (Dagstuhl Seminar 13121)}}, pages = {51--74}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {3}, number = {3}, editor = {Demaine, Erik D. and Fomin, Fedor V. and Hajiaghayi, MohammadTaghi and Thilikos, Dimitrios M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.3.51}, URN = {urn:nbn:de:0030-drops-40131}, doi = {10.4230/DagRep.3.3.51}, annote = {Keywords: Graph Minors, Treewidth, Graph algorithms, Parameterized Algorithms} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Minimum Fill-in is a fundamental and classical problem arising in sparse matrix computations. In terms of graphs it can be formulated as a problem of finding a triangulation of a given graph with the minimum number of edges. By the classical result of Rose, Tarjan, Lueker, and Ohtsuki from 1976, an inclusion minimal triangulation of a graph can be found in polynomial time but, as it was shown by Yannakakis in 1981, finding a triangulation with the minimum number of edges is NP-hard.
In this paper, we study the parameterized complexity of local search for the Minimum Fill-in problem in the following form: Given a triangulation H of a graph G, is there a better triangulation, i.e. triangulation with less edges than H, within a given distance from H? We prove that this problem is fixed-parameter tractable (FPT) being parameterized by the distance from the initial triangulation by providing an algorithm that in time O(f(k) |G|^{O(1)}) decides if a better triangulation of G can be obtained by swapping at most k edges of H.
Our result adds Minimum Fill-in to the list of very few problems for which local search is known to be FPT.

Fedor V. Fomin and Yngve Villanger. Searching for better fill-in. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 8-19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2013.8, author = {Fomin, Fedor V. and Villanger, Yngve}, title = {{Searching for better fill-in}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {8--19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha 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.2013.8}, URN = {urn:nbn:de:0030-drops-39187}, doi = {10.4230/LIPIcs.STACS.2013.8}, annote = {Keywords: Local Search, Parameterized Complexity, Fill-in, Triangulation, Chordal graph} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or deleting at most k edges. The motivation of the problem stems from various tasks in computational biology (Ben-Dor et al., Journal of Computational Biology 1999) and machine learning (Bansal et al., Machine Learning 2004). Although in general Correlation Clustering is APX-hard (Charikar et al., FOCS 2003), the version of the problem where the number of cliques may not exceed a prescribed constant p admits a PTAS (Giotis and Guruswami, SODA 2006).
We study the parameterized complexity of Correlation Clustering with this restriction on the number of cliques to be created. We give an algorithm that - in time O(2^{O(sqrt{pk})} + n+m) decides whether a graph G on n vertices and m edges can be transformed into a cluster graph with exactly p cliques by changing at most k adjacencies.
We complement these algorithmic findings by the following, surprisingly tight lower bound on the asymptotic behavior of our algorithm. We show that unless the Exponential Time Hypothesis (ETH) fails - for any constant 0 <= sigma <= 1, there is p = Theta(k^sigma) such that there is no algorithm deciding in time 2^{o(sqrt{pk})} n^{O(1)} whether an n-vertex graph G can be transformed into a cluster graph with at most p cliques by changing at most k adjacencies.
Thus, our upper and lower bounds provide an asymptotically tight analysis of the multivariate parameterized complexity of the problem for the whole range of values of p from constant to a linear function of k.

Fedor V. Fomin, Stefan Kratsch, Marcin Pilipczuk, Michal Pilipczuk, and Yngve Villanger. Tight bounds for Parameterized Complexity of Cluster Editing. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 32-43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2013.32, author = {Fomin, Fedor V. and Kratsch, Stefan and Pilipczuk, Marcin and Pilipczuk, Michal and Villanger, Yngve}, title = {{Tight bounds for Parameterized Complexity of Cluster Editing}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {32--43}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha 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.2013.32}, URN = {urn:nbn:de:0030-drops-39209}, doi = {10.4230/LIPIcs.STACS.2013.32}, annote = {Keywords: parameterized complexity, cluster editing, correlation clustering, subexponential algorithms, tight bounds} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We give the first linear kernels for Dominating Set and Connected Dominating Set problems on graphs excluding a fixed graph H as a topological minor.

Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, and Dimitrios M. Thilikos. Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 92-103, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2013.92, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Saurabh, Saket and Thilikos, Dimitrios M.}, title = {{Linear kernels for (connected) dominating set on graphs with excluded topological subgraphs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {92--103}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha 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.2013.92}, URN = {urn:nbn:de:0030-drops-39255}, doi = {10.4230/LIPIcs.STACS.2013.92}, annote = {Keywords: Parameterized complexity, kernelization, algorithmic graph minors, dominating set, connected dominating set} }

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

Cai and Yang initiated the systematic parameterized complexity study of the following set of problems around Eulerian graphs. For a given graph G and integer k, the task is to decide if G contains a (connected) subgraph with k vertices (edges) with all vertices of even (odd) degrees. They succeed to establish the parameterized complexity of all cases except two, when we ask about
- a connected k-edge subgraph with all vertices of odd degrees, the problem known as k-Edge Connected Odd Subgraph; and
- a connected k- vertex induced subgraph with all vertices of even degrees, the problem known as k-Vertex Eulerian Subgraph.
We resolve both open problems and thus complete the characterization of even/odd subgraph problems from parameterized complexity perspective. We show that k-Edge Connected Odd Subgraph is FPT and that k-Vertex Eulerian Subgraph is W[1]-hard.
Our FPT algorithm is based on a novel combinatorial result on the treewidth of minimal connected odd graphs with even amount of edges.

Fedor V. Fomin and Petr A. Golovach. Parameterized Complexity of Connected Even/Odd Subgraph Problems. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 432-440, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2012.432, author = {Fomin, Fedor V. and Golovach, Petr A.}, title = {{Parameterized Complexity of Connected Even/Odd Subgraph Problems}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {432--440}, 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.2012.432}, URN = {urn:nbn:de:0030-drops-33986}, doi = {10.4230/LIPIcs.STACS.2012.432}, annote = {Keywords: Parameterized complexity, Euler graph, even graph, odd graph, treewidth} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size O(k^{3/2}) in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation
ratios O(log{k}) on planar graphs and O(sqrt{k} log{k}) on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.

Fedor V. Fomin, Geevarghese Philip, and Yngve Villanger. Minimum Fill-in of Sparse Graphs: Kernelization and Approximation. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 164-175, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2011.164, author = {Fomin, Fedor V. and Philip, Geevarghese and Villanger, Yngve}, title = {{Minimum Fill-in of Sparse Graphs: Kernelization and Approximation}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {164--175}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.164}, URN = {urn:nbn:de:0030-drops-33451}, doi = {10.4230/LIPIcs.FSTTCS.2011.164}, annote = {Keywords: Minimum Fill-In, Approximation, Kernelization, Sparse graphs} }

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**Published in:** Dagstuhl Reports, Volume 1, Issue 2 (2011)

From February 14, 2012 to February 18, 2012, the Dagstuhl Seminar 11071
``Theory and Applications of Graph Searching Problems (GRASTA 2011)''
was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and open problems are put together in this paper. The first section describes the seminar topics and goals in general.
The second section contains the abstracts of the talks and the third section
includes the open problems presented during the seminar.

Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer, and Dimitrios M. Thilikos. Theory and Applications of Graph Searching Problems (GRASTA 2011) (Dagstuhl Seminar 11071). In Dagstuhl Reports, Volume 1, Issue 2, pp. 30-46, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@Article{fomin_et_al:DagRep.1.2.30, author = {Fomin, Fedor V. and Fraigniaud, Pierre and Kreutzer, Stephan and Thilikos, Dimitrios M.}, title = {{Theory and Applications of Graph Searching Problems (GRASTA 2011) (Dagstuhl Seminar 11071)}}, pages = {30--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2011}, volume = {1}, number = {2}, editor = {Fomin, Fedor V. and Fraigniaud, Pierre and Kreutzer, Stephan and Thilikos, Dimitrios M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.1.2.30}, URN = {urn:nbn:de:0030-drops-31534}, doi = {10.4230/DagRep.1.2.30}, annote = {Keywords: Graph Searching, Pursuit Evasion Games, Cop and Robers Games, Fugitive Search Games} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

We study a general class of problems called F-Deletion problems. In an F-Deletion problem, we are asked whether a subset of at most k
vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-Deletion problem when F contains a planar graph. We give
- a linear vertex kernel on graphs excluding t-claw K_(1,t), the star with t leaves, as an induced subgraph, where t is a fixed integer.
- an approximation algorithm achieving an approximation ratio of O(log^(3/2) OPT), where $OPT$ is the size of an optimal solution on general undirected graphs.
Finally, we obtain polynomial kernels for the case when F only contains graph theta_c as a minor for a fixed integer c. The graph theta_c consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, and Saket Saurabh. Hitting forbidden minors: Approximation and Kernelization. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 189-200, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2011.189, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Misra, Neeldhara and Philip, Geevarghese and Saurabh, Saket}, title = {{Hitting forbidden minors: Approximation and Kernelization}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {189--200}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.189}, URN = {urn:nbn:de:0030-drops-30103}, doi = {10.4230/LIPIcs.STACS.2011.189}, annote = {Keywords: kernelization} }

Document

**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs.
We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs.
We exemplify our approaches with two well studied problems. For the first problem, $k$-Leaf Out-Branching, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{\cO(\sqrt{k} \log k)} n+ n^{\cO(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{\cO(\sqrt{k} )}n + n^{\cO(1)}$.
The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely $k$-Internal Out-Branching, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{\cO(\sqrt{k} \log k)} + n^{\cO(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor.
Finally, we observe that for any $\ve>0$, the $k$-Directed Path problem is solvable in time $\cO((1+\ve)^k n^{f(\ve)})$, where $f$ is some function of $\ve$.
Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

Frederic Dorn, Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 251-262, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dorn_et_al:LIPIcs.STACS.2010.2459, author = {Dorn, Frederic and Fomin, Fedor V. and Lokshtanov, Daniel and Raman, Venkatesh and Saurabh, Saket}, title = {{Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {251--262}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2459}, URN = {urn:nbn:de:0030-drops-24599}, doi = {10.4230/LIPIcs.STACS.2010.2459}, annote = {Keywords: Parameterized Subexponential Algorithms, Directed Graphs, Out-Branching, Internal Out-Branching} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potential maximal cliques times $O(n^{O(t)})$ to find a maximum induced subgraph of treewidth t in G and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F.
Combined with an improved algorithm enumerating all potential maximal cliques in time $O(1.734601^n)$, this yields that both the problems are solvable in time $1.734601^n$ * $n^{O(t)}$.

Fedor V. Fomin and Yngve Villanger. Finding Induced Subgraphs via Minimal Triangulations. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 383-394, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2010.2470, author = {Fomin, Fedor V. and Villanger, Yngve}, title = {{Finding Induced Subgraphs via Minimal Triangulations}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {383--394}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2470}, URN = {urn:nbn:de:0030-drops-24708}, doi = {10.4230/LIPIcs.STACS.2010.2470}, annote = {Keywords: Bounded treewidth, minimal triangulation, moderately exponential time algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9511, Parameterized complexity and approximation algorithms (2010)

We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory – the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters.

Fedor V. Fomin, Petr Golovach, and Dimitrios M. Thilikos. Contraction Bidimensionality: the Accurate Picture. In Parameterized complexity and approximation algorithms. Dagstuhl Seminar Proceedings, Volume 9511, pp. 1-12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2010)

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@InProceedings{fomin_et_al:DagSemProc.09511.5, author = {Fomin, Fedor V. and Golovach, Petr and Thilikos, Dimitrios M.}, title = {{Contraction Bidimensionality: the Accurate Picture}}, booktitle = {Parameterized complexity and approximation algorithms}, pages = {1--12}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9511}, editor = {Erik D. Demaine and MohammadTaghi Hajiaghayi and D\'{a}niel Marx}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09511.5}, URN = {urn:nbn:de:0030-drops-25009}, doi = {10.4230/DagSemProc.09511.5}, annote = {Keywords: Paramerterized Algorithms, Bidimensionality, Graph Minors} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

A tournament $T = (V,A)$ is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on $n$ vertices and an integer parameter $k$, the {\sc Feedback Arc Set} problem asks whether thegiven digraph has a set of $k$ arcs whose removal results in an acyclicdigraph. The {\sc Feedback Arc Set} problem restricted to tournaments is knownas the {\sc $k$-Feedback Arc Set in Tournaments ($k$-FAST)} problem. In thispaper we obtain a linear vertex kernel for \FAST{}. That is, we give apolynomial time algorithm which given an input instance $T$ to \FAST{} obtains an equivalent instance $T'$ on $O(k)$ vertices. In fact, given any fixed $\epsilon > 0$, the kernelized instance has at most $(2 + \epsilon)k$ vertices.Our result improves the previous known bound of $O(k^2)$ on the kernel size for\FAST{}. Our kernelization algorithm solves the problem on a subclass of
tournaments in polynomial time and uses a known polynomial time approximation
scheme for \FAST.

Stéphane Bessy, Fedor V. Fomin, Serge Gaspers, Christophe Paul, Anthony Perez, Saket Saurabh, and Stéphan Thomassé. Kernels for Feedback Arc Set In Tournaments. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 37-47, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bessy_et_al:LIPIcs.FSTTCS.2009.2305, author = {Bessy, St\'{e}phane and Fomin, Fedor V. and Gaspers, Serge and Paul, Christophe and Perez, Anthony and Saurabh, Saket and Thomass\'{e}, St\'{e}phan}, title = {{Kernels for Feedback Arc Set In Tournaments}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {37--47}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2305}, URN = {urn:nbn:de:0030-drops-23055}, doi = {10.4230/LIPIcs.FSTTCS.2009.2305}, annote = {Keywords: Parameterized complexity, kernels, tournaments} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Partial Cover problems are optimization versions
of fundamental and well studied problems like {\sc Vertex Cover} and {\sc Dominating Set}.
Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number ($k$) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by $k$.
It was recently shown by Amini et. al. [{\em FSTTCS 08}\,] that {\sc Partial Vertex Cover} and {\sc Partial Dominating Set} are fixed parameter tractable on large classes of sparse graphs, namely $H$-minor free graphs,
which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time $2^{\cO(k)}n^{\cO(1)}$.

Fedor V. Fomin, Daniel Lokshtanov, Venkatesh Raman, and Saket Saurabh. Subexponential Algorithms for Partial Cover Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 193-201, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{fomin_et_al:LIPIcs.FSTTCS.2009.2318, author = {Fomin, Fedor V. and Lokshtanov, Daniel and Raman, Venkatesh and Saurabh, Saket}, title = {{Subexponential Algorithms for Partial Cover Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {193--201}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2318}, URN = {urn:nbn:de:0030-drops-23186}, doi = {10.4230/LIPIcs.FSTTCS.2009.2318}, annote = {Keywords: Partial cover problems, parameterized complexity, subexponential time algorithms, irrelevant vertex technique} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching, that is a rooted oriented spanning tree, with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics.
For the {\sc $k$-Leaf-Out-Branching} problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first ones separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding ``cheat kernelization'' raised by Mike Fellows and Jiong Guo independently.

Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh, and Yngve Villanger. Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 421-432, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{fernau_et_al:LIPIcs.STACS.2009.1843, author = {Fernau, Henning and Fomin, Fedor V. and Lokshtanov, Daniel and Raible, Daniel and Saurabh, Saket and Villanger, Yngve}, title = {{Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {421--432}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1843}, URN = {urn:nbn:de:0030-drops-18437}, doi = {10.4230/LIPIcs.STACS.2009.1843}, annote = {Keywords: Parameterized algorithms, Kernelization, Out-branching, Max-leaf, Lower bounds} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello (PODS'99, PODS'01) in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx in SODA'06, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. Computing each of these width parameters is known to be an NP-hard problem. Moreover, the (generalized) hypertree width of an n-vertex hypergraph cannot be approximated within a logarithmic factor unless P=NP. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class. This extends the results of Feige, Hajiaghayi, and Lee from STOC'05 on approximating treewidth of H-minor-free graphs.

Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Approximating Acyclicity Parameters of Sparse Hypergraphs. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 445-456, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2009)

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@InProceedings{fomin_et_al:LIPIcs.STACS.2009.1803, author = {Fomin, Fedor V. and Golovach, Petr A. and Thilikos, Dimitrios M.}, title = {{Approximating Acyclicity Parameters of Sparse Hypergraphs}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {445--456}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1803}, URN = {urn:nbn:de:0030-drops-18034}, doi = {10.4230/LIPIcs.STACS.2009.1803}, annote = {Keywords: Graph, Hypergraph, Hypertree width, Treewidth} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8431, Moderately Exponential Time Algorithms (2008)

From $19/10/2008$ to $24/10/2008$, the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Fedor V. Fomin, Kazuo Iwama, and Dieter Kratsch. 08431 Abstracts Collection – Moderately Exponential Time Algorithms. In Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, Volume 8431, pp. 1-22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{fomin_et_al:DagSemProc.08431.1, author = {Fomin, Fedor V. and Iwama, Kazuo and Kratsch, Dieter}, title = {{08431 Abstracts Collection – Moderately Exponential Time Algorithms}}, booktitle = {Moderately Exponential Time Algorithms}, pages = {1--22}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8431}, editor = {Fedor V. Fomin and Kazuo Iwama and Dieter Kratsch}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08431.1}, URN = {urn:nbn:de:0030-drops-18004}, doi = {10.4230/DagSemProc.08431.1}, annote = {Keywords: Algorithms, Exponential time algorithms, Graphs, SAT} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8431, Moderately Exponential Time Algorithms (2008)

The Dagstuhl seminar on Moderately Exponential Time Algorithms took place
from 19.10.08 to 24.10.08. The 54 participants came from 18 countries.
There were 27 talks and 2 open problem sessions. Talks were complemented
by intensive informal discussions, and many new research directions
and open problems will result from these discussions. The warm and encouraging Dagstuhl atmosphere stimulated new research projects.

Fedor V. Fomin, Kazuo Iwama, and Dieter Kratsch. 08431 Executive Summary – Moderately Exponential Time Algorithms. In Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, Volume 8431, pp. 1-2, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{fomin_et_al:DagSemProc.08431.2, author = {Fomin, Fedor V. and Iwama, Kazuo and Kratsch, Dieter}, title = {{08431 Executive Summary – Moderately Exponential Time Algorithms}}, booktitle = {Moderately Exponential Time Algorithms}, pages = {1--2}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8431}, editor = {Fedor V. Fomin and Kazuo Iwama and Dieter Kratsch}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08431.2}, URN = {urn:nbn:de:0030-drops-17976}, doi = {10.4230/DagSemProc.08431.2}, annote = {Keywords: Algorithms, NP-hard problems, Exact algorithms, Moderately Exponential Time Algorithms} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8431, Moderately Exponential Time Algorithms (2008)

Two problem sessions were part of the seminar on Moderately Exponential Time Algorithms. Some of the open problems presented at those sessions have been collected.

Fedor V. Fomin, Kazuo Iwama, Dieter Kratsch, Petteri Kaski, Mikko Koivisto, Lukasz Kowalik, Yoshio Okamoto, Johan van Rooij, and Ryan Williams. 08431 Open Problems – Moderately Exponential Time Algorithms. In Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, Volume 8431, pp. 1-8, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{fomin_et_al:DagSemProc.08431.3, author = {Fomin, Fedor V. and Iwama, Kazuo and Kratsch, Dieter and Kaski, Petteri and Koivisto, Mikko and Kowalik, Lukasz and Okamoto, Yoshio and van Rooij, Johan and Williams, Ryan}, title = {{08431 Open Problems – Moderately Exponential Time Algorithms}}, booktitle = {Moderately Exponential Time Algorithms}, pages = {1--8}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8431}, editor = {Fedor V. Fomin and Kazuo Iwama and Dieter Kratsch}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08431.3}, URN = {urn:nbn:de:0030-drops-17986}, doi = {10.4230/DagSemProc.08431.3}, annote = {Keywords: Algorithms, NP-hard problems, Moderately Exponential Time Algorithms} }

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

We prove the following result about approximating the maximum independent set in a graph. Informally, we show that any approximation algorithm with a "non-trivial" approximation ratio (as a function of the number of vertices of the input graph G) can be turned into an approximation algorithm achieving almost the same ratio, albeit as a function of the treewidth of G. More formally, we prove that for any function f, the existence of a polynomial time (n/f(n))-approximation algorithm yields the existence of a polynomial time O(tw⋅log{f(tw)}/f(tw))-approximation algorithm, where n and tw denote the number of vertices and the width of a given tree decomposition of the input graph. By pipelining our result with the state-of-the-art O(n ⋅ (log log n)²/log³n)-approximation algorithm by Feige (2004), this implies an O(tw⋅(log log tw)³/log³tw)-approximation algorithm.

Parinya Chalermsook, Fedor Fomin, Thekla Hamm, Tuukka Korhonen, Jesper Nederlof, and Ly Orgo. Polynomial-Time Approximation of Independent Set Parameterized by Treewidth. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 33:1-33:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chalermsook_et_al:LIPIcs.ESA.2023.33, author = {Chalermsook, Parinya and Fomin, Fedor and Hamm, Thekla and Korhonen, Tuukka and Nederlof, Jesper and Orgo, Ly}, title = {{Polynomial-Time Approximation of Independent Set Parameterized by Treewidth}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {33:1--33:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.33}, URN = {urn:nbn:de:0030-drops-186865}, doi = {10.4230/LIPIcs.ESA.2023.33}, annote = {Keywords: Maximum Independent Set, Treewidth, Approximation Algorithms, Parameterized Approximation} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

For a graph G, a set D subseteq V(G) is called a [1,j]-dominating set if every vertex in V(G) setminus D has at least one and at most j neighbors in D. A set D subseteq V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study [1,j]-Dominating Set on sparse graph classes from the perspective of parameterized complexity and prove the following results when the problem is parameterized by solution size:
- [1,j]-Dominating Set is W[1]-hard on d-degenerate graphs for d = j + 1;
- [1,j]-Dominating Set is FPT on nowhere dense graphs.
We also prove that the known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH). Finally, assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.

Mohsen Alambardar Meybodi, Fedor Fomin, Amer E. Mouawad, and Fahad Panolan. On the Parameterized Complexity of [1,j]-Domination Problems. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 34:1-34:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alambardarmeybodi_et_al:LIPIcs.FSTTCS.2018.34, author = {Alambardar Meybodi, Mohsen and Fomin, Fedor and Mouawad, Amer E. and Panolan, Fahad}, title = {{On the Parameterized Complexity of \lbrack1,j\rbrack-Domination Problems}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.34}, URN = {urn:nbn:de:0030-drops-99330}, doi = {10.4230/LIPIcs.FSTTCS.2018.34}, annote = {Keywords: \lbrack1, j\rbrack-dominating set, parameterized complexity, sparse graphs} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time.

Fedor Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 39:1-39:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fomin_et_al:LIPIcs.SoCG.2016.39, author = {Fomin, Fedor and Kolay, Sudeshna and Lokshtanov, Daniel and Panolan, Fahad and Saurabh, Saket}, title = {{Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {39:1--39:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.39}, URN = {urn:nbn:de:0030-drops-59310}, doi = {10.4230/LIPIcs.SoCG.2016.39}, annote = {Keywords: Rectilinear graphs, Steiner arborescence, parameterized algorithms} }

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Extended Abstract

**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

Covering problems are fundamental
classical problems in optimization, computer science and complexity
theory. Typically an input to these problems is a family of sets over
a finite universe and the goal is to cover the elements of the
universe with as few sets of the family as possible.
The variations of covering problems
include well known problems like Set Cover, Vertex Cover,
Dominating Set and Facility Location to name a few. Recently
there has been a lot of study on partial covering problems, a
natural generalization of covering problems. Here, the goal is not to
cover all the elements but to cover the specified number of
elements with the minimum number of sets.

Omid Amini, Fedor Fomin, and Saket Saurabh. Implicit Branching and Parameterized Partial Cover Problems (Extended Abstract). In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 1-12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2008)

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@InProceedings{amini_et_al:LIPIcs.FSTTCS.2008.1736, author = {Amini, Omid and Fomin, Fedor and Saurabh, Saket}, title = {{Implicit Branching and Parameterized Partial Cover Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {1--12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1736}, URN = {urn:nbn:de:0030-drops-17363}, doi = {10.4230/LIPIcs.FSTTCS.2008.1736}, annote = {Keywords: Implicit Branching, Parameterized Algorithms, Partial Dominating Set, Partial Vertex Cover, Local Treewidth} }

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