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**Published in:** Dagstuhl Reports, Volume 8, Issue 10 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 18421 "Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative".
Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph or particular satisfying assignments of logical expressions. Enumeration problems arise in a natural way in various fields of Computer Science, as, e.g., Artificial Intelligence and Data Mining, in Natural Sciences Engineering, Social Sciences, and Biology. The main challenge of the area of enumeration problems is that, contrary to decision, optimization and even counting problems, the output length of an enumeration problem is often exponential in the size of the input and cannot be neglected. This makes enumeration problems more challenging than many other types of algorithmic problems and demands development of specific techniques.
The principal goals of our Dagstuhl seminar were to increase the visibility of algorithmic enumeration within (Theoretical) Computer Science and to contribute to establishing it as an area of ``Algorithms and Complexity''. The seminar brought together researchers within the algorithms community, other fields of Computer Science and Computer Engineering, as well as researchers working on enumeration problems in other application areas, in particular Biology. The aim was to accelerate developments and discus new directions including algorithmic tools and hardness proofs.

Henning Fernau, Petr. A. Golovach, and Marie-France Sagot. Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative (Dagstuhl Seminar 18421). In Dagstuhl Reports, Volume 8, Issue 10, pp. 63-86, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{fernau_et_al:DagRep.8.10.63, author = {Fernau, Henning and Golovach, Petr. A. and Sagot, Marie-France}, title = {{Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative (Dagstuhl Seminar 18421)}}, pages = {63--86}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {8}, number = {10}, editor = {Fernau, Henning and Golovach, Petr. A. and Sagot, Marie-France}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.10.63}, URN = {urn:nbn:de:0030-drops-103483}, doi = {10.4230/DagRep.8.10.63}, annote = {Keywords: algorithms, input-sensitive enumeration, output-sensitive enumeration} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

We study the kernelization of exploration problems on temporal graphs. A temporal graph consists of a finite sequence of snapshot graphs 𝒢 = (G₁, G₂, … , G_L) that share a common vertex set but might have different edge sets. The non-strict temporal exploration problem (NS-TEXP for short) introduced by Erlebach and Spooner, asks if a single agent can visit all vertices of a given temporal graph where the edges traversed by the agent are present in non-strict monotonous time steps, i.e., the agent can move along the edges of a snapshot graph with infinite speed. The exploration must at the latest be completed in the last snapshot graph. The optimization variant of this problem is the k-arb NS-TEXP problem, where the agent’s task is to visit at least k vertices of the temporal graph. We show that under standard computational complexity assumptions, neither of the problems NS-TEXP nor k-arb NS-TEXP allow for polynomial kernels in the standard parameters: number of vertices n, lifetime L, number of vertices to visit k, and maximal number of connected components per time step γ; as well as in the combined parameters L+k, L + γ, and k+γ. On the way to establishing these lower bounds, we answer a couple of questions left open by Erlebach and Spooner.
We also initiate the study of structural kernelization by identifying a new parameter of a temporal graph p(𝒢) = ∑_{i=1}^L (|E(G_i)|) - |V(G)| + 1. Informally, this parameter measures how dynamic the temporal graph is. Our main algorithmic result is the construction of a polynomial (in p(𝒢)) kernel for the more general Weighted k-arb NS-TEXP problem, where weights are assigned to the vertices and the task is to find a temporal walk of weight at least k.

Emmanuel Arrighi, Fedor V. Fomin, Petr A. Golovach, and Petra Wolf. Kernelizing Temporal Exploration Problems. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arrighi_et_al:LIPIcs.IPEC.2023.1, author = {Arrighi, Emmanuel and Fomin, Fedor V. and Golovach, Petr A. and Wolf, Petra}, title = {{Kernelizing Temporal Exploration Problems}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {1:1--1:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara 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/LIPIcs.IPEC.2023.1}, URN = {urn:nbn:de:0030-drops-194201}, doi = {10.4230/LIPIcs.IPEC.2023.1}, annote = {Keywords: Temporal graph, temporal exploration, computational complexity, kernel} }

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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 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 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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Complete Volume

**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

LIPIcs, Volume 214, IPEC 2021, Complete Volume

16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 1-474, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{golovach_et_al:LIPIcs.IPEC.2021, title = {{LIPIcs, Volume 214, IPEC 2021, Complete Volume}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {1--474}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021}, URN = {urn:nbn:de:0030-drops-153828}, doi = {10.4230/LIPIcs.IPEC.2021}, annote = {Keywords: LIPIcs, Volume 214, IPEC 2021, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

Front Matter, Table of Contents, Preface, Conference Organization

16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2021.0, author = {Golovach, Petr A. and Zehavi, Meirav}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.0}, URN = {urn:nbn:de:0030-drops-153834}, doi = {10.4230/LIPIcs.IPEC.2021.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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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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**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 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

Petr A. Golovach, Christian Komusiewicz, Dieter Kratsch, and Van Bang Le. Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{golovach_et_al:LIPIcs.STACS.2021.37, author = {Golovach, Petr A. and Komusiewicz, Christian and Kratsch, Dieter and Le, Van Bang}, title = {{Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {37:1--37:18}, 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.2021.37}, URN = {urn:nbn:de:0030-drops-136823}, doi = {10.4230/LIPIcs.STACS.2021.37}, annote = {Keywords: enumeration problems, polynomial delay, output-sensitive algorithms, kernelization, structural parameterizations, matching cuts} }

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

We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results.
- For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known.
- Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

Steven Chaplick, Petr A. Golovach, Tim A. Hartmann, and Dušan Knop. Recognizing Proper Tree-Graphs. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chaplick_et_al:LIPIcs.IPEC.2020.8, author = {Chaplick, Steven and Golovach, Petr A. and Hartmann, Tim A. and Knop, Du\v{s}an}, title = {{Recognizing Proper Tree-Graphs}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.8}, URN = {urn:nbn:de:0030-drops-133118}, doi = {10.4230/LIPIcs.IPEC.2020.8}, annote = {Keywords: intersection graphs, H-graphs, recognition, fixed-parameter tractability} }

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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)

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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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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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 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 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In Clustering to Given Connectivities, we are given an n-vertex graph G, an integer k, and a sequence Lambda=<lambda_{1},...,lambda_{t}> of positive integers and we ask whether it is possible to remove at most k edges from G such that the resulting connected components are exactly t and their corresponding edge connectivities are lower-bounded by the numbers in Lambda. We prove that this problem, parameterized by k, is fixed parameter tractable, i.e., can be solved by an f(k)* n^{O(1)}-step algorithm, for some function f that depends only on the parameter k. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that we do not impose any restriction to the connectivity demands in Lambda.

Petr A. Golovach and Dimitrios M. Thilikos. Clustering to Given Connectivities. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2019.18, author = {Golovach, Petr A. and Thilikos, Dimitrios M.}, title = {{Clustering to Given Connectivities}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.18}, URN = {urn:nbn:de:0030-drops-114796}, doi = {10.4230/LIPIcs.IPEC.2019.18}, annote = {Keywords: graph clustering, edge connectivity, parameterized complexity} }

Document

**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} }

Document

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

A subset T subseteq V(G) of vertices of a graph G is said to be cyclable if G has a cycle C containing every vertex of T, and for a positive integer k, a graph G is k-cyclable if every subset of vertices of G of size at most k is cyclable. The Terminal Cyclability problem asks, given a graph G and a set T of vertices, whether T is cyclable, and the k-Cyclability problem asks, given a graph G and a positive integer k, whether G is k-cyclable. These problems are generalizations of the classical Hamiltonian Cycle problem. We initiate the study of these problems for graph classes that admit polynomial algorithms for Hamiltonian Cycle. We show that Terminal Cyclability can be solved in linear time for interval graphs, bipartite permutation graphs and cographs. Moreover, we construct certifying algorithms that either produce a solution, that is, a cycle, or output a graph separator that certifies a no-answer. We use these results to show that k-Cyclability can be solved in polynomial time when restricted to the aforementioned graph classes.

Christophe Crespelle, Carl Feghali, and Petr A. Golovach. Cyclability in Graph Classes. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{crespelle_et_al:LIPIcs.ISAAC.2019.16, author = {Crespelle, Christophe and Feghali, Carl and Golovach, Petr A.}, title = {{Cyclability in Graph Classes}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {16:1--16:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.16}, URN = {urn:nbn:de:0030-drops-115128}, doi = {10.4230/LIPIcs.ISAAC.2019.16}, annote = {Keywords: Cyclability, interval graphs, bipartite permutation graphs, cographs} }

Document

**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)

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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**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 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 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 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the Strong F-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In Strong F-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to destroy that strong copy of F in G.
We study Strong F-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when F =P_3. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization k - mu(G), where mu(G) is the maximum matching size of G. We conclude with some results on the parameterization of Strong F-closure by the number of edges of G that are not selected as strong.

Petr A. Golovach, Pinar Heggernes, Athanasios L. Konstantinidis, Paloma T. Lima, and Charis Papadopoulos. Parameterized Aspects of Strong Subgraph Closure. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golovach_et_al:LIPIcs.SWAT.2018.23, author = {Golovach, Petr A. and Heggernes, Pinar and Konstantinidis, Athanasios L. and Lima, Paloma T. and Papadopoulos, Charis}, title = {{Parameterized Aspects of Strong Subgraph Closure}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {23:1--23:13}, 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.2018.23}, URN = {urn:nbn:de:0030-drops-88490}, doi = {10.4230/LIPIcs.SWAT.2018.23}, annote = {Keywords: Strong triadic closure, Parameterized complexity, Forbidden subgraphs} }

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

Problems related to finding induced subgraphs satisfying given properties form one of the most studied areas within graph algorithms. Such problems have given rise to breakthrough results and led to development of new techniques both within the traditional P vs NP dichotomy and within parameterized complexity. The Pi-Subgraph problem asks whether an input graph contains an induced subgraph on at least k vertices satisfying graph property Pi. For many applications, it is desirable that the found subgraph has as few connections to the rest of the graph as possible, which gives rise to the Secluded Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and input t is a limit on the number of neighbors this subgraph has in the rest of the graph. This problem has been studied from a parameterized perspective, and unfortunately it turns out to be W[1]-hard for many graph properties Pi, even when parameterized by k+t. We show that the situation changes when we are looking for a connected induced subgraph satisfying Pi. In particular, we show that the Connected Secluded Pi-Subgraph problem is FPT when parameterized by just t for many important graph properties Pi.

Petr A. Golovach, Pinar Heggernes, Paloma T. Lima, and Pedro Montealegre. Finding Connected Secluded Subgraphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2017.18, author = {Golovach, Petr A. and Heggernes, Pinar and Lima, Paloma T. and Montealegre, Pedro}, title = {{Finding Connected Secluded Subgraphs}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {18:1--18:13}, 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.2017.18}, URN = {urn:nbn:de:0030-drops-85623}, doi = {10.4230/LIPIcs.IPEC.2017.18}, annote = {Keywords: Secluded subgraph, forbidden subgraphs, parameterized complexity} }

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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 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 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 of each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph is a graph that can be made planar by the removal of at most k vertices. We prove that the generalization of Square Root, in which we are given two subsets of edges prescribed to be in or out of a square root, respectively, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

Petr A. Golovach, Dieter Kratsch, Daniël Paulusma, and Anthony Stewart. A Linear Kernel for Finding Square Roots of Almost Planar Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{golovach_et_al:LIPIcs.SWAT.2016.4, author = {Golovach, Petr A. and Kratsch, Dieter and Paulusma, Dani\"{e}l and Stewart, Anthony}, title = {{A Linear Kernel for Finding Square Roots of Almost Planar Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.4}, URN = {urn:nbn:de:0030-drops-60333}, doi = {10.4230/LIPIcs.SWAT.2016.4}, annote = {Keywords: planar graphs, square roots, linear kernel} }

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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 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

The Plane Diameter Completion problem asks, given a plane graph G and a positive integer d, if it is a spanning subgraph of a plane graph H that has diameter at most d. We examine two variants of this problem where the input comes with another parameter k. In the first variant, called BPDC, k upper bounds the total number of edges to be added and in the second, called BFPDC, k upper bounds the number of additional edges per face. We prove that both problems are NP-complete, the first even for 3-connected graphs of face-degree at most 4 and the second even when k=1 on 3-connected graphs of face-degree at most 5. In this paper we give parameterized algorithms for both problems that run in O(n^{3})+2^{2^{O((kd)^2\log d)}} * n steps.

Petr A. Golovach, Clément Requilé, and Dimitrios M. Thilikos. Variants of Plane Diameter Completion. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 30-42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2015.30, author = {Golovach, Petr A. and Requil\'{e}, Cl\'{e}ment and Thilikos, Dimitrios M.}, title = {{Variants of Plane Diameter Completion}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {30--42}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.30}, URN = {urn:nbn:de:0030-drops-55697}, doi = {10.4230/LIPIcs.IPEC.2015.30}, annote = {Keywords: Planar graphs, graph modification problems, parameterized algorithms, dynamic programming, branchwidth} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Listing, generating or enumerating objects of specified type is one of the principal tasks in algorithmics. In graph algorithms one often enumerates vertex subsets satisfying a certain property. We study the enumeration of all minimal connected dominating sets of an input graph from various graph classes of bounded chordality. We establish enumeration algorithms as well as lower and upper bounds for the maximum number of minimal connected dominating sets in such graphs. In particular, we present algorithms to enumerate all minimal connected dominating sets of chordal graphs in time O(1.7159^n), of split graphs in time O(1.3803^n), and of AT-free, strongly chordal, and distance-hereditary graphs in time O^*(3^{n/3}), where n is the number of vertices of the input graph. Our algorithms imply corresponding upper bounds for the number of minimal connected dominating sets for these graph classes.

Petr A. Golovach, Pinar Heggernes, and Dieter Kratsch. Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 307-318, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{golovach_et_al:LIPIcs.IPEC.2015.307, author = {Golovach, Petr A. and Heggernes, Pinar and Kratsch, Dieter}, title = {{Enumerating Minimal Connected Dominating Sets in Graphs of Bounded Chordality}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {307--318}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.307}, URN = {urn:nbn:de:0030-drops-55925}, doi = {10.4230/LIPIcs.IPEC.2015.307}, annote = {Keywords: Minimal connected dominating set, exact algorithms, enumeration} }

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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 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0.
Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value.
As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

Konrad K. Dabrowski, Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Editing to Eulerian Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 97-108, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dabrowski_et_al:LIPIcs.FSTTCS.2014.97, author = {Dabrowski, Konrad K. and Golovach, Petr A. and van 't Hof, Pim and Paulusma, Daniel}, title = {{Editing to Eulerian Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {97--108}, 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.2014.97}, URN = {urn:nbn:de:0030-drops-48356}, doi = {10.4230/LIPIcs.FSTTCS.2014.97}, annote = {Keywords: Eulerian graphs, graph editing, polynomial algorithm} }

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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:** 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} }

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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:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-Free Graphs of Bounded Diameter. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brause_et_al:LIPIcs.ISAAC.2021.21, author = {Brause, Christoph and Golovach, Petr and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Partitioning H-Free Graphs of Bounded Diameter}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.21}, URN = {urn:nbn:de:0030-drops-154543}, doi = {10.4230/LIPIcs.ISAAC.2021.21}, annote = {Keywords: vertex partitioning problem, H-free, diameter, complexity dichotomy} }

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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 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:** 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} }

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