IA Scholar Query: Hitting topological minors is FPT.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSun, 20 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On the parameterized complexity of computing tree-partitions
https://scholar.archive.org/work/uxk6qxhywvdblgotkyqcrsbjdy
We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k^7) for G or reports that G has tree-partition width more than k, in time k^O(1)n^2. We can improve on the approximation factor or the dependence on n by sacrificing the dependence on k. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.Hans L. Bodlaender and Carla Groenland and Hugo Jacobwork_uxk6qxhywvdblgotkyqcrsbjdySun, 20 Nov 2022 00:00:00 GMTA Framework for Approximation Schemes on Disk Graphs
https://scholar.archive.org/work/otomykizzfcrvepuoxq6yroyoa
We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), P_k-Hitting for k∈{3,4,5}, Path Deletion, Pathwidth 1-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, Meirav Zehaviwork_otomykizzfcrvepuoxq6yroyoaFri, 04 Nov 2022 00:00:00 GMTHitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable
https://scholar.archive.org/work/hwtsnv2fqfcezcbl2bjtqxls7q
For a finite collection of graphs F, the F-TM-Deletion problem has as input an n-vertex graph G and an integer k and asks whether there exists a set S ⊆ V(G) with |S| ≤ k such that G ∖ S does not contain any of the graphs in F as a topological minor. We prove that for every such F, F-TM-Deletion is fixed parameter tractable on planar graphs. Our algorithm runs in a 2^𝒪(k^2)· n^2 time or, alternatively in 2^𝒪(k)· n^4 time. Our techniques can easily be extended to graphs that are embeddable on any fixed surface.Petr A. Golovach, Giannos Stamoulis, Dimitrios M. Thilikoswork_hwtsnv2fqfcezcbl2bjtqxls7qMon, 31 Oct 2022 00:00:00 GMTK-apices Of Minor-closed Graph Classes. I. Bounding The Obstructions
https://scholar.archive.org/work/geliif4ebrfzpgjcwpodot57di
Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G∖ S belongs to G. We denote by A_k ( G) the set of all graphs that are k-apices of G. We prove that every graph in the obstruction set of A_k ( G), i.e., the minor-minimal set of graphs not belonging to A_k ( G), has size at most 2^2^2^2^ poly(k), where poly is a polynomial function whose degree depends on the size of the minor-obstructions of G. This bound drops to 2^2^ poly(k) when G excludes some apex graph as a minor.Ignasi Sau, Giannos Stamoulis, Dimitrios M. Thilikoswork_geliif4ebrfzpgjcwpodot57diWed, 05 Oct 2022 00:00:00 GMTParameterized algorithmics for time-evolving structures: temporalizing and multistaging
https://scholar.archive.org/work/uyaoernzvvb7rmwysi2x6b5yzi
The thesis studies temporal graph problems and multistage problems. Since these problems typically are computationally hard, the focus is on developing fast exact (FPT-)algorithms. Temporal graph problems. A temporal graph is a graph whose edge set changes over time. Here, an edge at a specific time step is called time-edge. One of our main contributions is the introduction of a set of parameters tailored for temporal graph problems. We focus mainly on four problems on temporal graphs. Minimizing Reachability by Delaying. Given a temporal graph, a set of source vertices, and three integers k, r, and δ, the problem Minimizing Temporal Reachability by Delaying asks whether we can delay at most k time-edges by δ time steps (i.e., moving the edges δ time steps into the future) such that the sources can reach at most r vertices via temporal paths (i.e., paths using edges appearing in non-decreasing time-order). Our main contribution here is an algorithm running in O(r!k|G|) time, where |G| is the size of the temporal graph. This stands in contrast to the W[1]-hardness when parameterized by r for the problem of deleting instead of delaying time-edges. Restless Temporal Paths. A restless temporal path is a temporal path that can stay only a bounded amount of time at one vertex. Our main contribution here is a randomized algorithm to find a length-at-most-k restless temporal path from vertex s to vertex z in 4^ℓ |G|^O(1) time, where ℓ is the difference between k and the length of the shortest temporal path from s to z. Moreover, we show that finding these restless temporal paths is fixed-parameter tractable when parameterized by the timed feedback vertex number (that is, a temporal version of the classical feedback vertex number introduced in this thesis). This stands in contrast to the W[1]-hardness when parameterized by the feedback vertex number of the underlying graph. Temporal Separation. A temporal separator is a vertex set that intersects the vertices of all temporal paths between two distinguished vertices. We co [...]Philipp Zschoche, Technische Universität Berlin, Rolf Niedermeierwork_uyaoernzvvb7rmwysi2x6b5yziThu, 15 Sep 2022 00:00:00 GMTPlanarizing Graphs and their Drawings by Vertex Splitting
https://scholar.archive.org/work/cwbm5bksqnewzcqghgyadfas4i
The splitting number of a graph G=(V,E) is the minimum number of vertex splits required to turn G into a planar graph, where a vertex split removes a vertex v ∈ V, introduces two new vertices v_1, v_2, and distributes the edges formerly incident to v among its two split copies v_1, v_2. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in ℝ^2, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform fixed-parameter tractable (FPT) algorithm for the splitting number problem (without drawings). Then we show the NP-completeness of the splitting number problem for graph drawings, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.Martin Nöllenburg and Manuel Sorge and Soeren Terziadis and Anaïs Villedieu and Hsiang-Yun Wu and Jules Wulmswork_cwbm5bksqnewzcqghgyadfas4iThu, 08 Sep 2022 00:00:00 GMTHitting forbidden induced subgraphs on bounded treewidth graphs
https://scholar.archive.org/work/6iykiie74bdhdjptc5ydxh6xqy
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ⊆ V(G) such that G∖ S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) · n^O(1) assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^O(t^h-2) for every graph H on h ≥ 3 vertices, and that f_H(t) = 2^O(t) if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^Θ(t^h-2). We also show that f_H(t) = 2^Ω(t^h) when H=K_h,h, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_C_4(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h≤ 2 the problem can be solved in polynomial time; if h≥ 3, f_H(t) = 2^Θ(t) if H is a clique, and f_H(t) = 2^Θ(t^h-2) otherwise.Ignasi Sau, Uéverton S. Souzawork_6iykiie74bdhdjptc5ydxh6xqyThu, 08 Sep 2022 00:00:00 GMTVertex Deletion Parameterized by Elimination Distance and Even Less
https://scholar.archive.org/work/sdzr3cd7lrdmnjd5v32mha6lde
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · n^O(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · n^O(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost k^O(1) in time f(k) · n^O(1). This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczykwork_sdzr3cd7lrdmnjd5v32mha6ldeMon, 18 Jul 2022 00:00:00 GMTFixed-Parameter Tractability of Maximum Colored Path and Beyond
https://scholar.archive.org/work/vym4c25mvjhtngaqs7ucqbposm
We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored n-vertex undirected graph, vertices s and t, and an integer k, finds an (s,t)-path containing at least k different colors in time 2^k n^O(1). This is the first FPT algorithm for this problem, and it generalizes the algorithm of Björklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through k specified vertices. It also implies the first 2^k n^O(1) time algorithm for finding an (s,t)-path of length at least k. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an n-vertex undirected graph G, a matroid M whose elements correspond to the vertices of G and which is represented over a finite field of order q, a positive integer weight function on the vertices of G, two sets of vertices S,T ⊆ V(G), and integers p,k,w, and the task is to find p vertex-disjoint paths from S to T so that the union of the vertices of these paths contains an independent set of M of cardinality k and weight w, while minimizing the sum of the lengths of the paths. We give a 2^p+O(k^2 log (q+k)) n^O(1) w time randomized algorithm for this problem.Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Kirill Simonov, Giannos Stamouliswork_vym4c25mvjhtngaqs7ucqbposmFri, 15 Jul 2022 00:00:00 GMTTemporal graph exploration: restrictions and relaxations
https://scholar.archive.org/work/kxeukfp2efctjgg3en7gcwhpk4
This thesis considers the problem of exploring temporal graphs. A temporal graph G = hG1; :::;GLi of order n is a sequence of L undirected graphs (or layers) indexed by the timesteps t 2 f1; : : : ;Lg, such that V (G1) = V (G) and E(Gt) E(G) for some underlying graph G with order n. To explore G is to visit each vertex at least once via a sequence of edge-traversals (called an exploration schedule), with each consecutive edge traversed during a timestep strictly greater than the last. The arrival time of an the timestep during which the last unvisited vertex is reached for the first time.There exists an algorithm producing exploration schedules with arrival time O(n2) for any always-connected (i.e., Gt is connected for all t 2 f1; : : : ;Lg) temporal graph, and an infinite family F of always-connected temporal graphs for which any exploration schedule has arrival time (n2) [38, 86]. We isolate a number of characteristics held by the members of F and prove lower/upper bounds on the arrival time of exploration schedules for temporal graphs that are restricted from possessing them. First, we consider structural restrictions in which an input temporal graph has (1) degree upper bounded by in each layer; and (2) at most k edges 'missing' from the underlying graph in each layer; subquadratic upper bounds are proved in each case. We then consider 'relaxed' exploration schedules that can traverse a ?nite number of edges ( 1) in each timestep, focusing on the cases when 2 or n=k traversals are allowed. We also consider, from a complexity standpoint, a number of relaxed problem variants, in which (1) less than n vertices are required to be explored by a candidate, and (2) an unlimited but ?nite number of edge traversals can be made by a candidate exploration schedule, providing both FPT-membership results and hardness/NP-completeness results.Jakob T. Spoonerwork_kxeukfp2efctjgg3en7gcwhpk4Thu, 14 Jul 2022 00:00:00 GMTSubexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs
https://scholar.archive.org/work/5ealjkbcerh2titoxqrtbehdqi
We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs. In particular, we obtain the following results (where k is the solution-size parameter). 1. 2^O(√(k)log k)· n^O(1) time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a 2^O(√(k)log^4 k)· n^O(1) time algorithm for Edge Multiway Cut and a 2^O(r √(k)log k)· n^O(1) time algorithm for Vertex Multiway Cut, where r is the number of terminals to be separated; 3. a 2^O((r+√(k))log^4 (rk))· n^O(1) time algorithm for Edge Multicut and a 2^O((√(rk)+r) log (rk))· n^O(1) time algorithm for Vertex Multicut, where r is the number of terminal pairs to be separated; 4. a 2^O(√(k)log g log^4 k)· n^O(1) time algorithm for Group Feedback Edge Set and a 2^O(g √(k)log(gk))· n^O(1) time algorithm for Group Feedback Vertex Set, where g is the size of the group. 5. In addition, our approach also gives n^O(√(k)) time algorithms for all above problems with the exception of n^O(r+√(k)) time for Edge/Vertex Multicut and (ng)^O(√(k)) time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an h-almost-embeddable graph for any fixed constant h. In particular we show the following. Let G be an h-almost-embeddable graph for a constant h. Then for every p∈ℕ, there exist disjoint sets Z_1,...,Z_p ⊆ V(G) such that for every i ∈{1,...,p} and every Z'⊆ Z_i, the treewidth of G/(Z_i\ Z') is O(p+|Z'|). Here G/(Z_i\ Z') is the graph obtained from G by contracting edges with both endpoints in Z_i \ Z'.Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xuework_5ealjkbcerh2titoxqrtbehdqiMon, 04 Jul 2022 00:00:00 GMTTrue Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs
https://scholar.archive.org/work/rn2s4enrtvdf5a727x3wp5ae6a
We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set 𝒟 of n unit disks inducing a unit-disk graph G_𝒟 and a number p ∈ [n], one can partition 𝒟 into p subsets 𝒟₁,... ,𝒟_p such that for every i ∈ [p] and every 𝒟' ⊆ 𝒟_i, the graph obtained from G_𝒟 by contracting all edges between the vertices in 𝒟_i $1𝒟' admits a tree decomposition in which each bag consists of O(p+|𝒟'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} ⋅ n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} ⋅ n^{O(1)} time assuming the ETH.Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xue, Xavier Goaoc, Michael Kerberwork_rn2s4enrtvdf5a727x3wp5ae6aWed, 01 Jun 2022 00:00:00 GMTAdapting the Directed Grid Theorem into an FPT Algorithm
https://scholar.archive.org/work/fsm5w7jz7zbszku3h5ln7lwf6i
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most important tools in the field of structural graph theory, finding numerous applications in the design of algorithms for undirected graphs. An analogous version of the Grid Theorem in digraphs was conjectured by Johnson et al. [JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely, they showed that there is a function f(k) such that every digraph of directed tree-width at least f(k) contains a cylindrical grid of size k as a butterfly minor and stated that their proof can be turned into an XP algorithm, with parameter k, that either constructs a decomposition of the appropriate width, or finds the claimed large cylindrical grid as a butterfly minor. In this paper, we adapt some of the steps of the proof of Kawarabayashi and Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our main technical contributions are two FPT algorithms with parameter k. The first one either produces an arboreal decomposition of width 3k-2 or finds a haven of order k in a digraph D, improving on the original result for arboreal decompositions by Johnson et al. The second algorithm finds a well-linked set of order k in a digraph D of large directed tree-width. As tools to prove these results, we show how to solve a generalized version of the problem of finding balanced separators for a given set of vertices T in FPT time with parameter |T|, a result that we consider to be of its own interest.Victor Campos, Raul Lopes, Ana Karolinna Maia, Ignasi Sauwork_fsm5w7jz7zbszku3h5ln7lwf6iThu, 12 May 2022 00:00:00 GMTApplications and Techniques for Fast Machine Learning in Science
https://scholar.archive.org/work/jsedv4ikcrejljqwaiudndn6be
In this community review report, we discuss applications and techniques for fast machine learning (ML) in science—the concept of integrating powerful ML methods into the real-time experimental data processing loop to accelerate scientific discovery. The material for the report builds on two workshops held by the Fast ML for Science community and covers three main areas: applications for fast ML across a number of scientific domains; techniques for training and implementing performant and resource-efficient ML algorithms; and computing architectures, platforms, and technologies for deploying these algorithms. We also present overlapping challenges across the multiple scientific domains where common solutions can be found. This community report is intended to give plenty of examples and inspiration for scientific discovery through integrated and accelerated ML solutions. This is followed by a high-level overview and organization of technical advances, including an abundance of pointers to source material, which can enable these breakthroughs.Allison McCarn Deiana, Nhan Tran, Joshua Agar, Michaela Blott, Giuseppe Di Guglielmo, Javier Duarte, Philip Harris, Scott Hauck, Mia Liu, Mark S. Neubauer, Jennifer Ngadiuba, Seda Ogrenci-Memik, Maurizio Pierini, Thea Aarrestad, Steffen Bähr, Jürgen Becker, Anne-Sophie Berthold, Richard J. Bonventre, Tomás E. Müller Bravo, Markus Diefenthaler, Zhen Dong, Nick Fritzsche, Amir Gholami, Ekaterina Govorkova, Dongning Guo, Kyle J. Hazelwood, Christian Herwig, Babar Khan, Sehoon Kim, Thomas Klijnsma, Yaling Liu, Kin Ho Lo, Tri Nguyen, Gianantonio Pezzullo, Seyedramin Rasoulinezhad, Ryan A. Rivera, Kate Scholberg, Justin Selig, Sougata Sen, Dmitri Strukov, William Tang, Savannah Thais, Kai Lukas Unger, Ricardo Vilalta, Belina von Krosigk, Shen Wang, Thomas K. Warburtonwork_jsedv4ikcrejljqwaiudndn6beTue, 12 Apr 2022 00:00:00 GMTFurther Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems
https://scholar.archive.org/work/cvacapsf2rcyvjx3d53qx5qeca
For a positive integer c, a graph G is said to be c-closed if every pair of non-adjacent vertices in G have at most c-1 neighbours in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by Fox et al. [ICALP '18 and SICOMP '20]. Koana et al. [ESA '20] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set and (Threshold) Dominating Set, and showed that each of these problems admits a polynomial kernel, either w.r.t. the parameter k+c or w.r.t. the parameter k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable w.r.t. the parameter k+cl(G), whereas Partial Dominating Set, parameterized by k is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs, unless NP ⊆ co-NP/poly.Lawqueen Kanesh, Jayakrishnan Madathil, Sanjukta Roy, Abhishek Sahu, Saket Saurabh, Petra Berenbrink, Benjamin Monmegework_cvacapsf2rcyvjx3d53qx5qecaWed, 09 Mar 2022 00:00:00 GMTLossy Planarization: A Constant-Factor Approximate Kernelization for Planar Vertex Deletion
https://scholar.archive.org/work/377llva5ajasfplsbob5tng6gi
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = K_5, K_3,3. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial A-approximate kernelization, for some constant A > 1, based on the framework of lossy kernelization [Lokshtanov, Panolan, Ramanujan, and Saurabh, STOC'17]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G' on poly(k) vertices so that any B-approximate solution to G' can be lifted to an (A*B)-approximate solution to G, as long as A*B*OPT(G) <= k. In order to achieve this, we develop a framework for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time O(n^eps)-approximation algorithm, for any eps > 0, and a quasi-polynomial-time (log n)^O(1) approximation algorithm, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT^eps) and (log OPT)^O(1).Bart M. P. Jansen, Michał Włodarczykwork_377llva5ajasfplsbob5tng6giFri, 04 Feb 2022 00:00:00 GMTPlanarizing Graphs and their Drawings by Vertex Splitting
https://scholar.archive.org/work/tuxwc42hufa77lvrtdonosmzbq
The splitting number of a graph $G=(V,E)$ is the minimum number of vertex splits required to turn $G$ into a planar graph, where a vertex split removes a vertex $v \in V$, introduces two new vertices $v_1, v_2$, and distributes the edges formerly incident to $v$ among its two split copies $v_1, v_2$. The splitting number problem is known to be NP-complete. In this paper we shift focus to the splitting number of graph drawings in $\mathbb R^2$, where the new vertices resulting from vertex splits can be re-embedded into the existing drawing of the remaining graph. We first provide a non-uniform fixed-parameter tractable (FPT) algorithm for the splitting number problem (without drawings). Then we show the NP-completeness of the splitting number problem for graph drawings, even for its two subproblems of (1) selecting a minimum subset of vertices to split and (2) for re-embedding a minimum number of copies of a given set of vertices. For the latter problem we present an FPT algorithm parameterized by the number of vertex splits. This algorithm reduces to a bounded outerplanarity case and uses an intricate dynamic program on a sphere-cut decomposition.Soeren Nickel, Martin Nöllenburg, Manuel Sorge, Anaïs Villedieu, Hsiang-Yun Wu, Jules Wulmswork_tuxwc42hufa77lvrtdonosmzbqTue, 01 Feb 2022 00:00:00 GMTDeleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent
https://scholar.archive.org/work/izvuywq5wrdr5az7qxcittsq6m
For a graph class H, the graph parameters elimination distance to H (denoted by ed_ H) [Bulian and Dawar, Algorithmica, 2016], and H-treewidth (denoted by tw_ H) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class H. Here, the torso of a vertex set S in a graph G is the graph with vertex set S and an edge between two vertices u, v ∈ S if there is a path between u and v in G whose internal vertices all lie outside S. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class H that satisfies mild additional conditions. In fact, we show that for every hereditary graph class H satisfying mild additional conditions, with the exception of tw_ H parameterized by ed_ H, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to H) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing ed_ H and tw_ H. The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if H is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a "strong irrelevant vertex rule", then there exists a uniform FPT algorithm for ed_ H.Akanksha Agrawal, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Meirav Zehaviwork_izvuywq5wrdr5az7qxcittsq6mFri, 07 Jan 2022 00:00:00 GMTCycle structure and colorings of directed graphs
https://scholar.archive.org/work/4oljppo7pnh3vlrxkkkho6zd4m
This thesis deals with problems from the theory of finite directed graphs. A directed graph (digraph for short) is a binary relation whose domain has finite size. With that digraphs can be seen as a very general way of representing (possibly asymmetric) relations between pairs from a finite set of objects. Undoubtedly, such a generality allows to encode many structures by digraphs. This works particularly well if important properties of the structure at hand can be expressed as relations or connections between objects. To give some selected examples, let us mention road networks, electricity networks, radio networks, the world wide web, circuits in electronic devices, or neural networks. A main focus of the thesis at hand is the investigation of properties of one of the most fundamental objects all over graph theory, the so-called cycle (sometimes also called circuit). A cycle in a graph is determined by a closed alternating sequence of cyclically connected vertices and edges. In a graph of finite size one will typically see loads of distinct cycles of various types. Therefore cycles constitute an important and recurring motive in almost all branches of graph theory, for instance, they play important roles in structural graph theory, in the theory of flows on directed networks, in theoretical characterizations of graph classes, as well as in the theory of graph colorings. Additionally, cycles play a decisive role in numerous algorithmic problems and their solutions, such as in the Traveling Salesman Problem, algorithms for finding a largest matching in a given graph, in the max-flow problem, and also in subprocedures such as Kruskal's algorithm for finding a minimum weight spanning tree. For those reasons, a substantial amount of research in graph theory has specialised on the structure of cycles in graphs. In the first major part of this thesis we deal with cycles which occur in directed graphs, and prove several necessary and sufficient theoretical conditions for the existence of cycles of certain types. Additi [...]Raphael Mario Steiner, Technische Universität Berlin, Stefan Felsnerwork_4oljppo7pnh3vlrxkkkho6zd4mThu, 30 Dec 2021 00:00:00 GMT