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Two-Way and Multiway Partitioning of a Set of Intervals for Clique-Width Maximization

A. H. Farrahi, D.-T. Lee, M. Sarrafzadeh
1999 Algorithmica  
In this paper we study the problem of partitioning a set of intervals in order to maximize the sum of the clique-widths of the partitions.  ...  G i v en a set S of intervals, it is straightforward to compute its clique-interval and clique-width.  ...  Objective: Find a k-way partitioning P = S 1 ; S 2 ; : : : ; S k o f S that maximizes the clique-width of P, w P = P k i=1 w S i . k-CWM : k-way partitioning of an interval set for clique-width maximization  ... 
doi:10.1007/pl00009257 fatcat:ums76np4wzckfjfxwiwq4bhfla

Page 6627 of Mathematical Reviews Vol. , Issue 99j [page]

1999 Mathematical Reviews  
[Lee, Der Tsai] (1-NW-EL; Evanston, IL); Sarrafzadeh, M. (1-NW-EL; Evanston, IL) Two-way and multiway partitioning of a set of intervals for clique-width maximization.  ...  The intersection of these intervals is called the clique- interval, and its length is called the clique-width.  ... 

Quick Separation in Chordal and Split Graphs

Pranabendu Misra, Fahad Panolan, Ashutosh Rai, Saket Saurabh, Roohani Sharma, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct  ...  t_i : i ∈ [𝓁]} of size at most k such that for every vertex pair (s_i,t_i), s_i and t_i are in two different connected components of G-S.  ...  Recall that a graph G is a split graph if and only if V (G) can be partitioned into two parts: C and I, such that the set G[C] is a clique and G[I] is an independent set.  ... 
doi:10.4230/lipics.mfcs.2020.70 dblp:conf/mfcs/MisraP00S20 fatcat:zvxz7xaj5vairbvhncasrxqze4

Structural Parameterizations with Modulator Oblivion [article]

Ashwin Jacob, Fahad Panolan, Venkatesh Raman, Vibha Sahlot
2020 arXiv   pre-print
Instead, we construct a tree decomposition of the given graph in time 2^O(k)n^O(1) where each bag is a union of four cliques and O(k) vertices.  ...  One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm.  ...  Now for each maximal clique we use 11 the O(2k km) algorithm for node multiway cut.  ... 
arXiv:2002.09972v1 fatcat:m5rmbaghtjfwfgnvmftp2kq4o4

Bounding the mim‐width of hereditary graph classes

Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, Daniël Paulusma
2021 Journal of Graph Theory  
We also prove a number of new results showing that, for certain H1 and H2, the class of (H1, H2)-free graphs has unbounded mim-width. Boundedness of clique-width implies boundedness of mim-width.  ...  We prove that for a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H1, H2)-free graphs.  ...  It shows how we can bound the mim-width of a graph in terms of the mim-width of the graphs induced by blocks of a partition of the vertex set and the mim-width between any two of the parts.  ... 
doi:10.1002/jgt.22730 fatcat:xk56ne2wdrcoza34vlcvybep3q

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width [article]

Benjamin Bergougnoux, Charis Papadopoulos, Jan Arne Telle
2022 arXiv   pre-print
The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for  ...  SFVS intersects all cycles containing a vertex of a given set.  ...  For clique-width, our meta-algorithm implies that we can solve SFVS and NMC in time 1) where k is the clique-width of a given clique-width expression.  ... 
arXiv:1910.00887v8 fatcat:iuxxwa6gdncmrfowgnqm4ax32i

Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

Jiong Guo, Falk Hüffner, Erhan Kenar, Rolf Niedermeier, Johannes Uhlmann
2008 European Journal of Operational Research  
It is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair.  ...  Complementing and refining previous results from the literature, we provide several NP-completeness and (fixedparameter) tractability results for restricted classes of graphs such as trees, interval graphs  ...  In particular, we are grateful to a referee who pointed out a great simplification of the proof of Theorem 5.  ... 
doi:10.1016/j.ejor.2007.02.014 fatcat:n2kpld7wovcjney3cz2yrbjta4

Bounding the Mim-Width of Hereditary Graph Classes [article]

Nick Brettell, Jake Horsfield, Andrea Munaro, Giacomo Paesani, Daniel Paulusma
2020 arXiv   pre-print
For a given graph H, the class of H-free graphs has bounded mim-width if and only if it has bounded clique-width. We show that the same is not true for (H_1,H_2)-free graphs.  ...  We also prove a number of new results showing that, for certain H_1 and H_2, the class of (H_1,H_2)-free graphs has unbounded mim-width.  ...  It shows how we can bound the mim-width of a graph in terms of the mim-width of the graphs induced by blocks of a partition of the vertex set and the mim-width between any two of the parts.  ... 
arXiv:2004.05018v3 fatcat:327nsviufjgm7pu6iyudggiave

A survey of parameterized algorithms and the complexity of edge modification [article]

Christophe Crespelle, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach
2020 arXiv   pre-print
The survey provides an overview of the developing area of parameterized algorithms for graph modification problems.  ...  We concentrate on edge modification problems, where the task is to change a small number of adjacencies in a graph in order to satisfy some required property.  ...  We thank Marcin Pilipczuk, William Lochet, and Dekel Tsur for helpful comments. References  ... 
arXiv:2001.06867v2 fatcat:6ogysat6n5drpbxsvs63zjmega

Computing Subset Transversals in H-Free Graphs [article]

Nick Brettell, Matthew Johnson, Giacomo Paesani, Daniël Paulusma
2021 arXiv   pre-print
We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to H-free graphs, that is  ...  As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for (sP_1+P_4)-free graphs for every s≥ 1.  ...  We thank three anonymous reviewers for helpful comments that improved the presentation of several proofs.  ... 
arXiv:2005.13938v3 fatcat:m234ciidwrclhf4hmpaxo2jsja

Vertex Deletion Parameterized by Elimination Distance and Even Less [article]

Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczyk
2022 arXiv   pre-print
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  ...  We consider two classes of parameterizations which are relaxations of either treedepth of treewidth.  ...  If for any two distinct Q, Q ∈ P(P, V (G)), the intervals I(V (Q)) and I(V (Q )) would overlap, then either the paths were not maximal subpaths of P , or P would not be chordless.  ... 
arXiv:2103.09715v4 fatcat:vv3kluwdtjhqdhaflntr6msvdi

Polynomial-Time Algorithms for the Subset Feedback Vertex Set Problem on Interval Graphs and Permutation Graphs [chapter]

Charis Papadopoulos, Spyridon Tzimas
2017 Lecture Notes in Computer Science  
Here we give the first polynomial-time algorithms for the problem on two unrelated subclasses of AT-free graphs: interval graphs and permutation graphs.  ...  Given a vertex-weighted graph G = (V, E) and a set S ⊆ V , a subset feedback vertex set X is a set of the vertices of G such that the graph induced by V \ X has no cycle containing a vertex of S.  ...  Theorem 5 . 1 . 51 The number of maximal S-forests of a co-bipartite graph is at most 16n 4 and these can be enumerated in time O(n 4 ).or even on bounded linear clique-width graphs, has not been resolved  ... 
doi:10.1007/978-3-662-55751-8_30 fatcat:zl7n67a34beyzfvirj73zq7sgm

Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation [article]

Meike Hatzel and Lars Jaffke and Paloma T. Lima and Tomáš Masařík and Marcin Pilipczuk and Roohani Sharma and Manuel Sorge
2022 arXiv   pre-print
This problem, given a directed graph G, pairs of vertices (called terminals) (s_1,t_1), (s_2,t_2), and (s_3,t_3), and an integer k, asks to find a set of at most k non-terminal vertices in G that intersect  ...  We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona  ...  A division D of M is a pair (D R , D C ), where D R and D C are partitions of the rows and columns into intervals of consecutive rows and intervals of consecutive columns, respectively.  ... 
arXiv:2207.07425v1 fatcat:qd63kas2xrbzbkjustsybrxudu

Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams [article]

Dániel Marx, Michał Pilipczuk
2015 arXiv   pre-print
We focus on the Voronoi diagram of a hypothetical solution of k objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then  ...  The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs.  ...  The graph Rad(H) is a bipartite multigraph, where one partite set consists of V (H), the vertices of H, and the second partite set is F (H), the faces of H.  ... 
arXiv:1504.05476v1 fatcat:2fz7pzmuubafpbdzehccv4zlyi

Distributed Evaluation of Subgraph Queries Using Worstcase Optimal LowMemory Dataflows [article]

Khaled Ammar, Frank McSherry, Semih Salihoglu, Manas Joglekar
2018 arXiv   pre-print
We study the problem of finding and monitoring fixed-size subgraphs in a continually changing large-scale graph.  ...  We present the first approach that (i) performs worst-case optimal computation and communication, (ii) maintains a total memory footprint linear in the number of input edges, and (iii) scales down per-worker  ...  LFTJ-Inc uses another set of indices called sensitivity indices which, for each prefix p, store the set of intervals in the extensions of p such that any update to these intervals could result in the output  ... 
arXiv:1802.03760v1 fatcat:cjlrsnng3zep3k2m3sequqpzxm
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