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

1999
*
Algorithmica
*

In this paper we study the problem

doi:10.1007/pl00009257
fatcat:ums76np4wzckfjfxwiwq4bhfla
*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*...##
###
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

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

An important special case

doi:10.4230/lipics.mfcs.2020.70
dblp:conf/mfcs/MisraP00S20
fatcat:zvxz7xaj5vairbvhncasrxqze4
*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*. ...##
###
Structural Parameterizations with Modulator Oblivion
[article]

2020
*
arXiv
*
pre-print

Instead, we construct

arXiv:2002.09972v1
fatcat:m5rmbaghtjfwfgnvmftp2kq4o4
*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. ...##
###
Bounding the mim‐width of hereditary graph classes

2021
*
Journal of Graph Theory
*

We also prove

doi:10.1002/jgt.22730
fatcat:xk56ne2wdrcoza34vlcvybep3q
*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. ...##
###
Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-width
[article]

2022
*
arXiv
*
pre-print

The

arXiv:1910.00887v8
fatcat:iuxxwa6gdncmrfowgnqm4ax32i
*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. ...##
###
Complexity and exact algorithms for vertex multicut in interval and bounded treewidth graphs

2008
*
European Journal of Operational Research
*

It is defined as: given an undirected graph

doi:10.1016/j.ejor.2007.02.014
fatcat:n2kpld7wovcjney3cz2yrbjta4
*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. ...##
###
Bounding the Mim-Width of Hereditary Graph Classes
[article]

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. ...

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

2020
*
arXiv
*
pre-print

The survey provides an overview

arXiv:2001.06867v2
fatcat:6ogysat6n5drpbxsvs63zjmega
*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 ...##
###
Computing Subset Transversals in H-Free Graphs
[article]

2021
*
arXiv
*
pre-print

We study the computational complexity

arXiv:2005.13938v3
fatcat:m234ciidwrclhf4hmpaxo2jsja
*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. ...##
###
Vertex Deletion Parameterized by Elimination Distance and Even Less
[article]

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. ...

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

2017
*
Lecture Notes in Computer Science
*

Here we give the first polynomial-time algorithms

doi:10.1007/978-3-662-55751-8_30
fatcat:zl7n67a34beyzfvirj73zq7sgm
*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 ...##
###
Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation
[article]

2022
*
arXiv
*
pre-print

This problem, given

arXiv:2207.07425v1
fatcat:qd63kas2xrbzbkjustsybrxudu
*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. ...##
###
Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams
[article]

2015
*
arXiv
*
pre-print

We focus on the Voronoi diagram

arXiv:1504.05476v1
fatcat:2fz7pzmuubafpbdzehccv4zlyi
*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. ...##
###
Distributed Evaluation of Subgraph Queries Using Worstcase Optimal LowMemory Dataflows
[article]

2018
*
arXiv
*
pre-print

We study the problem

arXiv:1802.03760v1
fatcat:cjlrsnng3zep3k2m3sequqpzxm
*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 ...
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