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Polynomial Treedepth Bounds in Linear Colorings
[article]

2018
*
arXiv
*
pre-print

We establish a

arXiv:1802.09665v4
fatcat:sxvjaosnqrf3pncztwliuvz4lu
*polynomial*upper*bound*on the*treedepth**in*general graphs, and give tighter*bounds**in*trees and interval graphs via constructive*coloring*algorithms. ... Low-*treedepth**colorings*are an important tool for algorithms that exploit structure*in*classes of*bounded*expansion; they guarantee subgraphs that use few*colors*have*bounded**treedepth*. ... Acknowledgments The authors would like to thank Felix Reidl and Fernando Sánchez-Villaamil for bringing these*colorings*to our attention and several anonymous reviewers for their helpful suggestion. ...##
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Polynomial Treedepth Bounds in Linear Colorings

2020
*
Algorithmica
*

We establish a

doi:10.1007/s00453-020-00760-0
fatcat:6df2hicsfra3baazchzkd2aime
*polynomial*upper*bound*on the*treedepth**in*general graphs, and give tighter*bounds**in*trees and interval graphs via constructive*coloring*algorithms. ... Low-*treedepth**colorings*are an important tool for algorithms that exploit structure*in*classes of*bounded*expansion; they guarantee subgraphs that use few*colors*have*bounded**treedepth*. ... The p-*linear**colorings*are computable*in**polynomial*time and require a constant number of*colors**in*classes of*bounded*expansion, while inducing graphs of*bounded**treedepth*for all small sets of*colors*...##
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Improved bounds for the excluded-minor approximation of treedepth
[article]

2019
*
arXiv
*
pre-print

We also show an application of our techniques for approximation algorithms of

arXiv:1904.13077v2
fatcat:yfpigc5rlfch3adcvmj3jdptpm
*treedepth*: given a graph G of*treedepth*k and treewidth t, one can*in**polynomial*time compute a*treedepth*decomposition of ... The main technical ingredient*in*our result is a proof that every tree of*treedepth*d contains a subcubic subtree of*treedepth*at least d ·log_3 ((1+√(5))/2). ... [7] provided a*polynomial*relation between the*treedepth*and the minimum number of*colors**in*a*linear**coloring*; by replacing their usage of [6] by our result (and using an improved*bound*for the excluded ...##
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Improved Bounds for the Excluded-Minor Approximation of Treedepth

2021
*
SIAM Journal on Discrete Mathematics
*

We also show an application for approximation algorithms of

doi:10.1137/19m128819x
fatcat:osouriiacrf27krqachtpicfrq
*treedepth*: given a graph G of*treedepth*k and treewidth t, one can*in**polynomial*time compute a*treedepth*decomposition of G of width O(kt log ... The main technical ingredient*in*our result is a proof that every tree of*treedepth*d contains a subcubic subtree of*treedepth*at least d · log 3 ((1 + √ 5)/2). ... [5] provided a*polynomial*relation between the*treedepth*and the minimum number of*colors**in*a*linear**coloring*; by replacing their usage of [4] by our result (and using an improved*bound*for the excluded ...##
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Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems
[chapter]

2017
*
Lecture Notes in Computer Science
*

We generalize earlier ad-hoc results by showing that if F is a class of graphs whose (q + 1)-

doi:10.1007/978-3-319-57586-5_29
fatcat:gz32xfndkngfhdkf3qi44hpjpa
*colorable*members have*bounded**treedepth*, then there exists some ε > 0 such that q-*Coloring*can be solved*in*... is known to be solvable*in**polynomial*time. ... -)*Coloring*if and only if the (q + 1)-*colorable*members of F have*bounded**treedepth*. ...##
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Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems
[article]

2017
*
arXiv
*
pre-print

We generalize earlier ad-hoc results by showing that if F is a class of graphs whose (q+1)-

arXiv:1701.06985v1
fatcat:oax7x4bduja3zeqrhad37jnof4
*colorable*members have*bounded**treedepth*, then there exists some ε > 0 such that q-*Coloring*can be solved*in*time ... is known to be solvable*in**polynomial*time. ... -)*Coloring*if and only if the (q + 1)-*colorable*members of F have*bounded**treedepth*. ...##
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The complexity landscape of decompositional parameters for ILP

2018
*
Artificial Intelligence
*

Integer

doi:10.1016/j.artint.2017.12.006
fatcat:a7pxontrvzc2bkigjghntxxpte
*Linear*Programming (ILP) can be seen as the archetypical problem for NPcomplete optimization problems, and a wide range of problems*in*artificial intelligence are solved*in*practice via a translation ...*In*particular, we show that ILP is fixed-parameter tractable when parameterized by the*treedepth*of the constraint matrix and the maximum absolute value of any coefficient occurring*in*the ILP instance ... The authors also thank the anonymous reviewers for many insightful suggestions and comments-and*in*particular for helping improve Theorem 13. ...##
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The Complexity Landscape of Decompositional Parameters for ILP
[article]

2018
*
arXiv
*
pre-print

Integer

arXiv:1809.00585v1
fatcat:yuy3tu2v7jeurhlfyeqqqi27hq
*Linear*Programming (ILP) can be seen as the archetypical problem for NP-complete optimization problems, and a wide range of problems*in*artificial intelligence are solved*in*practice via a translation ...*In*particular, we show that ILP is fixed-parameter tractable when parameterized by the*treedepth*of the constraint matrix and the maximum absolute value of any coefficient occurring*in*the ILP instance ... The authors also thank the anonymous reviewers for many insightful suggestions and comments-and*in*particular for helping improve Theorem 13. ...##
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Graph Homomorphism Polynomials: Algorithms and Complexity
[article]

2020
*
arXiv
*
pre-print

We discover that,

arXiv:2011.04778v2
fatcat:23bzgro5snf6bazzsykubzzfou
*in*the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such*polynomial*families are exactly characterized by the*treedepth*, the pathwidth, ... These*polynomials*have received a lot of attention recently for their crucial role*in*several new algorithms for counting and detecting graph patterns, and also for obtaining natural*polynomial*families ... So, we first show that for proving lower*bounds*for homomorphism*polynomials*, it is sufficient to consider the*colored*subgraph isomorphism*polynomial*. ...##
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Computing treedepth in polynomial space and linear fpt time
[article]

2022
*
arXiv
*
pre-print

The running time is 2^O(d^2)· n^O(1) and the space usage is

arXiv:2205.02656v1
fatcat:f3q3uae5dnhxzn43r2swb4slsm
*polynomial**in*n. ... The*treedepth*of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent*in*G is*bound*by the ancestor/descendant ... Simultaneously achieving time complexity*linear**in*n and*polynomial*space complexity is a property that is desired from an algorithm for computing the*treedepth*of a graph. ...##
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Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

2018
*
Algorithms
*

Since graphs of

doi:10.3390/a11070098
fatcat:4diwb3o3fzfn7jz6joixddfbry
*bounded**treedepth*are more restricted than graphs of*bounded*treeor pathwidth, we are interested*in*the algorithmic utility of this additional structure. ... Specifically, we design two novel algorithms for DOMINATING SET on graphs of*treedepth*d: A pure branching algorithm that runs*in*time d O(d 2 ) · n and uses space O(d 3 log d + d log n) and a hybrid of ... We further demonstrate that*treedepth*allows non-DP*linear*-time algorithms that only use*polynomial*space*in*the depth of the provided decomposition. ...##
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An Experimental Evaluation of a Bounded Expansion Algorithmic Pipeline
[article]

2018
*
arXiv
*
pre-print

Previous work has suggested that the structural restrictions of graphs from classes of

arXiv:1712.06690v2
fatcat:cr2mkxxvgnfwfhrhq42qt4fnli
*bounded*expansion--locally dense pockets*in*a globally sparse graph--naturally coincide with common properties of ... From there, we establish viability of the*bounded*expansion framework by demonstrating that*in*some scenarios CONCUSS achieves run times competitive with a popular algorithm for subgraph isomorphism counting ... Acknowledgments This work was supported*in*part by the DARPA GRAPHS Program and the Gordon & Betty Moore Foundation's Data-Driven Discovery Initiative through Grants SPAWAR-N66001-14-1-4063 and GBMF4560 ...##
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Kernelization Using Structural Parameters on Sparse Graph Classes
[article]

2015
*
arXiv
*
pre-print

Meta-theorems for

arXiv:1302.6863v3
fatcat:vctxcjvj55hkhmplzxawjj4d5q
*polynomial*(*linear*) kernels have been the subject of intensive research*in*parameterized complexity. ... More specifically, we show that graph problems that have finite integer index (FII) have*linear*kernels on graphs of*bounded*expansion when parameterized by the size of a modulator to constant-*treedepth*... Such a*coloring*is called a p-*treedepth**coloring*and can be computed*in**linear*time [27] . Here we choose p = 2 d and obtain such a*coloring*for G using n p*colors*. ...##
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Kernelization using structural parameters on sparse graph classes

2017
*
Journal of computer and system sciences (Print)
*

We prove that graph problems with nite integer index have

doi:10.1016/j.jcss.2016.09.002
fatcat:4fcere7danhlth7bjvzv2agx7u
*linear*kernels on graphs of*bounded*expansion when parameterized by the size of a modulator to constant-*treedepth*graphs. ... We only require the problems to have FII on graphs of constant*treedepth*. ... Such a*coloring*is called a p-*treedepth**coloring*and can be computed*in**linear*time [28] . Here we choose p = 2 d and obtain such a*coloring*for G using n p*colors*. ...##
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Kernelization Using Structural Parameters on Sparse Graph Classes
[chapter]

2013
*
Lecture Notes in Computer Science
*

We prove that graph problems with nite integer index have

doi:10.1007/978-3-642-40450-4_45
fatcat:nkzftiqszfea5e5474tuxixa2m
*linear*kernels on graphs of*bounded*expansion when parameterized by the size of a modulator to constant-*treedepth*graphs. ... We only require the problems to have FII on graphs of constant*treedepth*. ... Such a*coloring*is called a p-*treedepth**coloring*and can be computed*in**linear*time [28] . Here we choose p = 2 d and obtain such a*coloring*for G using n p*colors*. ...
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