Filters








195 Hits in 2.0 sec

Polynomial Treedepth Bounds in Linear Colorings [article]

Jeremy Kun, Michael P. O'Brien, Marcin Pilipczuk, Blair D. Sullivan
2018 arXiv   pre-print
We establish a 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.  ... 
arXiv:1802.09665v4 fatcat:sxvjaosnqrf3pncztwliuvz4lu

Polynomial Treedepth Bounds in Linear Colorings

Jeremy Kun, Michael P. O'Brien, Marcin Pilipczuk, Blair D. Sullivan
2020 Algorithmica  
We establish a 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  ... 
doi:10.1007/s00453-020-00760-0 fatcat:6df2hicsfra3baazchzkd2aime

Improved bounds for the excluded-minor approximation of treedepth [article]

Wojciech Czerwiński and Wojciech Nadara and Marcin Pilipczuk
2019 arXiv   pre-print
We also show an application of our techniques for approximation algorithms of 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  ... 
arXiv:1904.13077v2 fatcat:yfpigc5rlfch3adcvmj3jdptpm

Improved Bounds for the Excluded-Minor Approximation of Treedepth

Wojciech Czerwiński, Wojciech Nadara, Marcin Pilipczuk
2021 SIAM Journal on Discrete Mathematics  
We also show an application for approximation algorithms of 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  ... 
doi:10.1137/19m128819x fatcat:osouriiacrf27krqachtpicfrq

Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems [chapter]

Lars Jaffke, Bart M. P. Jansen
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)-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.  ... 
doi:10.1007/978-3-319-57586-5_29 fatcat:gz32xfndkngfhdkf3qi44hpjpa

Fine-Grained Parameterized Complexity Analysis of Graph Coloring Problems [article]

Lars Jaffke, Bart M. P. Jansen
2017 arXiv   pre-print
We generalize earlier ad-hoc results by showing that if F is a class of graphs whose (q+1)-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.  ... 
arXiv:1701.06985v1 fatcat:oax7x4bduja3zeqrhad37jnof4

The complexity landscape of decompositional parameters for ILP

Robert Ganian, Sebastian Ordyniak
2018 Artificial Intelligence  
Integer 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.  ... 
doi:10.1016/j.artint.2017.12.006 fatcat:a7pxontrvzc2bkigjghntxxpte

The Complexity Landscape of Decompositional Parameters for ILP [article]

Robert Ganian, Sebastian Ordyniak
2018 arXiv   pre-print
Integer 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.  ... 
arXiv:1809.00585v1 fatcat:yuy3tu2v7jeurhlfyeqqqi27hq

Graph Homomorphism Polynomials: Algorithms and Complexity [article]

Balagopal Komarath, Anurag Pandey, C. S. Rahul
2020 arXiv   pre-print
We discover that, 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.  ... 
arXiv:2011.04778v2 fatcat:23bzgro5snf6bazzsykubzzfou

Computing treedepth in polynomial space and linear fpt time [article]

Wojciech Nadara, Michał Pilipczuk, Marcin Smulewicz
2022 arXiv   pre-print
The running time is 2^O(d^2)· n^O(1) and the space usage is 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.  ... 
arXiv:2205.02656v1 fatcat:f3q3uae5dnhxzn43r2swb4slsm

Width, Depth, and Space: Tradeoffs between Branching and Dynamic Programming

Li-Hsuan Chen, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil
2018 Algorithms  
Since graphs of 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.  ... 
doi:10.3390/a11070098 fatcat:4diwb3o3fzfn7jz6joixddfbry

An Experimental Evaluation of a Bounded Expansion Algorithmic Pipeline [article]

Michael P. O'Brien, Blair D. Sullivan
2018 arXiv   pre-print
Previous work has suggested that the structural restrictions of graphs from classes of 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  ... 
arXiv:1712.06690v2 fatcat:cr2mkxxvgnfwfhrhq42qt4fnli

Kernelization Using Structural Parameters on Sparse Graph Classes [article]

Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar
2015 arXiv   pre-print
Meta-theorems for 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.  ... 
arXiv:1302.6863v3 fatcat:vctxcjvj55hkhmplzxawjj4d5q

Kernelization using structural parameters on sparse graph classes

Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar
2017 Journal of computer and system sciences (Print)  
We prove that graph problems with nite integer index have 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.  ... 
doi:10.1016/j.jcss.2016.09.002 fatcat:4fcere7danhlth7bjvzv2agx7u

Kernelization Using Structural Parameters on Sparse Graph Classes [chapter]

Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Sebastian Ordyniak, Felix Reidl, Peter Rossmanith, Fernando Sánchez Villaamil, Somnath Sikdar
2013 Lecture Notes in Computer Science  
We prove that graph problems with nite integer index have 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.  ... 
doi:10.1007/978-3-642-40450-4_45 fatcat:nkzftiqszfea5e5474tuxixa2m
« Previous Showing results 1 — 15 out of 195 results