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Subexponential time algorithms for finding small tree and path decompositions
[article]

2016
*
arXiv
*
pre-print

The Minimum Size

arXiv:1601.02415v2
fatcat:iu2zhpgpjffmdl34vhkujzc4nm
*Tree**Decomposition*(MSTD)*and*Minimum Size*Path**Decomposition*(MSPD) problems ask*for*a given n-vertex graph G*and*integer k, what is the minimum number of bags of a*tree**decomposition*... We present*algorithms*that solve both problems*for*fixed k in 2^O(n/ n)*time**and*show that they cannot be solved in 2^o(n / n)*time*, assuming the Exponential*Time*Hypothesis. ... We thank the anonymous referees*for*their detailed*and*helpful comments. ...##
###
Subexponential Time Algorithms for Finding Small Tree and Path Decompositions
[chapter]

2015
*
Lecture Notes in Computer Science
*

The Minimum Size

doi:10.1007/978-3-662-48350-3_16
fatcat:74rujxgshnhqnmqwds4zk55x4m
*Tree**Decomposition*(MSTD)*and*Minimum Size*Path**Decomposition*(MSPD) problems ask*for*a given n-vertex graph G*and*integer k, what is the minimum number of bags of a*tree**decomposition*... We present*algorithms*that solve both problems*for*fixed k in 2 O(n/ log n)*time**and*show that they cannot be solved in 2 o(n/ log n)*time*, assuming the Exponential*Time*Hypothesis. ... We thank the anonymous referees*for*their detailed*and*helpful comments. ...##
###
Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

2016
*
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
*

*for*graphs of bounded treewidth, this result gives a versatile technique

*for*obtaining (randomized)

*subexponential*parameterized

*algorithms*

*for*problems on planar graphs, usually with running

*time*bound ... k-

*PATH*, VERTEX COVER LOCAL SEARCH,

*and*SUBGRAPH ISOMORPHISM, among others. ...

*for*pointing out the similarities between our clustering procedure

*and*tools from the theory of metric embeddings. ...

##
###
Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

2019
*
International Symposium on Parameterized and Exact Computation
*

This leads,

doi:10.4230/lipics.ipec.2019.23
dblp:conf/iwpec/NovotnaOPRLW19
fatcat:lcgp6expunh4rlp3jjysmw6lsa
*for*example, to the following corollaries*for*specific classes C*and*D: a largest induced forest in a Pt-free graph can be found in 2Õ (n 2/3 )*time*,*for*every fixed t;*and*a largest induced ... ACM Subject Classification Theory of computation → Design*and*analysis of*algorithms*; Theory of computation → Problems, reductions*and*completeness ... Therefore,*for*the purpose of verifying property 3, we can assume that a*tree**decomposition*of width O(w) is additionally provided 23:6*Subexponential**Algorithms**Finding*Large Sparse Subgraphs on input ...##
###
Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs

2012
*
Theoretical Computer Science
*

We use these

doi:10.1016/j.tcs.2011.09.014
fatcat:253avdexpzhetkbpp24762fsnm
*algorithms*to derive faster 2-approximations; furthermore, we present the first PTASes*and**subexponential*FPTalgorithms*for*independent set*and*vertex cover on these graph classes using a ... Our technique applies, in particular, to problems such as Steiner*tree*, (directed) subgraph with a property, (directed) longest*path*,*and*(connected/independent) dominating set, on some or all proper minor-closed ... We improved the best known FPT-*algorithms**for*k-Steiner*Tree**and*Directed k-*Path*in H-minor-free graphs that were previously only known to be in EPT even on planar graphs (in the case of Steiner*Tree*). ...##
###
Subexponential-time algorithms for finding large induced sparse subgraphs
[article]

2019
*
arXiv
*
pre-print

This leads,

arXiv:1910.01082v1
fatcat:25wctiirxbdn7iuuwgrltscxeq
*for*example, to the following corollaries*for*specific classes C*and*D: * a largest induced forest in a P_t-free graph can be found in 2^Õ(n^2/3)*time*,*for*every fixed t;*and** a largest induced ... ;*and** the considered problem admits a single-exponential fixed-parameter*algorithm*when parameterized by the treewidth of the input graph. ... It is known that*small*balanced separators can be used to construct*tree**decompositions*of*small*width, as made explicit in the following lemma. Lemma 1 ([11] ). ...##
###
Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

2020
*
Algorithmica
*

This leads,

doi:10.1007/s00453-020-00745-z
pmid:34720297
pmcid:PMC8550324
fatcat:3xx4rhvhbvaq3ewppedpfhmuqu
*for*example, to the following corollaries*for*specific classes C*and*D :a largest induced forest in a P t -free graph can be found in 2 O ~ ( n 2 / 3 )*time*,*for*every fixed t; anda largest ... Let C*and*D be hereditary graph classes. Consider the following problem: given a graph G ∈ D ,*find*a largest, in terms of the number of vertices, induced subgraph of G that belongs to C . ... It is known that*small*balanced separators can be used to construct*tree**decompositions*of*small*width, as made explicit in the following lemma. ...##
###
Faster Approximation Schemes and Parameterized Algorithms on H-Minor-Free and Odd-Minor-Free Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

We use these

doi:10.1007/978-3-642-15155-2_56
fatcat:zno6rurjkre3dasjvjchdcbizu
*algorithms*to derive faster 2-approximations; furthermore, we present the first PTASes*and**subexponential*FPT-*algorithms**for*independent set*and*vertex cover on these graph classes using a ... Our technique applies, in particular, to problems such as Steiner*tree*, (directed) subgraph with a property, (directed) longest*path*,*and*(connected/independent) dominating set, on some or all proper minor-closed ... I would like to thank Holger Dell, Martin Grohe,*and*Matthias Mnich*for*helpful discussions*and*comments on earlier versions of this work. ...##
###
The Square Root Phenomenon in Planar Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

*Subexponential*

*algorithm*

*for*3-Coloring Can we achieve even better running

*time*(e.g., 2 O( 3 √ n) ) on planar graphs? There is no 2 o(m) -

*time*

*algorithm*

*for*m-clause 3SAT. ... We use this observation to

*find*a

*path*of length at least k on planar graphs: Set w := 5 √ k.

*Find*an O(1)-approximate

*tree*

*decomposition*. ... Input: A k × k matrix

*and*a set of pairs S i,j ⊆ [D] × [D]

*for*each cell.

*Find*: A pair s i,j ∈ S i,j

*for*each cell such that Vertical neighbors agree in the 1st coordinate. ...

##
###
The Square Root Phenomenon in Planar Graphs
[chapter]

2013
*
Lecture Notes in Computer Science
*

*Subexponential*

*algorithm*

*for*3-Coloring Can we achieve even better running

*time*(e.g., 2 O( 3 √ n) ) on planar graphs? There is no 2 o(m) -

*time*

*algorithm*

*for*m-clause 3SAT. ... We use this observation to

*find*a

*path*of length at least k on planar graphs: Set w := 5 √ k.

*Find*an O(1)-approximate

*tree*

*decomposition*. ... Input: A k × k matrix

*and*a set of pairs S i,j ⊆ [D] × [D]

*for*each cell.

*Find*: A pair s i,j ∈ S i,j

*for*each cell such that Vertical neighbors agree in the 1st coordinate. ...

##
###
Subexponential Time Algorithms for Embedding H-Minor Free Graphs

2016
*
International Colloquium on Automata, Languages and Programming
*

We show that,

doi:10.4230/lipics.icalp.2016.9
dblp:conf/icalp/BodlaenderNZ16
fatcat:msf7om5zsvht3hflwrweoewfmq
*for*any fixed graph H*and*> 0, if P is H-Minor Free*and*G has treewidth tw, We also show that this is optimal, in the sense that the existence of an*algorithm**for*one of these problems running ... in 2 o(n/ log n)*time*would contradict the Exponential*Time*Hypothesis. ... They give a 2 O(n/ log n) -*time**algorithm**for**finding**tree**decompositions*with few bags*and*a matching lower bound (based on the Exponential*Time*Hypothesis),*and*a 2 O(n/ log n)*algorithm**for*determining ...##
###
H-colouring P_t-free graphs in subexponential time
[article]

2019
*
arXiv
*
pre-print

On the other hand, no

arXiv:1803.05396v3
fatcat:jkbqftbapzfc3lbhfuvucc52oq
*subexponential**time**algorithm*exists*for*4-colourability of P_t-free graphs assuming the Exponential*Time*Hypothesis. ... We show that the generating function*for*(list) graph homomorphisms from G to H can be calculated in*subexponential**time*2^O(√(tn(n)))*for*n=|V(G)| in the class of P_t-free graphs G. ... The pathwidth of G is defined as the minimum of max i |X i | − 1 over the*path*-*decompositions*of G. Let T be a rooted*tree*with root r.*For*a vertex v, let T v denote the*path*of T between r*and*v. ...##
###
Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument
[article]

2022
*
arXiv
*
pre-print

]

arXiv:2203.04836v1
fatcat:aoalxcrsc5gfnf2ijr6rk63kfq
*and*provide a*subexponential*-*time**algorithm*with improved running*time*2^𝒪(√(n)log n)*and*a quasipolynomial-*time*approximation scheme with improved running*time*2^𝒪(ε^-1log^5 n). ... The Gyárfás'*path*argument, a powerful tool that is the main building block*for*many*algorithms*in P_t-free graphs, ensures that given an n-vertex P_t-free graph, in polynomial*time*we can*find*a set P ... We expect it is a first step to get a quasipolynomial-*time**algorithm**for*MWIS in S t,t,t -free graphs, similarly as Theorem 1 is an essential ingredient of the*algorithms**for*P t -free graphs [15, 23] ...##
###
Parameterized Algorithms and Data Reduction for Safe Convoy Routing

2018
*
Algorithmic Approaches for Transportation Modeling, Optimization, and Systems
*

*For*graphs with constant crossing number, we provide a

*subexponential*2 O( √ n) -

*time*

*algorithm*

*and*prove a matching lower bound. ... Regarding

*tree*-like graphs, we obtain a 2 O(tw) • 2 • n-

*time*

*algorithm*

*for*graphs of treewidth tw, show that there is no problem kernel with size polynomial in tw, yet show a problem kernel with size polynomial ... the

*path*length,

*and*so can our

*subexponential*-

*time*

*algorithms*in planar graphs (Theorem 2.1). ...

##
###
Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

2020
*
European Symposium on Algorithms
*

of

doi:10.4230/lipics.esa.2020.49
dblp:conf/esa/FominG20a
fatcat:dh4jo3kb5rc3xgw7rqfted5doi
*subexponential*parameterized*algorithms**for*most of the structural parameters*and*optimization problems is highly unlikely. ... Besides*subexponential**time**algorithms*, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). ... Acknowledgements We thank Torstein Strømme, Daniel Lokshtanov,*and*Pranabendu Misra*for*fruitful discussions on the topic of this paper. ...
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