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The Complexity of Packing Edge-Disjoint Paths
[article]

2019
*
arXiv
*
pre-print

We introduce and study

arXiv:1910.00440v1
fatcat:wm2hcglaqngalbldsc57ljg6cu
*the**complexity**of**Path**Packing*. Given a graph G and a list*of**paths*,*the*task is to embed*the**paths**edge*-*disjoint*in G. ...*Packing**paths**of*length two is polynomial time solvable, while*packing**paths**of*length three is NP-hard. ... Since*packing**edge*-*disjoint*and vertex-*disjoint*triangles is NP-hard for planar graphs,*the*parameterized*complexity*is studied [7] . ...##
###
The Complexity of Packing Edge-Disjoint Paths

2019
*
International Symposium on Parameterized and Exact Computation
*

We introduce and study

doi:10.4230/lipics.ipec.2019.10
dblp:conf/iwpec/DreierFHKRTW19
fatcat:usj4jariyncldcolhvejshtku4
*the**complexity**of**Path**Packing*. Given a graph G and a list*of**paths*,*the*task is to embed*the**paths**edge*-*disjoint*in G. ... Finally, even*the*spacial case Exact*Path**Packing*where*the**paths*have to cover every*edge*in G exactly once is already NP-hard for two*paths*on 4-regular graphs. ... Since*packing**edge*-*disjoint*and vertex-*disjoint*triangles is NP-hard for planar graphs,*the*parameterized*complexity*is studied [7] . ...##
###
Kernelization of Cycle Packing with Relaxed Disjointness Constraints

2018
*
SIAM Journal on Discrete Mathematics
*

While

doi:10.1137/17m1136614
fatcat:2bb622lgo5fy3i7ooe3at4ybse
*the*Pairwise*Disjoint*Cycle*Packing*problem admits a polynomial kernel for all t ≥ 1,*the*kernelization*complexity**of*Almost*Disjoint*Cycle*Packing*reveals an interesting spectrum*of*upper and lower ... A key result in*the*field*of*kernelization, a subfield*of*parameterized*complexity*, states that*the*classic*Disjoint*Cycle*Packing*problem, i.e. finding k vertex*disjoint*cycles in a given graph G, admits ... As we shall see,*the*kernelization*complexity*landscape for Almost*Disjoint*Cycle*Packing*is much more diverse than that*of*Pairwise*Disjoint*Cycle*Packing*. ...##
###
Packing (2^k+1-1)-order perfect binary trees into (k+1)-connected graph
[article]

2013
*
arXiv
*
pre-print

We prove that in G'

arXiv:1309.3825v1
fatcat:qb6uyasudvdblbxafhk3pw3ede
*the*largest number*of*vertex-*disjoint*subgraphs isomorphic to T_k is equal to*the*smallest number*of*vertices that cover all subgraphs isomorphic to T_k. ...*Packing*problem is to find in G*the*largest number*of*independent subgraphs each*of*which is isomorphic to H. Let U⊂V. If*the*graph G-U has no subgraph isomorphic to H, U is a cover*of*G. ... A key process in*the*algorithm is to detect*the*blocks*of*a connected graph.*The*T 1*packing*algorithm has*complexity**of*O(n 2 ) and*the*T 2*packing*algorithm has*complexity**of*O(n 4 ). ...##
###
Edge-disjoint odd cycles in 4-edge-connected graphs

2016
*
Journal of combinatorial theory. Series B (Print)
*

We show that, for any ε > 0, if k = O((log log log n) 1/2−ε ), then

doi:10.1016/j.jctb.2015.12.002
fatcat:ur67y2kfrvfulf4rnh7yorekqy
*the**edge*-*disjoint*k odd cycle*packing*in G can be solved in polynomial time*of*n. ACM Subject Classification G.2.2 Graph Theory ... Finding*edge*-*disjoint*odd cycles is one*of**the*most important problems in graph theory, graph algorithm and combinatorial optimization. ... As we have already seen before,*the**edge*-*disjoint*k odd cycle*packing*is a generalization*of**the*k*edge*-*disjoint**paths*problem. ...##
###
Edge-packing of graphs and network reliability

1988
*
Discrete Mathematics
*

*Edge*-

*packings*

*of*graphs An

*edge*-

*packing*

*of*a multigraph G is a collection

*of*

*edge*-

*disjoint*subgraphs

*of*G. ...

*The*three problems

*of*most interest to us here are edgepackings by spanning trees, by s, t-

*paths*, and by s, t-cuts. We first review

*the*use

*of*

*edge*-

*packings*in

*the*reliability context. ... Acknowledgements Thanks to University

*of*Auckland and University

*of*Toronto for hospitality while this paper was written, and to NSERC Canada for financial support. ...

##
###
Edge-Packings of Graphs and Network Reliability
[chapter]

1988
*
Annals of Discrete Mathematics
*

*Edge*-

*packings*

*of*graphs An

*edge*-

*packing*

*of*a multigraph G is a collection

*of*

*edge*-

*disjoint*subgraphs

*of*G. ...

*The*three problems

*of*most interest to us here are edgepackings by spanning trees, by s, t-

*paths*, and by s, t-cuts. We first review

*the*use

*of*

*edge*-

*packings*in

*the*reliability context. ... Acknowledgements Thanks to University

*of*Auckland and University

*of*Toronto for hospitality while this paper was written, and to NSERC Canada for financial support. ...

##
###
Edge-disjoint packings of graphs

1994
*
Discrete Applied Mathematics
*

In this paper we study two types

doi:10.1016/0166-218x(92)00153-d
fatcat:xjlp5zevqfb5dkq4d62it7f6ja
*of**edge*-*disjoint**packings**of*graphs. ... We show that if G has at most two*edges*then*the*induced*edge*-*disjoint*G*packing*problem belongs to P, whereas for all other graphs G*the*problem is NP-complete. ...*The*rest*of*this section deals with*edge*-*disjoint*induced*packing*problems; we show that we can determine*the**complexity*status*of*all remaining such problems. ...##
###
Parameterized Complexity of $$(A,\ell )$$-Path Packing
[chapter]

2020
*
Lecture Notes in Computer Science
*

Given a graph G = (V, E), A ⊆ V , and integers k and ,

doi:10.1007/978-3-030-48966-3_4
fatcat:g5va7c7tt5fm5ltbvsjmdmfdqu
*the*(A, )-*Path**Packing*problem asks to find k vertex-*disjoint**paths**of*length that have endpoints in A and internal points in V \ A. ... We study*the*parameterized*complexity**of*this problem with parameters |A|, , k, treewidth, pathwidth, and their combinations. We present sharp*complexity*contrasts with respect to these parameters. ... Given G and A, A-*Path**Packing*is*the*problem*of*finding*the*maximum number*of*vertex-*disjoint*A-*paths*in G. ...##
###
The P k Partition Problem and Related Problems in Bipartite Graphs
[chapter]

2007
*
Lecture Notes in Computer Science
*

More precisely, we prove that

doi:10.1007/978-3-540-69507-3_36
fatcat:x3qfgf2hifaddkfx7gzn2u53lu
*the*problem consisting*of*deciding if a graph*of*nk vertices has n vertex*disjoint*simple*paths*{P1, · · · , Pn} such that each*path*Pi has k vertices is NP-complete, even ... In this paper, we continue*the*investigation proposed in [15] about*the*approximability*of*P k partition problems, but focusing here on their*complexity*. ... In other words, we want to know if there exists n vertex*disjoint*simple*paths**of*length k in G. ...##
###
Distributed Connectivity Decomposition
[article]

2013
*
arXiv
*
pre-print

(II) A decomposition

arXiv:1311.5317v1
fatcat:74qs7lgzmff6tmnzcgopksmel4
*of*each undirected graph with*edge*-connectivity λ into (fractionally)*edge*-*disjoint*weighted spanning trees with total weight λ-1/2(1-ε), in O(D+√(nλ)) rounds. ... We also show round*complexity*lower bounds*of*Ω̃(D+√(n/k)) and Ω̃(D+√(n/λ)) for*the*above two decompositions, using techniques*of*[Das Sarma et al., STOC'11]. ... λ ′*edge*-*disjoint*spanning trees-which we call a spanning tree*packing**of*size λ ′ -then for each pair*of*vertices we get λ ′*edge*-*disjoint**paths*, one through each tree. ...##
###
An Algorithm for Packing Connectors

1998
*
Journal of combinatorial theory. Series B (Print)
*

As a corollary, we obtain an O( (n; m) + nm) time algorithm for nding a maximum number

doi:10.1006/jctb.1998.1863
fatcat:isgqcnpkojblncqgqnxhldniay
*of*S{T connectors, where (n; m) denotes*the**complexity**of*nding a maximum number*of**edge**disjoint*spanning trees in ... On*the*other hand, if G is bipartite with colour classes S and T, then an S{T connector is an*edge*cover*of*G (a set*of**edges*covering all vertices). ... Here, (x; y) denotes*the**complexity**of*nding*the*maximum number*of**edge*-*disjoint*spanning trees in a graph on x vertices and y*edges*. ...##
###
Parameterized Complexity of (A,ℓ)-Path Packing
[article]

2020
*
arXiv
*
pre-print

Given a graph G = (V,E), A ⊆ V, and integers k and ℓ,

arXiv:2008.03448v1
fatcat:qqhkwar3xjex3hvckwytlduske
*the*(A,ℓ)-*Path**Packing*problem asks to find k vertex-*disjoint**paths**of*length ℓ that have endpoints in A and internal points in V ∖ A. ... We study*the*parameterized*complexity**of*this problem with parameters |A|, ℓ, k, treewidth, pathwidth, and their combinations. We present sharp*complexity*contrasts with respect to these parameters. ... Given G and A, A-*Path**Packing*is*the*problem*of*finding*the*maximum number*of*vertex-*disjoint*A-*paths*in G. ...##
###
Distributed connectivity decomposition

2014
*
Proceedings of the 2014 ACM symposium on Principles of distributed computing - PODC '14
*

A distributed O(log n)-approximation

doi:10.1145/2611462.2611491
dblp:conf/podc/Censor-HillelGK14
fatcat:dpytrsr765gurdxrj24mfek44i
*of*vertex connectivity with round*complexity**of*O(D + √ n). ... with total weight λ−1 We also show round*complexity*lower bounds*of*Ω(D + n k ) andΩ(D + n λ ) for*the*above two decompositions, using techniques*of*[Das Sarma et al., STOC'11]. ... λ*edge*-*disjoint*spanning trees-which we call a spanning tree*packing**of*size λ -then for each pair*of*vertices we get λ*edge*-*disjoint**paths*, one through each tree. ...##
###
Counting Paths and Packings in Halves
[article]

2009
*
arXiv
*
pre-print

It is shown that one can count k-

arXiv:0904.3093v1
fatcat:asuyur4l45d2tpsekdyx5cp2yy
*edge**paths*in an n-vertex graph and m-set k-*packings*on an n-element universe, respectively, in time n k/2 and n mk/2, up to a factor polynomial in n, k, and m; in polynomial ... These are implications*of*a more general result: given two set families on an n-element universe, one can count*the**disjoint*pairs*of*sets in*the*Cartesian product*of**the*two families with (n ℓ) basic ... This research was supported in part by*the*Academy*of*Finland, Grants 117499 (P.K.) and 125637 (M.K.). ...
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