2,156 research outputs found
Irrelevant vertices for the planar Disjoint Paths Problem
The Disjoint Paths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),…,(sk,tk)(s1,t1),…,(sk,tk), whether there is a collection of k pairwise vertex-disjoint paths linking sisi and titi, for i=1,…,ki=1,…,k. In their f(k)⋅n3f(k)⋅n3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k)g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem , whose – very technical – proof gives a function g(k)g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k)=O(k3/2⋅2k)g(k)=O(k3/2⋅2k). Our bound is radically better than the bounds known for general graphs
Decompositions and Algorithms for the Disjoint Paths Problem in Planar Graphs
Στο πρόβλημα των Διακεκριμενων Μονοπατιων μας ζητείται να εξετάσουμε, δεδομένου ενός γραφήματος G και ενος συνόλου k ζευγών τερματικών,αν τα ζεύγη των τερματικών μπορούν να συνδεθούν με διακεκριμένα μονοπάτια. Στα "Graph Minors", μια σειρά 23 εργασιών μεταξύ 1984 και 2011, οι Neil Robertson και Paul D. Seymour, ανάμεσα σε άλλα σπουδαία αποτελέσματα που επηρέασαν βαθιά την Θεωρία Γραφημάτων, παρουσίασαν έναν f(k)*n^3 αλγόριθμο για το πρόβλημα των Διακεκριμενων Μονοπατιων. Για να το καταφέρουν αυτό, εισήγαγαν την "τεχνκή της άσχετης κορυφής" σύμφωνα με την οποία σε κάθε στιγμιότυπο δεντροπλάτους μεγαλύτερου του g(k) υπάρχει μια "άσχετη" κορυφή της οποίας η αφαίρεση δημιουργεί ένα ισοδύναμο στιγμιότυπο του προβλήματος.
Εδώ μελετάμε το πρόβλημα σε επίπεδα γραφήματα και αποδεικνύουμε ότι για κάθε σταθερό k κάθε στιγμιότυπο του προβλήματος των Διακεκριμενων Μονοπατιων σε επιπεδα γραφηματα μπορεί να μετασχηματιστεί σε ένα ισοδύναμο που έχει φραγμένο δενδροπλάτος, αφαιρώντας ταυτόχρονα ένα σύνολο κορυφών από το δεδομένο επίπεδο γράφημα. Ως συνέπεια αυτού, το πρόβλημα των Διακεκριμένων Μονοπατιών σε επίπεδα γραφήματα μπορεί να λυθεί σε γραμμικό χρόνο για κάθε σταθερό πλήθος τερματικών.> In the Disjoint Paths Problem, given a graph G and a set of k pairs of terminals, we ask whether the pairs of terminals can be linked by pairwise disjoint paths.
> In the Graph Minors series of 23 papers between 1984 and 2011, Neil Robertson and Paul D. Seymour, among other great results that heavily influenced Graph Theory, provided an f(k)\cdot n^{3} algorithm for the Disjoint Paths Problem. To achieve this, they introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem.
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> We study the problem in planar graphs and we prove that for every fixed k every instance of the Planar Disjoint Paths Problem can be transformed to an equivalent one that has bounded treewidth, by simultaneously discarding a set of vertices of the given planar graph. As a consequence the Planar Disjoint Paths Problem can be solved in linear time for every fixed number of terminals
Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion
The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016].
Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands
Finding topological subgraphs is fixed-parameter tractable
We show that for every fixed undirected graph , there is a
time algorithm that tests, given a graph , if contains as a
topological subgraph (that is, a subdivision of is subgraph of ). This
shows that topological subgraph testing is fixed-parameter tractable, resolving
a longstanding open question of Downey and Fellows from 1992. As a corollary,
for every we obtain an time algorithm that tests if there is
an immersion of into a given graph . This answers another open question
raised by Downey and Fellows in 1992
Reconfiguration on sparse graphs
A vertex-subset graph problem Q defines which subsets of the vertices of an
input graph are feasible solutions. A reconfiguration variant of a
vertex-subset problem asks, given two feasible solutions S and T of size k,
whether it is possible to transform S into T by a sequence of vertex additions
and deletions such that each intermediate set is also a feasible solution of
size bounded by k. We study reconfiguration variants of two classical
vertex-subset problems, namely Independent Set and Dominating Set. We denote
the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete
on graphs of bounded bandwidth and W[1]-hard parameterized by k on general
graphs. We show that ISR is fixed-parameter tractable parameterized by k when
the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we
answer positively an open question concerning the parameterized complexity of
the problem on graphs of bounded treewidth. Moreover, our techniques generalize
recent results showing that ISR is fixed-parameter tractable on planar graphs
and graphs of bounded degree. For DSR, we show the problem fixed-parameter
tractable parameterized by k when the input graph does not contain large
bicliques, a class of graphs which includes graphs of bounded degeneracy and
nowhere-dense graphs
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