145 research outputs found
Half-Integral Linkages in Highly Connected Directed Graphs
We study the half-integral k-Directed Disjoint Paths Problem (1/2 kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k=2, and the input graph is L-strongly connected, for any L >= 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in highly connected digraphs (even with k as part of the input).
Specifically, we show that there is an absolute constant c such that for each k >= 2 there exists L(k) such that 1/2 kDDPP is solvable in time O(|V(G)|^c) for a L(k)-strongly connected directed graph G. As the function L(k) grows rather quickly, we also show that 1/2 kDDPP is solvable in time O(|V(G)|^{f(k)}) in (36k^3+2k)-strongly connected directed graphs. We show that for each epsilon<1, deciding half-integral feasibility of kDDPP instances is NP-complete when k is given as part of the input, even when restricted to graphs with strong connectivity epsilon k
New Menger-like dualities in digraphs and applications to half-integral linkages
We present new min-max relations in digraphs between the number of paths
satisfying certain conditions and the order of the corresponding cuts. We
define these objects in order to capture, in the context of solving the
half-integral linkage problem, the essential properties needed for reaching a
large bramble of congestion two (or any other constant) from the terminal set.
This strategy has been used ad-hoc in several articles, usually with lengthy
technical proofs, and our objective is to abstract it to make it applicable in
a simpler and unified way. We provide two proofs of the min-max relations, one
consisting in applying Menger's Theorem on appropriately defined auxiliary
digraphs, and an alternative simpler one using matroids, however with worse
polynomial running time.
As an application, we manage to simplify and improve several results of
Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding
half-integral linkages in digraphs. Concerning the former, besides being
simpler, our proof provides an almost optimal bound on the strong connectivity
of a digraph for it to be half-integrally feasible under the presence of a
large bramble of congestion two (or equivalently, if the directed tree-width is
large, which is the hard case). Concerning the latter, our proof uses brambles
as rerouting objects instead of cylindrical grids, hence yielding much better
bounds and being somehow independent of a particular topology.
We hope that our min-max relations will find further applications as, in our
opinion, they are simple, robust, and versatile to be easily applicable to
different types of routing problems in digraphs
Graph Theory
This workshop focused on recent developments in graph theory. These included in particular recent breakthroughs on nowhere-zero flows in graphs, width parameters, applications of graph sparsity in algorithms, and matroid structure results
The Directed Grid Theorem
The grid theorem, originally proved by Robertson and Seymour in Graph Minors
V in 1986, is one of the most central results in the study of graph minors. It
has found numerous applications in algorithmic graph structure theory, for
instance in bidimensionality theory, and it is the basis for several other
structure theorems developed in the graph minors project.
In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [Reed
97, Johnson, Robertson, Seymour, Thomas 01]), independently, conjectured an
analogous theorem for directed graphs, i.e. the existence of a function f : N
-> N such that every digraph of directed tree-width at least f(k) contains a
directed grid of order k. In an unpublished manuscript from 2001, Johnson,
Robertson, Seymour and Thomas give a proof of this conjecture for planar
digraphs. But for over a decade, this was the most general case proved for the
Reed, Johnson, Robertson, Seymour and Thomas conjecture.
Only very recently, this result has been extended to all classes of digraphs
excluding a fixed undirected graph as a minor (see [Kawarabayashi, Kreutzer
14]). In this paper, nearly two decades after the conjecture was made, we are
finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas
conjecture in full generality and to prove the directed grid theorem.
As consequence of our results we are able to improve results in Reed et al.
in 1996 [Reed, Robertson, Seymour, Thomas 96] (see also [Open Problem Garden])
on disjoint cycles of length at least l and in [Kawarabayashi, Kobayashi,
Kreutzer 14] on quarter-integral disjoint paths. We expect many more
algorithmic results to follow from the grid theorem.Comment: 43 pages, 21 figure
Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs
We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h))
Adapting the Directed Grid Theorem into an FPT Algorithm
The Grid Theorem of Robertson and Seymour [JCTB, 1986], is one of the most
important tools in the field of structural graph theory, finding numerous
applications in the design of algorithms for undirected graphs. An analogous
version of the Grid Theorem in digraphs was conjectured by Johnson et al.
[JCTB, 2001], and proved by Kawarabayashi and Kreutzer [STOC, 2015]. Namely,
they showed that there is a function such that every digraph of directed
tree-width at least contains a cylindrical grid of size as a
butterfly minor and stated that their proof can be turned into an XP algorithm,
with parameter , that either constructs a decomposition of the appropriate
width, or finds the claimed large cylindrical grid as a butterfly minor. In
this paper, we adapt some of the steps of the proof of Kawarabayashi and
Kreutzer to improve this XP algorithm into an FPT algorithm. Towards this, our
main technical contributions are two FPT algorithms with parameter . The
first one either produces an arboreal decomposition of width or finds a
haven of order in a digraph , improving on the original result for
arboreal decompositions by Johnson et al. The second algorithm finds a
well-linked set of order in a digraph of large directed tree-width. As
tools to prove these results, we show how to solve a generalized version of the
problem of finding balanced separators for a given set of vertices in FPT
time with parameter , a result that we consider to be of its own interest.Comment: 31 pages, 9 figure
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