15,247 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
Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size . After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph there
is no family of cycles such that every vertex of is in at most four of
the cycles, then there exists a feedback vertex set in of size .
Furthermore, a variant of our proof shows that if in a directed graph there
is no family of cycles such that every vertex of is in at most two of
the cycles, then there exists a feedback vertex set in of size .
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers and , if a directed graph has directed treewidth
, then one can find in a family of
subgraphs, each of directed treewidth at least , such that every vertex of
is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19
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
Half-integral Erd\H{o}s-P\'osa property of directed odd cycles
We prove that there exists a function
such that every digraph contains either directed odd cycles where every
vertex of is contained in at most two of them, or a vertex set of size
at most hitting all directed odd cycles. This extends the half-integral
Erd\H{o}s-P\'osa property of undirected odd cycles, proved by Reed [Mangoes and
blueberries. Combinatorica 1999], to digraphs.Comment: 16 pages, 5 figure
A unified half-integral Erd\H{o}s-P\'{o}sa theorem for cycles in graphs labelled by multiple abelian groups
Erd\H{o}s and P\'{o}sa proved in 1965 that there is a duality between the
maximum size of a packing of cycles and the minimum size of a vertex set
hitting all cycles. Such a duality does not hold if we restrict to odd cycles.
However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to
half-integral packing. We prove a far-reaching generalisation of the theorem of
Reed; if the edges of a graph are labelled by finitely many abelian groups,
then there is a duality between the maximum size of a half-integral packing of
cycles whose values avoid a fixed finite set for each abelian group and the
minimum size of a vertex set hitting all such cycles.
A multitude of natural properties of cycles can be encoded in this setting,
for example cycles of length at least , cycles of length modulo ,
cycles intersecting a prescribed set of vertices at least times, and cycles
contained in given -homology classes in a graph embedded on a
fixed surface. Our main result allows us to prove a duality theorem for cycles
satisfying a fixed set of finitely many such properties.Comment: 28 pages, 4 figure
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