7,759 research outputs found
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
Charting the Algorithmic Complexity of Waypoint Routing
Modern computer networks support interesting new routing models in which traffic flows from a source sto a destination t can be flexibly steered through a sequence of waypoints, such as (hardware) middleboxes or (virtualized) network functions (VNFs), to create innovative network services like service chains or segment routing. While the benefits and technological challenges of providing such routing models have been articulated and studied intensively over the last years, less is known about the underlying algorithmic traffic routing problems.
The goal of this paper is to provide the network community with an overview of algorithmic techniques for waypoint routing and also inform about limitations due to computational hardness. In particular, we put the waypoint routing problem into perspective with respect to classic graph theoretical problems. For example, we find that while computing a shortest path from a source s to a destination t is simple (e.g., using Dijkstra's algorithm), the problem of finding a shortest route from s to t via a single waypoint already features a deep combinatorial structure.</jats:p
On disjoint paths in acyclic planar graphs
We give an algorithm with complexity for the integer
multiflow problem on instances with an acyclic planar digraph
and Eulerian. Here, is a polynomial function, , and is the maximum request . When is
fixed, this gives a polynomial algorithm for the arc-disjoint paths problem
under the same hypothesis
Walking Through Waypoints
We initiate the study of a fundamental combinatorial problem: Given a
capacitated graph , find a shortest walk ("route") from a source to a destination that includes all vertices specified by a set
: the \emph{waypoints}. This waypoint routing problem
finds immediate applications in the context of modern networked distributed
systems. Our main contribution is an exact polynomial-time algorithm for graphs
of bounded treewidth. We also show that if the number of waypoints is
logarithmically bounded, exact polynomial-time algorithms exist even for
general graphs. Our two algorithms provide an almost complete characterization
of what can be solved exactly in polynomial-time: we show that more general
problems (e.g., on grid graphs of maximum degree 3, with slightly more
waypoints) are computationally intractable
Parameterizing Path Partitions
We study the algorithmic complexity of partitioning the vertex set of a given
(di)graph into a small number of paths. The Path Partition problem (PP) has
been studied extensively, as it includes Hamiltonian Path as a special case.
The natural variants where the paths are required to be either \emph{induced}
(Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition,
SPP), have received much less attention. Both problems are known to be
NP-complete on undirected graphs; we strengthen this by showing that they
remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP
remains \NP-hard on undirected bipartite graphs. When parameterized by the
natural parameter ``number of paths'', both SPP and IPP are shown to be
W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected
graphs for the same parameter, as well as for other special subclasses of
directed graphs (IPP is known to be NP-hard on undirected graphs, even for two
paths). On the positive side, we show that for undirected graphs, both problems
are in FPT, parameterized by neighborhood diversity. We also give an explicit
algorithm for the vertex cover parameterization of PP. When considering the
dual parameterization (graph order minus number of paths), all three variants,
IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the
mentioned neighborhood diversity and dual parameterization results to directed
graphs; here, we need to define a proper novel notion of directed neighborhood
diversity. As we also show, most of our results also transfer to the case of
covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of
the CIAC 2023 conferenc
Vertex Disjoint Path in Upward Planar Graphs
The -vertex disjoint paths problem is one of the most studied problems in
algorithmic graph theory. In 1994, Schrijver proved that the problem can be
solved in polynomial time for every fixed when restricted to the class of
planar digraphs and it was a long standing open question whether it is
fixed-parameter tractable (with respect to parameter ) on this restricted
class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered
the question positively. Despite the importance of this result (and the
brilliance of their proof), it is of rather theoretical importance. Their proof
technique is both technically extremely involved and also has at least double
exponential parameter dependence. Thus, it seems unrealistic that the algorithm
could actually be implemented. In this paper, therefore, we study a smaller
class of planar digraphs, the class of upward planar digraphs, a well studied
class of planar graphs which can be drawn in a plane such that all edges are
drawn upwards. We show that on the class of upward planar digraphs the problem
(i) remains NP-complete and (ii) the problem is fixed-parameter tractable.
While membership in FPT follows immediately from \cite{CMPP}'s general result,
our algorithm has only single exponential parameter dependency compared to the
double exponential parameter dependence for general planar digraphs.
Furthermore, our algorithm can easily be implemented, in contrast to the
algorithm in \cite{CMPP}.Comment: 14 page
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