20 research outputs found
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
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
Finding k partially disjoint paths in a directed planar graph
The {\it partially disjoint paths problem} is: {\it given:} a directed graph,
vertices , and a set of pairs from
, {\it find:} for each a directed path
such that if then and are disjoint.
We show that for fixed , this problem is solvable in polynomial time if
the directed graph is planar. More generally, the problem is solvable in
polynomial time for directed graphs embedded on a fixed compact surface.
Moreover, one may specify for each edge a subset of
prescribing which of the paths are allowed to traverse this edge
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
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A Structural Theorem For Shortest Vertex-Disjoint Paths Computation in Planar Graphs
Given k terminal pairs (s₁,t₁),(s₂,t₂),..., (s[subscript k],t[subscript k]) in an edge-weighted graph G, the k Shortest Vertex-Disjoint Paths problem is to find a collection P₁, P₂,..., P[subscript k] of vertex-disjoint paths with minimum total length, where P[subscript i] is an s[subscript i]-to-t[subscript i] path. As a special case of the multi-commodity flow problem, computing vertex disjoint paths has found several applications, for example in VLSI design, or network routing. In this thesis we describe a Structural Theorem for a special case of the Shortest Vertex-Disjoint Paths problem in undirected planar graphs where the terminal vertices are on the boundary of the outer face. At a high level, our Structural Theorem guarantees that the i[superscript th] path of the k Shortest Vertex-Disjoint paths does not cross j[superscript th] (j ≠ i) path of the k-1 Vertex-Disjoint Paths problem