5,426 research outputs found
Algorithms for flows and disjoint paths in planar graphs
In this dissertation we describe several algorithms for computing flows, connectivity, and disjoint paths in planar graphs. In all cases, the algorithms are either the first polynomial-time algorithms or are faster than all previously-known algorithms.
First, we describe algorithms for the maximum flow problem in directed planar graphs with integer capacities on both vertices and arcs and with multiple sources and sinks. The algorithms are the first to solve the problem in near-linear time when the number of terminals is fixed and the capacities are polynomially bounded. As a byproduct, we get the first algorithm to solve the vertex-disjoint S-T paths problem in near-linear time when the number of terminals is fixed but greater than 2. We also modify our algorithms to handle real capacities in near-linear time when they are three terminals.
Second, we describe algorithms to compute element-connectivity and a related structure called the reduced graph. We show that global element-connectivity in planar graphs can be found in linear time if the terminals can be covered by O(1) faces. We also show that the reduced graph can be computed in subquadratic time in planar graphs if the number of terminals is fixed.
Third, we describe algorithms for solving or approximately solving the vertex-disjoint paths problem when we want to minimize the total length of the paths. For planar graphs, we describe: (1) an exact algorithm for the case of four pairs of terminals on a single face; and (2) a k-approximation algorithm for the case of k pairs of terminals on a single face.
Fourth, we describe algorithms and a hardness result for the ideal orientation problem. We show that the problem is NP-hard in planar graphs. On the other hand, we show that the problem is polynomial-time solvable in planar graphs when the number of terminals is fixed, the terminals are all on the same face, and no two of the terminal pairs cross. We also describe an algorithm for serial instances of a generalization of the ideal orientation problem called the k-min-sum orientation problem
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
Hitting forbidden minors: Approximation and Kernelization
We study a general class of problems called F-deletion problems. In an
F-deletion problem, we are asked whether a subset of at most vertices can
be deleted from a graph such that the resulting graph does not contain as a
minor any graph from the family F of forbidden minors.
We obtain a number of algorithmic results on the F-deletion problem when F
contains a planar graph. We give (1) a linear vertex kernel on graphs excluding
-claw , the star with leves, as an induced subgraph, where
is a fixed integer. (2) an approximation algorithm achieving an approximation
ratio of , where is the size of an optimal solution on
general undirected graphs. Finally, we obtain polynomial kernels for the case
when F contains graph as a minor for a fixed integer . The graph
consists of two vertices connected by parallel edges. Even
though this may appear to be a very restricted class of problems it already
encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback
Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is
based on a non-trivial application of protrusion techniques, previously used
only for problems on topological graph classes
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
Bidimensionality and EPTAS
Bidimensionality theory is a powerful framework for the development of
metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to
obtain sub-exponential time parameterized algorithms for problems on H-minor
free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for
bidimensional problems, and subsequently improved these results to EPTASs.
Fomin et. al related the theory to the existence of linear kernels for
parameterized problems. In this paper we revisit bidimensionality theory from
the perspective of approximation algorithms and redesign the framework for
obtaining EPTASs to be more powerful, easier to apply and easier to understand.
Two of the most widely used approaches to obtain PTASs on planar graphs are
the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and
Hajiaghayi strengthened both approaches using bidimensionality and obtained
EPTASs for a multitude of problems. We unify the two strenghtened approaches to
combine the best of both worlds. At the heart of our framework is a
decomposition lemma which states that for "most" bidimensional problems, there
is a polynomial time algorithm which given an H-minor-free graph G as input and
an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n
X is f(e). Here, OPT is the objective function value of the problem in question
and f is a function depending only on e. This allows us to obtain EPTASs on
(apex)-minor-free graphs for all problems covered by the previous framework, as
well as for a wide range of packing problems, partial covering problems and
problems that are neither closed under taking minors, nor contractions. To the
best of our knowledge for many of these problems including cycle packing,
vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no
EPTASs on planar graphs were previously known
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