689 research outputs found
Minimum multicuts and Steiner forests for Okamura-Seymour graphs
We study the problem of finding minimum multicuts for an undirected planar
graph, where all the terminal vertices are on the boundary of the outer face.
This is known as an Okamura-Seymour instance. We show that for such an
instance, the minimum multicut problem can be reduced to the minimum-cost
Steiner forest problem on a suitably defined dual graph. The minimum-cost
Steiner forest problem has a 2-approximation algorithm. Hence, the minimum
multicut problem has a 2-approximation algorithm for an Okamura-Seymour
instance.Comment: 6 pages, 1 figur
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Secluded Connectivity Problems
Consider a setting where possibly sensitive information sent over a path in a
network is visible to every {neighbor} of the path, i.e., every neighbor of
some node on the path, thus including the nodes on the path itself. The
exposure of a path can be measured as the number of nodes adjacent to it,
denoted by . A path is said to be secluded if its exposure is small. A
similar measure can be applied to other connected subgraphs, such as Steiner
trees connecting a given set of terminals. Such subgraphs may be relevant due
to considerations of privacy, security or revenue maximization. This paper
considers problems related to minimum exposure connectivity structures such as
paths and Steiner trees. It is shown that on unweighted undirected -node
graphs, the problem of finding the minimum exposure path connecting a given
pair of vertices is strongly inapproximable, i.e., hard to approximate within a
factor of for any (under an
appropriate complexity assumption), but is approximable with ratio
, where is the maximum degree in the graph. One of
our main results concerns the class of bounded-degree graphs, which is shown to
exhibit the following interesting dichotomy. On the one hand, the minimum
exposure path problem is NP-hard on node-weighted or directed bounded-degree
graphs (even when the maximum degree is 4). On the other hand, we present a
polynomial algorithm (based on a nontrivial dynamic program) for the problem on
unweighted undirected bounded-degree graphs. Likewise, the problem is shown to
be polynomial also for the class of (weighted or unweighted) bounded-treewidth
graphs
Complete Acyclic Colorings
We study two parameters that arise from the dichromatic number and the
vertex-arboricity in the same way that the achromatic number comes from the
chromatic number. The adichromatic number of a digraph is the largest number of
colors its vertices can be colored with such that every color induces an
acyclic subdigraph but merging any two colors yields a monochromatic directed
cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest
number of colors that can be used such that every color induces a forest but
merging any two yields a monochromatic cycle. We study the relation between
these parameters and their behavior with respect to other classical parameters
such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
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