77,168 research outputs found
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
The Complexity of Counting Homomorphisms to Cactus Graphs Modulo 2
A homomorphism from a graph G to a graph H is a function from V(G) to V(H)
that preserves edges. Many combinatorial structures that arise in mathematics
and computer science can be represented naturally as graph homomorphisms and as
weighted sums of graph homomorphisms. In this paper, we study the complexity of
counting homomorphisms modulo 2. The complexity of modular counting was
introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who
famously introduced a problem for which counting modulo 7 is easy but counting
modulo 2 is intractable. Modular counting provides a rich setting in which to
study the structure of homomorphism problems. In this case, the structure of
the graph H has a big influence on the complexity of the problem. Thus, our
approach is graph-theoretic. We give a complete solution for the class of
cactus graphs, which are connected graphs in which every edge belongs to at
most one cycle. Cactus graphs arise in many applications such as the modelling
of wireless sensor networks and the comparison of genomes. We show that, for
some cactus graphs H, counting homomorphisms to H modulo 2 can be done in
polynomial time. For every other fixed cactus graph H, the problem is complete
for the complexity class parity-P which is a wide complexity class to which
every problem in the polynomial hierarchy can be reduced (using randomised
reductions). Determining which H lead to tractable problems can be done in
polynomial time. Our result builds upon the work of Faben and Jerrum, who gave
a dichotomy for the case in which H is a tree.Comment: minor change
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
Computing hypergraph width measures exactly
Hypergraph width measures are a class of hypergraph invariants important in
studying the complexity of constraint satisfaction problems (CSPs). We present
a general exact exponential algorithm for a large variety of these measures. A
connection between these and tree decompositions is established. This enables
us to almost seamlessly adapt the combinatorial and algorithmic results known
for tree decompositions of graphs to the case of hypergraphs and obtain fast
exact algorithms.
As a consequence, we provide algorithms which, given a hypergraph H on n
vertices and m hyperedges, compute the generalized hypertree-width of H in time
O*(2^n) and compute the fractional hypertree-width of H in time
O(m*1.734601^n).Comment: 12 pages, 1 figur
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Approximating acyclicity parameters of sparse hypergraphs
The notions of hypertree width and generalized hypertree width were
introduced by Gottlob, Leone, and Scarcello in order to extend the concept of
hypergraph acyclicity. These notions were further generalized by Grohe and
Marx, who introduced the fractional hypertree width of a hypergraph. All these
width parameters on hypergraphs are useful for extending tractability of many
problems in database theory and artificial intelligence. In this paper, we
study the approximability of (generalized, fractional) hyper treewidth of
sparse hypergraphs where the criterion of sparsity reflects the sparsity of
their incidence graphs. Our first step is to prove that the (generalized,
fractional) hypertree width of a hypergraph H is constant-factor sandwiched by
the treewidth of its incidence graph, when the incidence graph belongs to some
apex-minor-free graph class. This determines the combinatorial borderline above
which the notion of (generalized, fractional) hypertree width becomes
essentially more general than treewidth, justifying that way its functionality
as a hypergraph acyclicity measure. While for more general sparse families of
hypergraphs treewidth of incidence graphs and all hypertree width parameters
may differ arbitrarily, there are sparse families where a constant factor
approximation algorithm is possible. In particular, we give a constant factor
approximation polynomial time algorithm for (generalized, fractional) hypertree
width on hypergraphs whose incidence graphs belong to some H-minor-free graph
class
Structured Connectivity Augmentation
We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping phi:V(H)->V(G) if every edge uv of F is either an edge of G, or phi^{-1}(u)phi^{-1}(v) is an edge of H. Thus F contains both G and H as subgraphs, and the edge set of F is the union of the edge sets of G and phi(H). We consider the following optimization problem. Given graphs G, H, and a weight function omega assigning non-negative weights to pairs of vertices of V(G), the task is to find phi of minimum weight omega(phi)=sum_{xyin E(H)}omega(phi(x)phi(y)) such that the edge connectivity of the superposition F of G and H with respect to phi is higher than the edge connectivity of G. Our main result is the following ``dichotomy\u27\u27 complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when Hin C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when Hin C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems
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