229 research outputs found
The bidimensionality theory and its algorithmic applications
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(kÂČ), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into Lâ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D
Bidimensionality of Geometric Intersection Graphs
Let B be a finite collection of geometric (not necessarily convex) bodies in
the plane. Clearly, this class of geometric objects naturally generalizes the
class of disks, lines, ellipsoids, and even convex polygons. We consider
geometric intersection graphs GB where each body of the collection B is
represented by a vertex, and two vertices of GB are adjacent if the
intersection of the corresponding bodies is non-empty. For such graph classes
and under natural restrictions on their maximum degree or subgraph exclusion,
we prove that the relation between their treewidth and the maximum size of a
grid minor is linear. These combinatorial results vastly extend the
applicability of all the meta-algorithmic results of the bidimensionality
theory to geometrically defined graph classes
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
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
Contraction-Bidimensionality of Geometric Intersection Graphs
Given a graph G, we define bcg(G) as the minimum k for which G can be contracted to the uniformly triangulated grid Gamma_k. A graph class G has the SQGC property if every graph G in G has treewidth O(bcg(G)c) for some 1 <= c < 2. The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a general family of graph classes that satisfy the SQGC property and includes bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for several intersection graph classes of 2-dimensional geometrical objects
Contraction Bidimensionality: the Accurate Picture
We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory -- the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters
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