285 research outputs found
Large induced subgraphs via triangulations and CMSO
We obtain an algorithmic meta-theorem for the following optimization problem.
Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an
integer. For a given graph G, the task is to maximize |X| subject to the
following: there is a set of vertices F of G, containing X, such that the
subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X)
models \phi.
Some special cases of this optimization problem are the following generic
examples. Each of these cases contains various problems as a special subcase:
1) "Maximum induced subgraph with at most l copies of cycles of length 0
modulo m", where for fixed nonnegative integers m and l, the task is to find a
maximum induced subgraph of a given graph with at most l vertex-disjoint cycles
of length 0 modulo m.
2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\
containing a planar graph, the task is to find a maximum induced subgraph of a
given graph containing no graph from \Gamma\ as a minor.
3) "Independent \Pi-packing", where for a fixed finite set of connected
graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G
with the maximum number of connected components, such that each connected
component of G[F] is isomorphic to some graph from \Pi.
We give an algorithm solving the optimization problem on an n-vertex graph G
in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential
maximal cliques in G and f is a function depending of t and \phi\ only. We also
show how a similar running time can be obtained for the weighted version of the
problem. Pipelined with known bounds on the number of potential maximal
cliques, we deduce that our optimization problem can be solved in time
O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with
polynomial number of minimal separators
On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi
Qualitative calculi play a central role in representing and reasoning about
qualitative spatial and temporal knowledge. This paper studies distributive
subalgebras of qualitative calculi, which are subalgebras in which (weak)
composition distributives over nonempty intersections. It has been proven for
RCC5 and RCC8 that path consistent constraint network over a distributive
subalgebra is always minimal and globally consistent (in the sense of strong
-consistency) in a qualitative sense. The well-known subclass of convex
interval relations provides one such an example of distributive subalgebras.
This paper first gives a characterisation of distributive subalgebras, which
states that the intersection of a set of relations in the subalgebra
is nonempty if and only if the intersection of every two of these relations is
nonempty. We further compute and generate all maximal distributive subalgebras
for Point Algebra, Interval Algebra, RCC5 and RCC8, Cardinal Relation Algebra,
and Rectangle Algebra. Lastly, we establish two nice properties which will play
an important role in efficient reasoning with constraint networks involving a
large number of variables.Comment: Adding proof of Theorem 2 to appendi
Sequences of regressions and their independences
Ordered sequences of univariate or multivariate regressions provide
statistical models for analysing data from randomized, possibly sequential
interventions, from cohort or multi-wave panel studies, but also from
cross-sectional or retrospective studies. Conditional independences are
captured by what we name regression graphs, provided the generated distribution
shares some properties with a joint Gaussian distribution. Regression graphs
extend purely directed, acyclic graphs by two types of undirected graph, one
type for components of joint responses and the other for components of the
context vector variable. We review the special features and the history of
regression graphs, derive criteria to read all implied independences of a
regression graph and prove criteria for Markov equivalence that is to judge
whether two different graphs imply the same set of independence statements.
Knowledge of Markov equivalence provides alternative interpretations of a given
sequence of regressions, is essential for machine learning strategies and
permits to use the simple graphical criteria of regression graphs on graphs for
which the corresponding criteria are in general more complex. Under the known
conditions that a Markov equivalent directed acyclic graph exists for any given
regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited
discussion paper in the journal TES
Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing
Characteristic imsets are 0-1 vectors which correspond to Markov equivalence
classes of directed acyclic graphs. The study of their convex hull, named the
characteristic imset polytope, has led to new and interesting geometric
perspectives on the important problem of causal discovery. In this paper we
begin the study of the associated toric ideal. We develop a new generalization
of the toric fiber product, which we call a quasi-independence gluing, and show
that under certain combinatorial homogeneity conditions, one can iteratively
compute a Gr\"obner basis via lifting. For faces of the characteristic imset
polytope associated to trees, we apply this technique to compute a Gr\"obner
basis for the associated toric ideal. We end with a study of the characteristic
ideal of the cycle and propose directions for future work.Comment: 19 pages, 7 figure
Global minimization of polynomial integral functionals
We describe a `discretize-then-relax' strategy to globally minimize integral
functionals over functions in a Sobolev space satisfying prescribed
Dirichlet boundary conditions. The strategy applies whenever the integral
functional depends polynomially on and its derivatives, even if it is
nonconvex. The `discretize' step uses a bounded finite-element scheme to
approximate the integral minimization problem with a convergent hierarchy of
polynomial optimization problems over a compact feasible set, indexed by the
decreasing size of the finite-element mesh. The `relax' step employs sparse
moment-SOS relaxations to approximate each polynomial optimization problem with
a hierarchy of convex semidefinite programs, indexed by an increasing
relaxation order . We prove that, as and ,
solutions of such semidefinite programs provide approximate minimizers that
converge in to the global minimizer of the original integral functional
if this is unique. We also report computational experiments that show our
numerical strategy works well even when technical conditions required by our
theoretical analysis are not satisfied.Comment: 22 pages, 9 figure
An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy
Tolerance graphs model interval relations in such a way that intervals can tolerate a certain degree of overlap without being in conflict. This class of graphs has attracted many research efforts, mainly due to its interesting structure and its numerous applications, especially in DNA sequence analysis and resource allocation, among others. In one of the most natural generalizations of tolerance graphs, namely multitolerance graphs, two tolerances are allowed for each interval—one from the left and one from the right side of the interval. Then, in its interior part, every interval tolerates the intersection with others by an amount that is a convex combination of its two border-tolerances. In the comparison of DNA sequences between different organisms, the natural interpretation of this model lies on the fact that, in some applications, we may want to treat several parts of the genomic sequences differently. That is, we may want to be more tolerant at some parts of the sequences than at others. These two tolerances for every interval—together with their convex hull—define an infinite number of the so called tolerance-intervals, which make the multitolerance model inconvenient to cope with. In this article we introduce the first non-trivial intersection model for multitolerance graphs, given by objects in the 3-dimensional space called trapezoepipeds. Apart from being important on its own, this new intersection model proves to be a powerful tool for designing efficient algorithms. Given a multitolerance graph with n vertices and m edges along with a multitolerance representation, we present algorithms that compute a minimum coloring and a maximum clique in optimal O(nlogn) time, and a maximum weight independent set in O(m+nlogn) time. Moreover, our results imply an optimal O(nlogn) time algorithm for the maximum weight independent set problem on tolerance graphs, thus closing the complexity gap for this problem. Additionally, by exploiting more the new 3D-intersection model, we completely classify multitolerance graphs in the hierarchy of perfect graphs. The resulting hierarchy of classes of perfect graphs is complete, i.e. all inclusions are strict
Chordal Graphs and Their Relatives: Algorithms and Applications
While the problem of generating random graphs has received much attention, the problem of generating graphs for specific classes has not been studied much. In this dissertation, we propose schemes for generating chordal graphs, weakly chordal graphs, and strongly chordal graphs. We also present semi-dynamic algorithms for chordal graphs and strongly chordal graphs. As an application of a completion technique for chordal graphs, we also discuss a 1-round algorithm for approximate point placement in the plane in an adversarial model where the distance query graph presented to the adversary is chordal. The proposed generation algorithms take the number of vertices, n, and the number of edges, m, as input and produces a graph in a given class as output. The generation method either starts with a tree or a complete graph. We then insert additional edges in the tree or delete edges from the complete graph. Our algorithm ensures that the graph properties are preserved after each edge is inserted or deleted. We have also proposed algorithms to generate weakly chordal graphs and strongly chordal graphs from an arbitrary graph as input. In this case, we ensure the graph properties will be achieved on the termination of the conversion process. We have also proposed a semi-dynamic algorithm for edge-deletion in a chordal graph. To the best of our knowledge, no study has been done for the problem of dynamic algorithms for strongly chordal graphs. To address this gap, we have also proposed a semi-dynamic algorithm for edge-deletions and a semi-dynamic algorithm for edge-insertions in strongly chordal graphs
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