26,421 research outputs found
EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails
The Clique Problem in Ray Intersection Graphs
Ray intersection graphs are intersection graphs of rays, or halflines, in the
plane. We show that any planar graph has an even subdivision whose complement
is a ray intersection graph. The construction can be done in polynomial time
and implies that finding a maximum clique in a segment intersection graph is
NP-hard. This solves a 21-year old open problem posed by Kratochv\'il and
Ne\v{s}et\v{r}il.Comment: 12 pages, 7 figure
Approximating Minimum Independent Dominating Sets in Wireless Networks
We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in a graph that is also independent. It thus combines the advantages of both structures, and there are many applications that rely on these two structures e.g. in the area of wireless ad hoc networks. The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1+")- pproximate minimum dominating set, or a certificate showing that the input graph does not reflect a wireless network
Algebraic Methods of Classifying Directed Graphical Models
Directed acyclic graphical models (DAGs) are often used to describe common
structural properties in a family of probability distributions. This paper
addresses the question of classifying DAGs up to an isomorphism. By considering
Gaussian densities, the question reduces to verifying equality of certain
algebraic varieties. A question of computing equations for these varieties has
been previously raised in the literature. Here it is shown that the most
natural method adds spurious components with singular principal minors, proving
a conjecture of Sullivant. This characterization is used to establish an
algebraic criterion for isomorphism, and to provide a randomized algorithm for
checking that criterion. Results are applied to produce a list of the
isomorphism classes of tree models on 4,5, and 6 nodes. Finally, some evidence
is provided to show that projectivized DAG varieties contain useful information
in the sense that their relative embedding is closely related to efficient
inference
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