A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research

Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author

Between Subgraph Isomorphism and Maximum Common Subgraph

When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

Unit Grid Intersection Graphs: Recognition and Properties

It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs (UGIGs), the recognition remains hard, as well as for all graph classes contained inbetween. The result holds even when considering only graphs with arbitrarily large girth. Furthermore, we ask the question of representing UGIGs on grids of minimal size. We show that the UGIGs that can be represented in a square of side length 1+epsilon, for a positive epsilon no greater than 1, are exactly the orthogonal ray graphs, and that there exist families of trees that need an arbitrarily large grid

DOPE: Distributed Optimization for Pairwise Energies

We formulate an Alternating Direction Method of Mul-tipliers (ADMM) that systematically distributes the computations of any technique for optimizing pairwise functions, including non-submodular potentials. Such discrete functions are very useful in segmentation and a breadth of other vision problems. Our method decomposes the problem into a large set of small sub-problems, each involving a sub-region of the image domain, which can be solved in parallel. We achieve consistency between the sub-problems through a novel constraint that can be used for a large class of pair-wise functions. We give an iterative numerical solution that alternates between solving the sub-problems and updating consistency variables, until convergence. We report comprehensive experiments, which demonstrate the benefit of our general distributed solution in the case of the popular serial algorithm of Boykov and Kolmogorov (BK algorithm) and, also, in the context of non-submodular functions.Comment: Accepted at CVPR 201

FO Model Checking of Geometric Graphs

Over the past two decades the main focus of research into first-order (FO) model checking algorithms has been on sparse relational structures - culminating in the FPT algorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs. On contrary to that, except the case of locally bounded clique-width only little is currently known about FO model checking of dense classes of graphs or other structures. We study the FO model checking problem for dense graph classes definable by geometric means (intersection and visibility graphs). We obtain new nontrivial FPT results, e.g., for restricted subclasses of circular-arc, circle, box, disk, and polygon-visibility graphs. These results use the FPT algorithm by Gajarsk\'y et al. for FO model checking of posets of bounded width. We also complement the tractability results by related hardness reductions

A strategy for the visual recognition of objects in an industrial environment.

This thesis is concerned with the problem of recognizing industrial objects rapidly and flexibly. The system design is based on a general strategy that consists of a generalized local feature detector, an extended learning algorithm and the use of unique structure of the objects. Thus, the system is not designed to be limited to the industrial environment. The generalized local feature detector uses the gradient image of the scene to provide a feature description that is insensitive to a range of imaging conditions such as object position, and overall light intensity. The feature detector is based on a representative point algorithm which is able to reduce the data content of the image without restricting the allowed object geometry. Thus, a major advantage of the local feature detector is its ability to describe and represent complex object structure. The reliance on local features also allows the system to recognize partially visible objects. The task of the learning algorithm is to observe the feature description generated by the feature detector in order to select features that are reliable over the range of imaging conditions of interest. Once a set of reliable features is found for each object, the system finds unique relational structure which is later used to recognize the objects. Unique structure is a set of descriptions of unique subparts of the objects of interest. The present implementation is limited to the use of unique local structure. The recognition routine uses these unique descriptions to recognize objects in new images. An important feature of this strategy is the transference of a large amount of processing required for graph matching from the recognition stage to the learning stage, which allows the recognition routine to execute rapidly. The test results show that the system is able to function with a significant level of insensitivity to operating conditions; The system shows insensitivity to its 3 main assumptions -constant scale, constant lighting, and 2D images- displaying a degree of graceful degradation when the operating conditions degrade. For example, for one set of test objects, the recognition threshold was reached when the absolute light level was reduced by 70%-80%, or the object scale was reduced by 30%-40%, or the object was tilted away from the learned 2D plane by 300-400. This demonstrates a very important feature of the learning strategy: It shows that the generalizations made by the system are not only valid within the domain of the sampled set of images, but extend outside this domain. The test results also show that the recognition routine is able to execute rapidly, requiring 10ms-500ms (on a PDP11/24 minicomputer) in the special case when ideal operating conditions are guaranteed. (Note: This does not include pre-processing time). This thesis describes the strategy, the architecture and the implementation of the vision system in detail, and gives detailed test results. A proposal for extending the system to scale independent 3D object recognition is also given

Claw-free t-perfect graphs can be recognised in polynomial time

A graph is called t-perfect if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect