Efficient search of combinatorial maps using signatures

In this paper, we address the problem of computing canonical representations of n-dimensional combinatorial maps and of using them for efficiently searching for a map in a database. We define two combinatorial map signatures: the first one has a quadratic space complexity and may be used to decide an isomorphism with a new map in linear time whereas the second one has a linear space complexity and may be used to decide an isomorphism in quadratic time. We show that these signatures can be used to efficiently search for a map in a database.     

Solving subgraph isomorphism problems with constraint programming

The subgraph isomorphism problem consists in deciding if there exists a copy of a pattern graph in a target graph. We introduce in this paper a global constraint and an associated filtering algorithm to solve this problem within the context of constraint programming. The main idea of the filtering algorithm is to label every node with respect to its relationships with other nodes of the graph, and to define a partial order on these labels in order to express compatibility of labels for subgraph isomorphism. This partial order over labels is used to filter domains. Labelings can also be strengthened by adding information from the labels of neighbors. Such a strengthening can be applied iteratively until a fixpoint is reached. Practical experiments illustrate that our new filtering approach is more effective on difficult instances of scale free graphs than state-of-the-art algorithms and other constraint programming approaches.     

A parametric filtering algorithm for the graph isomorphism problem

We introduce a new filtering algorithm, called IDL(d)-filtering, for a global constraint dedicated to the graph isomorphism problem—the goal of which is to decide if two given graphs have an identical structure. The basic idea of IDL(d)-filtering is to label every vertex with respect to its relationships with other vertices around it in the graph, and to use these labels to filter domains by removing values that have different labels. IDL(d)-filtering is parameterized by a positive integer value d which gives a limit on the distance between a vertex to be labelled and the set of vertices considered to build its label. We experimentally compare different instantiations of IDL(d)-filtering with state-of-the-art dedicated algorithms and show that IDL(d)-filtering is more efficient on regular sparse graphs and competitive on other kinds of graphs.     

Polynomial algorithms for subisomorphism of nD open combinatorial maps

Combinatorial maps describe the subdivision of objects in cells, and incidence and adjacency relations between cells, and they are widely used to model 2D and 3D images. However, there is no algorithm for comparing combinatorial maps, which is an important issue for image processing and analysis. In this paper, we address two basic comparison problems, i.e., map isomorphism, which involves deciding if two maps are equivalent, and submap isomorphism, which involves deciding if a copy of a pattern map may be found in a target map. We formally define these two problems for nD open combinatorial maps, we give polynomial time algorithms for solving them, and we illustrate their interest and feasibility for searching patterns in 2D and 3D images, as any child would aim to do when he searches Wally in Martin Handford’s books.     

Ants can solve constraint satisfaction problems

We describe a novel incomplete approach for solving constraint satisfaction problems (CSPs) based on the ant colony optimization (ACO) metaheuristic. The idea is to use artificial ants to keep track of promising areas of the search space by laying trails of pheromone. This pheromone information is used to guide the search, as a heuristic for choosing values to be assigned to variables. We first describe the basic ACO algorithm for solving CSPs and we show how it can be improved by combining it with local search techniques. Then, we introduce a preprocessing step, the goal of which is to favor a larger exploration of the search space at a lower cost, and we show that it allows ants to find better solutions faster. Finally, we evaluate our approach on random binary problems     

Concurrent Cooperating Solvers over Reals

Systems combining an interval narrowing solver and a linear programming solver can tackle constraints over the reals that none of these solvers can handle on their own. In this paper we introduce a cooperating scheme where an interval narrowing solver and a linear programming solver work concurrently. Information exchanged by the solvers is therefore handled as soon as it becomes available. Moreover, to improve the pruning, the linear programming solver computes the actual range of values of each variable with respect to the subset of linear constraints. To validate the proposed architecture a prototype system-named CCC-has been developed. Several examples are given to illustrate the gain in speed and precision we can expect with CCC.     

AllDifferent-based filtering for subgraph isomorphism

The subgraph isomorphism problem involves deciding if there exists a copy of a pattern graph in a target graph. This problem may be solved by a complete tree search combined with filtering techniques that aim at pruning branches that do not contain solutions. We introduce a new filtering algorithm based on local all different constraints. We show that this filtering is stronger than other existing filterings — i.e., it prunes more branches — and that it is also more efficient — i.e., it allows one to solve more instances quicker.