Samba Ndojh Ndiaye

Computing and exploiting tree-decompositions for solving constraint networks

Methods exploiting tree-decompositions seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. However, they have not shown a real practical interest yet. In this paper, we study several methods for computing a rough optimal tree-decomposition and assess their relevance for solving CSPs.

CP models for maximum common subgraph problems

The distance between two graphs is usually defined by means of the size of a largest common subgraph. This common subgraph may be an induced subgraph, obtained by removing nodes, or a partial subgraph, obtained by removing arcs and nodes. In this paper, we introduce two soft CSPs which model these two maximum common subgraph problems in a unified framework. We also introduce and compare different CP models, corresponding to different levels of constraint propagation.

Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

The maximum common subgraph problem is to find the largest subgraph common to two given graphs. This problem can be solved either by constraint-based search, or by reduction to the maximum clique problem. We evaluate these two models using modern algorithms, and see that the best choice depends mainly upon whether the graphs have labelled edges. We also study a variant of this problem where the subgraph is required to be connected. We introduce a filtering algorithm for this property and show that it may be combined with a restricted branching technique for the constraint-based approach. We show how to implement a similar branching technique in clique-inspired algorithms. Finally, we experimentally compare approaches for the connected version, and see again that the best choice depends on whether graphs have labels.

Dynamic management of heuristics for solving structured CSPs

This paper deals with the problem of solving efficiently structured CSPs. It is well known that (hyper)tree-decompositions offer the best approaches from a theoretical viewpoint, but from the practical viewpoint, these methods do not offer efficient algorithms. Therefore, we introduce here a framework founded on coverings of CSP by acyclic hypergraphs. We study their properties and relations, and evaluate theoretically their interest with respect to the solving of structured problems. This framework does not define a new decomposition, but makes easier a dynamic management of the CSP structure during the search, and so an exploitation of dynamic variables ordering heuristics in the solving method. Thus, we provide a new complexity result which outperforms significantly the previous one given in the literature about heuristics for solving a decomposed problem. Finally, we present experimental results to assess the practical interest of these notions.

On the notion of cycles in hypergraphs

The notion of hypergraph cyclicity is crucial in numerous fields of application of hypergraph theory (e.g. in computer science, in relational database theory and constraint programming). Surprisingly, while this notion has been well studied during last thirty years, no relevant definition of cycles in hypergraphs has been proposed by the community. In this paper, we propose a definition of cycles in hypergraphs, @a-cycle based on the same principle in graph theory, meaning that a hypergraph is acyclic iff it does not contain an @a-cycle. This result completes the theory of the mostly used notion of hypergraph acyclicity, the @a-acyclicity.

Extending to Soft and Preference Constraints a Framework for Solving Efficiently Structured Problems

This paper deals with the problem of solving efficiently structured COPs (constraints optimization problems). The formalism based on COPs allows to represent numerous real-life problems defined using constraints and to manage preferences and soft constraints. In spite of theoretical results, has discarded (hyper)tree-decompositions for the benefit of coverings by acyclic hypergraphs in the CSP area. We extend here this work to constraint optimization. We first study these coverings from a theoretical viewpoint. Then we exploit them in a framework aiming not to define a new decomposition, but to make easier a dynamic management of the structure during the search (unlike most of structural methods which usually exploit the structure statically), and so the use of dynamic variable ordering heuristics. Thus, we provide a new complexity result which outperforms significantly the previous one given in the literature. Finally, we assess the practical interest of these notions.

A Comparison of Decomposition Methods for the Maximum Common Subgraph Problem

The maximum common subgraph problem is an NP-hard problem which is very difficult to solve with exact approaches. To speed up the solution process, we may decompose it into independent subproblems which are solved in parallel. We describe a new decomposition method which exploits the structure of the problem to decompose it. We compare this structural decomposition with domain-based decompositions, which basically split variable domains. Experimental results show us that the structural decomposition leads to better speedups on two classes of instances, and to worse speedups on one class of instances.

Combined Strategies for Decomposition-Based Methods for Solving CSPs

In this paper, we consider theoretical and practicalmethods based on decompositions of constraint networks. We exploit the fact that decomposition-based methods can be used considering two steps. The first step is related to the (hyper)graphical decomposition (e.g. Tree-Decomposition [16] or Hypertree-Decomposition [7]) while the second step exploits the decomposition to solve the CSPs. Thanks to this approach, we define then hybrid methods which can be optimal from a theoretical viewpoint while being efficient in practice. The complexity analysis of these combined methods allows us to give a more detailed presentation of the Constraint Tractability Hierarchy introduced in [7]. Finally, we justify our approach with experimental results.

A New Evaluation of Forward Checking and Its Consequences on Efficiency of Tools for Decomposition of CSPs

In this paper, a new evaluation of the complexity of forward checking for solving non-binary CSPs with finite domains is proposed. Unlike what is done usually, it does not consider the size of domains, but the size of the relations associated to the constraints. It may lead sometimes to define better complexity bounds. By using this first result, we show that the tractability hierarchy proposed in [6] which compares different methods based on a decomposition of constraint networks can be seen from a new viewpoint.

Observations from Parallelising Three Maximum Common (Connected) Subgraph Algorithms

We discuss our experiences adapting three recent algorithms for maximum common (connected) subgraph problems to exploit multi-core parallelism. These algorithms do not easily lend themselves to parallel search, as the search trees are extremely irregular, making balanced work distribution hard, and runtimes are very sensitive to value-ordering heuristic behaviour. Nonetheless, our results show that each algorithm can be parallelised successfully, with the threaded algorithms we create being clearly better than the sequential ones. We then look in more detail at the results, and discuss how speedups should be measured for this kind of algorithm. Because of the difficulty in quantifying an average speedup when so-called anomalous speedups (superlinear and sublinear) are common, we propose a new measure called aggregate speedup.