This paper addresses the weighted vertex coloring problem (WVCP) which is an NP-hard variant of the graph coloring problem with various applications. Given a vertex-weighted graph, the problem consists of partitioning vertices in independent sets (colors) so as to minimize the sum of the maximum weights of the colors. We first present an iterative procedure to reduce the size of WVCP instances and prove new upper bounds on the objective value and the number of colors. Alternative constraint programming models are then introduced which rely on primal and dual encodings of the problem and use symmetry breaking constraints. A large number of experiments are conducted on benchmark instances. We analyze the impact of using specific bounds to reduce the search space and speed up the exact resolution of instances. New optimality proofs are reported for some benchmark instances.

A new causal discovery method, Structural Agnostic Modeling (SAM), is presented in this paper. Leveraging both conditional independencies and distributional asymmetries, SAM aims to find the underlying causal structure from observational data. The approach is based on a game between different players estimating each variable distribution conditionally to the others as a neural net, and an adversary aimed at discriminating the generated data against the original data. A learning criterion combining distribution estimation, sparsity and acyclicity constraints is used to enforce the optimization of the graph structure and parameters through stochastic gradient descent. SAM is extensively experimentally validated on synthetic and real data.

This paper presents a new open source Python framework for causal discovery from observational data and domain background knowledge, aimed at causal graph and causal mechanism modeling. The cdt package implements an end-to-end approach, recovering the direct dependencies (the skeleton of the causal graph) and the causal relationships between variables. It includes algorithms from the `Bnlearn' and `Pcalg' packages, together with algorithms for pairwise causal discovery such as ANM.