Diverse combinatorial optimization problems can be modeled as instances of the Pseudo-boolean Optimization (PBO) problem. The Predict-and-Search (PaS) framework is a powerful technique applied to Mixed Integer Linear Programming (MIP) that uses Graph Neural Networks (GNNs) to predict candidate variable values for guiding the search process of an optimization solver. Current PaS implementations rely on heuristically chosen labels during training and produce fast, good solutions, sacrificing optimality guarantees. We present Backbone-based Predict and Search (BackPaS), a specialized PaS framework for PBO. Our main contribution is redefining the GNN prediction task to identify backbones—literals fixed across all optimal solutions. Predicting these critical variables accelerates the search toward optimality if the network’s prediction is correct. BackPaS uses a specialized GNN architecture built on a literal-based bipartite graph to predict backbone membership and polarity, formulated as a multi-class classification problem. Then, a parameterized adaptive trust region incorporates these GNN predictions to adjust the solver’s search space. Empirically, we demonstrate that BackPaS effectively learns backbone patterns across PBO benchmarks, including Maximum Independent Set (MIS), Minimum Vertex Cover (MVC), and Combinatorial Auctions (CA). We trained the model on small instances for which we computed their backbones. When tested on much larger instances (up to 6 × the training size), our method achieves significant performance gains compared to the commercial solver Gurobi and the state-of-the-art PaS implementation ConPaS, substantially improving its anytime behavior and demonstrating strong generalization capabilities.
The Quadratic Assignment Problem (QAP) consists of finding a permutation that minimizes a quadratic objective function. Exact methods generally rely on a branch-and-bound procedure, the efficiency of which depends heavily on the quality of its lower bound. In integer linear programming, several bounds have been investigated, exhibiting different trade-offs between speed and quality. The Gilmore-Lawler bound appears to be the most commonly used in practice. It requires solving a linear assignment problem (LAP) for each variable-value pair. We show how to obtain this bound using Singleton Node Consistency (SNC) and LAP. In Cost Function Networks (CFNs), we propose a reformulation that transforms the result of applying LAP to a given variable-value pair into cost functions of arity 1 and 2, which can be added to the original problem. Combined with existing lower bounds for CFNs, including EDAC and a recent CFN propagator for AllDifferent, this method (SNC-LAP-GLB), used as a preprocessing, significantly increases the initial lower bound and accelerates the search, resulting in competitive results on the QAPLIB benchmark. We then propose an extension of the AllDifferent propagator for the Global Cardinality Constraint. It allows us to exploit variable symmetries on some challenging QAPLIB instances, thus improving the results.
We consider the problem of determining the minimax regret of a set of utility vectors, where the minimax regret is defined with respect to a given space of possible weight vectors. We present two novel approaches for this problem. The first one improves on an existing method based on the set of undominated utility vectors, by adding maximum regret upper bounds in order to reduce the number of linear programming computations. Our second approach uses a branch-and-bound algorithm that can be applied when the utility vectors are given in the form of sub-utility functions over a set of Boolean variables. Both of our approaches make heavy use of two particular sets of weight vectors, one that approximates the convex hull of the weight space from within, and the other from without. We show that our two approaches complement each other well, with each one having its own combinations of parameters for which it outperforms the other.
We consider a dynamic stochastic capacity problem in which tasks arrive according to a Poisson Process. Release times, stochastic deadlines and masses characterize tasks. A task might be scheduled within its time windows at a given time slot of bounded mass capacity. Each non-empty slot induces a deterministic setup cost. We aim to define a dynamic scheduling policy that maximizes the expected mass of scheduled tasks minus the setup costs over a long time horizon. This type of problem has many applications in planning, scheduling, and yield management. We use a rolling horizon approach: at each step, a particular case of the Multiple Knapsack with Restriction Assignment must be solved. We present and benchmark myopic heuristics, lookahead heuristics, and the exact solution provided by integer linear programming. We provide an extensive comparison in our dynamic and stochastic context. Finally, rather than searching for the most robust heuristic, we devise an algorithm selection process, based on random forests, to select the most effective heuristic for a given instance, considering its parameters. This achieves significant improvements.
Modern combinatorial solvers can be understood as searching for proofs of unsatisfiability or optimality. The proof system implemented by a solver, therefore, fundamentally shapes solver performance. While propositional resolution is simple and complete for propositional formulas, no existing practical resolution-based system offers unrestricted complete reasoning over integer linear inequalities. We introduce hypercube linear resolution, a new proof system that is both sound and complete for integer linear reasoning. Hypercube linear resolution integrates propositional resolution with Fourier resolution through a new constraint type, the hypercube linear constraint, which captures linear relations within a discrete hypercube. Our main contribution is the theory of the proof system, establishing its soundness and completeness. Our proof system generalises propositional and Fourier resolution, and can be seen as an extended cutting planes proof system. We also provide a conflict-driven search algorithm that exploits the system in practice. Our preliminary experiments demonstrate that the new system reduces conflicts compared to propositional resolution. This shows that the structure captured by hypercube linear constraints can be exploited to improve constraint solving.
