We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structures called \textit{zero-suppressed binary decision diagrams} (ZDDs), which can compactly represent sets of combinatorial strategies. Our algorithm numerically approximates the equilibria of CCGs, which we can differentiate with respect to parameters of CCGs by automatic differentiation. With the resulting derivatives, we can apply gradient-based methods to Stackelberg models of CCGs. Our method is tailored to induce Nesterov's acceleration and can fully utilize the empirical compactness of ZDDs. These technical advantages enable us to deal with CCGs with a vast number of combinatorial strategies. Experiments on real-world network design instances demonstrate the practicality of our method.

Therefore, our algorithms can be extended to more general semiring based graphical models. We will fix all typos and presentation issues. We will extend the discussion in the paper to clarify this point.

Weighted model counting computes the sum of the rational-valued weights associated with the satisfying assignments for a Boolean formula, where the weight of an assignment is given by the product of the weights assigned to the positive and negated variables comprising the assignment. Weighted model counting finds applications across a variety of domains including probabilistic reasoning and quantitative risk assessment. Most weighted model counting programs operate by (explicitly or implicitly) converting the input formula into a form that enables arithmetic evaluation, using multiplication for conjunctions and addition for disjunctions. Performing this evaluation using floating-point arithmetic can yield inaccurate results, and it cannot quantify the level of precision achieved. Computing with rational arithmetic gives exact results, but it is costly in both time and space. This paper describes how to combine multiple numeric representations to efficiently compute weighted model counts that are guaranteed to achieve a user-specified precision. When all weights are nonnegative, we prove that the precision loss of arithmetic evaluation using floating-point arithmetic can be tightly bounded. We show that supplementing a standard IEEE double-precision representation with a separate 64-bit exponent, a format we call extended-range double (ERD), avoids the underflow and overflow issues commonly encountered in weighted model counting. For problems with mixed negative and positive weights, we show that a combination of interval floating-point arithmetic and rational arithmetic can achieve the twin goals of efficiency and guaranteed precision. For our evaluations, we have devised especially challenging formulas and weight assignments, demonstrating the robustness of our approach.

Safety-critical controllers of complex systems are hard to construct manually. Automated approaches such as controller synthesis or learning provide a tempting alternative but usually lack explainability. To this end, learning decision trees (DTs) have been prevalently used towards an interpretable model of the generated controllers. However, DTs do not exploit shared decision-making, a key concept exploited in binary decision diagrams (BDDs) to reduce their size and thus improve explainability. In this work, we introduce predicate decision diagrams (PDDs) that extend BDDs with predicates and thus unite the advantages of DTs and BDDs for controller representation. We establish a synthesis pipeline for efficient construction of PDDs from DTs representing controllers, exploiting reduction techniques for BDDs also for PDDs.

We introduce an enumeration-free method based on mathematical programming to precisely characterize various properties such as fairness or sparsity within the set of "good models", known as Rashomon set. This approach is generically applicable to any hypothesis class, provided that a mathematical formulation of the model learning task exists. It offers a structured framework to define the notion of business necessity and evaluate how fairness can be improved or degraded towards a specific protected group, while remaining within the Rashomon set and maintaining any desired sparsity level. We apply our approach to two hypothesis classes: scoring systems and decision diagrams, leveraging recent mathematical programming formulations for training such models. As seen in our experiments, the method comprehensively and certifiably quantifies trade-offs between predictive performance, sparsity, and fairness. We observe that a wide range of fairness values are attainable, ranging from highly favorable to significantly unfavorable for a protected group, while staying within less than 1% of the best possible training accuracy for the hypothesis class. Additionally, we observe that sparsity constraints limit these trade-offs and may disproportionately harm specific subgroups. As we evidenced, thoroughly characterizing the tensions between these key aspects is critical for an informed and accountable selection of models.

Decision diagrams (DDs) have emerged as an efficient tool for simulating quantum circuits due to their capacity to exploit data redundancies in quantum states and quantum operations, enabling the efficient computation of probability amplitudes. However, their application in quantum machine learning (QML) has remained unexplored. This paper introduces variational decision diagrams (VDDs), a novel graph structure that combines the structural benefits of DDs with the adaptability of variational methods for efficiently representing quantum states. We investigate the trainability of VDDs by applying them to the ground state estimation problem for transverse-field Ising and Heisenberg Hamiltonians. Analysis of gradient variance suggests that training VDDs is possible, as no signs of vanishing gradients--also known as barren plateaus--are observed. This work provides new insights into the use of decision diagrams in QML as an alternative to design and train variational ans\"atze.

We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structures called \textit{zero-suppressed binary decision diagrams} (ZDDs), which can compactly represent sets of combinatorial strategies. Our algorithm numerically approximates the equilibria of CCGs, which we can differentiate with respect to parameters of CCGs by automatic differentiation.

Recent advancements in machine learning have accelerated its widespread adoption across various real-world applications. However, in safety-critical domains, the deployment of machine learning models is riddled with challenges due to their complexity, lack of interpretability, and absence of formal guarantees regarding their behavior. In this paper, we introduce a verification framework tailored for Bayesian networks, designed to address these drawbacks. Our framework comprises two key components: (1) a two-step compilation and encoding scheme that translates Bayesian networks into Boolean logic literals, and (2) formal verification queries that leverage these literals to verify various properties encoded as constraints. Specifically, we introduce two verification queries: if-then rules (ITR) and feature monotonicity (FMO). We benchmark the efficiency of our verification scheme and demonstrate its practical utility in real-world scenarios.

Decision diagrams (DDs) are powerful tools to represent effectively propositional formulas, which are largely used in many domains, in particular in formal verification and in knowledge compilation. Some forms of DDs (e.g., OBDDs, SDDs) are canonical, that is, (under given conditions on the atom list) they univocally represent equivalence classes of formulas. Given the limited expressiveness of propositional logic, a few attempts to leverage DDs to SMT level have been presented in the literature. Unfortunately, these techniques still suffer from some limitations: most procedures are theory-specific; some produce theory DDs (T-DDs) which do not univocally represent T-valid formulas or T-inconsistent formulas; none of these techniques provably produces theory-canonical T-DDs, which (under given conditions on the T-atom list) univocally represent T-equivalence classes of formulas. Also, these procedures are not easy to implement, and very few implementations are actually available. In this paper, we present a novel very-general technique to leverage DDs to SMT level, which has several advantages: it is very easy to implement on top of an AllSMT solver and a DD package, which are used as blackboxes; it works for every form of DDs and every theory, or combination thereof, supported by the AllSMT solver; it produces theory-canonical T-DDs if the propositional DD is canonical. We have implemented a prototype tool for both T-OBDDs and T-SDDs on top of OBDD and SDD packages and the MathSAT SMT solver. Some preliminary empirical evaluation supports the effectiveness of the approach.

The branch-and-bound algorithm based on decision diagrams introduced by Bergman et al. in 2016 is a framework for solving discrete optimization problems with a dynamic programming formulation. It works by compiling a series of bounded-width decision diagrams that can provide lower and upper bounds for any given subproblem. Eventually, every part of the search space will be either explored or pruned by the algorithm, thus proving optimality. This paper presents new ingredients to speed up the search by exploiting the structure of dynamic programming models. The key idea is to prevent the repeated expansion of nodes corresponding to the same dynamic programming states by querying expansion thresholds cached throughout the search. These thresholds are based on dominance relations between partial solutions previously found and on the pruning inequalities of the filtering techniques introduced by Gillard et al. in 2021. Computational experiments show that the pruning brought by this caching mechanism allows significantly reducing the number of nodes expanded by the algorithm. This results in more benchmark instances of difficult optimization problems being solved in less time while using narrower decision diagrams.