We introduce a new framework for data denoising, partially inspired by martingale optimal transport. For a given noisy distribution (the data), our approach involves finding the closest distribution to it among all distributions which 1) have a particular prescribed structure (expressed by requiring they lie in a particular domain), and 2) are self-consistent with the data. We show that this amounts to maximizing the variance among measures in the domain which are dominated in convex order by the data. For particular choices of the domain, this problem and a relaxed version of it, in which the self-consistency condition is removed, are intimately related to various classical approaches to denoising. We prove that our general problem has certain desirable features: solutions exist under mild assumptions, have certain robustness properties, and, for very simple domains, coincide with solutions to the relaxed problem. We also introduce a novel relationship between distributions, termed Kantorovich dominance, which retains certain aspects of the convex order while being a weaker, more robust, and easier-to-verify condition. Building on this, we propose and analyze a new denoising problem by substituting the convex order in the previously described framework with Kantorovich dominance. We demonstrate that this revised problem shares some characteristics with the full convex order problem but offers enhanced stability, greater computational efficiency, and, in specific domains, more meaningful solutions. Finally, we present simple numerical examples illustrating solutions for both the full convex order problem and the Kantorovich dominance problem.

Counterfactuals are widely used to explain ML model predictions by providing alternative scenarios for obtaining the more desired predictions. They can be generated by a variety of methods that optimize different, sometimes conflicting, quality measures and produce quite different solutions. However, choosing the most appropriate explanation method and one of the generated counterfactuals is not an easy task. Instead of forcing the user to test many different explanation methods and analysing conflicting solutions, in this paper, we propose to use a multi-stage ensemble approach that will select single counterfactual based on the multiple-criteria analysis. It offers a compromise solution that scores well on several popular quality measures. This approach exploits the dominance relation and the ideal point decision aid method, which selects one counterfactual from the Pareto front. The conducted experiments demonstrated that the proposed approach generates fully actionable counterfactuals with attractive compromise values of the considered quality measures.

Dominance breaking is a powerful technique in improving the solving efficiency of Constraint Optimization Problems (COPs) by removing provably suboptimal solutions with additional constraints. While dominance breaking is effective in a range of practical problems, it is usually problem specific and requires human insights into problem structures to come up with correct dominance breaking constraints. Recently, a framework is proposed to generate nogood constraints automatically for dominance breaking, which formulates nogood generation as solving auxiliary Constraint Satisfaction Problems (CSPs). However, the framework uses a pattern matching approach to synthesize the auxiliary generation CSPs from the specific forms of objectives and constraints in target COPs, and is only applicable to a limited class of COPs. This paper proposes a novel rewriting system to derive constraints for the auxiliary generation CSPs automatically from COPs with nested function calls, significantly generalizing the original framework. In particular, the rewriting system exploits functional constraints flattened from nested functions in a high-level modeling language. To generate more effective dominance breaking nogoods and derive more relaxed constraints in generation CSPs, we further characterize how to extend the system with rewriting rules exploiting function properties, such as monotonicity, commutativity, and associativity, for specific functional constraints. Experimentation shows significant runtime speedup using the dominance breaking nogoods generated by our proposed method. Studying patterns of generated nogoods also demonstrates that our proposal can reveal dominance relations in the literature and discover new dominance relations on problems with ineffective or no known dominance breaking constraints.

In this preference elicitation setting, our focus is on determining the parameters of a decision model that accurately captures the pairwise preferences of a Decision Maker (DM) over subsets, by comparing subsets of elements. The preferences are depicted using a highly adaptable model whose versatility stems from its ability to incorporate positive or negative synergies between elements [24]. Moreover, we provide an ordinally robust approach, in the sense that the preferences we infer do not rely on arbitrarily specified parameter values, but on the set of all parameter values that are compatible with the observed preferences. Importantly, another distinctive feature of our approach is its ability to learn the parameter set itself (not only the values of parameters). The preference model we consider can be used in different contexts, depending on the nature of the subsets we are comparing. The subsets are represented by binary vectors, showing the presence or absence of an element in the subset. The elements of a subset can be for example: individuals (in the comparison of coalitions, teams, etc.), binary attributes (in the comparison of multiattribute alternatives), objects (in the comparison of subsets in a subset choice problem), etc. For illustration, a toy example of such an elicitation context could be a coffee shop trying to determine its customers' favorite frozen yogurt flavor combination by offering them to test a small number of flavor combinations rather than having them taste each combination.

Among the many merits of Roman Słowiński in his so long and so rich scientific carrier, we have to consider his pioneering approach to the use of artificial intelligence methodologies to decision support, and, in particular, to Multiple Criteria Decision Aiding (MCDA) (for an updated state of the art see [48]). In this perspective, the proposal and the development of the Dominance-based Rough Set Approach (DRSA) is a cornerstone in the domain. The DRSA basic idea of a decision support procedure based on a decision model expressed in natural language and obtained from simple preference information in terms of exemplary decisions has attracted the interest of experts and it is now considered one of the three main approaches to MCDA, together with the classical Multiple Attribute Utility Theory (MAUT) [58] and the outranking approach [75]. In fact, DRSA is not a mere application to MCDA of concepts and tools already proposed and developed in the domain of artificial intelligence, knowledge discovery, data mining and machine learning. Indeed, consideration of preference orders typical for MCDA problems required a reformulation of many important concepts and methodologies, so that DRSA became a methodology viable and interesting per se also in these domains. Consequently, after more or less 25 years from the proposal of DRSA, we try to present a first assessment taking into consideration the basic ideas and the main developments.

The exploitation of dominance relations in constraint optimization problems can lead to dramatic reductions in search space. We propose an automatic method to detect some of the dominance relations manually identified by Chu and Stuckey for optimization problems, and to construct the associated dominance breaking constraints. Experimental results show that the method is able to find several dominance relations and to generate effective dominance breaking constraints.

In the field of pattern mining, a negative sequential pattern is specified by means of a sequence consisting of events to occur and of other events, called negative events, to be absent. For instance, containment of the pattern $\langle a\ \neg b\ c\rangle$ arises with an occurrence of a and a subsequent occurrence of c but no occurrence of b in between. This article is to shed light on the ambiguity of such a seemingly intuitive notation and we identify eight possible semantics for the containment relation between a pattern and a sequence. These semantics are illustrated and formally studied, in particular we propose dominance and equivalence relations between them. Also we prove that support is anti-monotonic for some of these semantics. Some of the results are discussed with the aim of developing algorithms to extract efficiently frequent negative patterns.

Constraint Satisfaction Problems (CSPs) typically have many solutions that satisfy all constraints. Often though, some solutions are preferred over others, that is, some solutions dominate other solutions. We present solution dominance as a formal framework to reason about such settings. We define Constraint Dominance Problems (CDPs) as CSPs with a dominance relation, that is, a preorder over the solutions of the CSP. This framework captures many well-known variants of constraint satisfaction, including optimization, multi-objective optimization, Max-CSP, minimal models, minimum correction subsets as well as optimization over CP-nets and arbitrary dominance relations. We extend MiniZinc, a declarative language for modeling CSPs, to CDPs by introducing dominance nogoods; these can be derived from dominance relations in a principled way. A generic method for solving arbitrary CDPs incrementally calls a CSP solver and is compatible with any existing solver that supports MiniZinc. This encourages experimenting with different solution dominance relations for a problem, as well as comparing different solvers without having to modify their implementations.

In optimal planning as heuristic search, admissible pruning techniques are paramount. One idea is dominance pruning, identifying states "better than" other states. Prior approaches are limited to simple dominance notions, like "more STRIPS facts true" or "higher resource supply". We apply simulation, well-known in model checking, to compute much more general dominance relations based on comparing transition behavior across states. We do so effectively by expressing state-space simulations through the composition of simulations on orthogonal projections. We show how simulation can be made more powerful by intertwining it with a notion of label dominance. Our experiments show substantial improvements across several IPC benchmark domains.

We increase the usefulness of Chu and Stuckey's work by automating it, that is, by developing a method to (a) automatically The exploitation of dominance relations in constraint identify symmetries for a given problem, and (b) optimization problems can lead to dramatic automatically construct the associated dominance breaking reductions in search space. We propose an automatic constraints. Note that these dominance breaking constraints method to detect some of the dominance relations are not symmetry breaking constraints, as the key element manually identified by Chu and Stuckey for is for f(σ(θ)) to be better. Further, the symmetries need to optimization problems, and to construct the associated be detected for the inherent satisfaction problem -- that is, dominance breaking constraints. Experimental the problem without the objective function -- or, otherwise, results show that the method is able to find several f(σ(θ)) will be equal to f(θ), not better.