arXiv subjects

Olivier Spanjaard

Publications and source records attributed to Olivier Spanjaard.

11 recordsLinked to original sources

arXiv2023

Robust Ordinal Regression for Subsets Comparisons with Interactions

This paper is dedicated to a robust ordinal method for learning the preferences of a decision maker between subsets. The decision model, derived from Fishburn and LaValle (1996) and whose parameters we learn, is general enough to be compatible with any strict weak order on subsets, thanks to the consideration of possible interactions between elements. Moreover, we accept not to predict some preferences if the available preference data are not compatible with a reliable prediction. A predicted preference is considered reliable if all the simplest models (Occam's razor) explaining the preference data agree on it. Following the robust ordinal regression methodology, our predictions are based on an uncertainty set encompassing the possible values of the model parameters. We define a robust ordinal dominance relation between subsets and we design a procedure to determine whether this dominance relation holds. Numerical tests are provided on synthetic and real-world data to evaluate the richness and reliability of the preference predictions made.

cs.AI↗

arXiv2022

Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_\infty$ norms

We present various results about Euclidean preferences in the plane under $\ell_1$, $\ell_2$ and $\ell_{\infty}$ norms. When there are four candidates, we show that the maximal size (in terms of the number of pairwise distinct preferences) of Euclidean preference profiles in the plane under norm $\ell_1$ or $\ell_{\infty}$ is 19. Whatever the number of candidates, we prove that at most four distinct candidates can be ranked in last position of a two-dimensional Euclidean preference profile under norm $\ell_1$ or $\ell_\infty$, which generalizes the case of one-dimensional Euclidean preferences (for which it is well known that at most two candidates can be ranked last). We generalize this result to $2^d$ (resp. $2d$) for $\ell_1$ (resp. $\ell_\infty$) for $d$-dimensional Euclidean preferences. We also establish that the maximal size of a two-dimensional Euclidean preference profile on $m$ candidates under norm $\ell_1$ is in $Θ(m^4)$, i.e., the same order of magnitude as under norm $\ell_2$. Finally, we provide a new proof that two-dimensional Euclidean preference profiles under norm $\ell_2$ for four candidates can be characterized by three voter-maximal two-dimensional Euclidean profiles. This proof is a simpler alternative to that proposed by Kamiya et al. in Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47(2):379-400.

math.MG↗

arXiv2022

Cautious Learning of Multiattribute Preferences

This paper is dedicated to a cautious learning methodology for predicting preferences between alternatives characterized by binary attributes (formally, each alternative is seen as a subset of attributes). By "cautious", we mean that the model learned to represent the multi-attribute preferences is general enough to be compatible with any strict weak order on the alternatives, and that we allow ourselves not to predict some preferences if the data collected are not compatible with a reliable prediction. A predicted preference will be considered reliable if all the simplest models (following Occam's razor principle) explaining the training data agree on it. Predictions are based on an ordinal dominance relation between alternatives [Fishburn and LaValle, 1996]. The dominance relation relies on an uncertainty set encompassing the possible values of the parameters of the multi-attribute utility function. Numerical tests are provided to evaluate the richness and the reliability of the predictions made.

cs.AI↗

arXiv2022

Weighted majority tournaments and Kemeny ranking with 2-dimensional Euclidean preferences

The assumption that voters' preferences share some common structure is a standard way to circumvent NP-hardness results in social choice problems. While the Kemeny ranking problem is NP-hard in the general case, it is known to become easy if the preferences are 1-dimensional Euclidean. In this note, we prove that the Kemeny ranking problem remains NP-hard for $k$-dimensional Euclidean preferences with $k\!\ge\!2$ under norms $\ell_1$, $\ell_2$ and $\ell_\infty$, by showing that any weighted tournament (resp. weighted bipartite tournament) with weights of same parity (resp. even weights) is inducible as the weighted majority tournament of a profile of 2-Euclidean preferences under norm $\ell_2$ (resp. $\ell_1,\ell_{\infty}$), computable in polynomial time. More generally, this result regarding weighted tournaments implies, essentially, that hardness results relying on the (weighted) majority tournament that hold in the general case (e.g., NP-hardness of Slater ranking) are still true for 2-dimensional Euclidean preferences.

cs.DM↗

arXiv2022

Beyond Pairwise Comparisons in Social Choice: A Setwise Kemeny Aggregation Problem

In this paper, we advocate the use of setwise contests for aggregating a set of input rankings into an output ranking. We propose a generalization of the Kemeny rule where one minimizes the number of k-wise disagreements instead of pairwise disagreements (one counts 1 disagreement each time the top choice in a subset of alternatives of cardinality at most k differs between an input ranking and the output ranking). After an algorithmic study of this k-wise Kemeny aggregation problem, we introduce a k-wise counterpart of the majority graph. This graph reveals useful to divide the aggregation problem into several sub-problems, which enables to speed up the exact computation of a consensus ranking. By introducing a k-wise counterpart of the Spearman distance, we also provide a 2-approximation algorithm for the k-wise Kemeny aggregation problem. We conclude with numerical tests.

cs.AI↗

arXiv2020

Bayesian preference elicitation for multiobjective combinatorial optimization

We introduce a new incremental preference elicitation procedure able to deal with noisy responses of a Decision Maker (DM). The originality of the contribution is to propose a Bayesian approach for determining a preferred solution in a multiobjective decision problem involving a combinatorial set of alternatives. We assume that the preferences of the DM are represented by an aggregation function whose parameters are unknown and that the uncertainty about them is represented by a density function on the parameter space. Pairwise comparison queries are used to reduce this uncertainty (by Bayesian revision). The query selection strategy is based on the solution of a mixed integer linear program with a combinatorial set of variables and constraints, which requires to use columns and constraints generation methods. Numerical tests are provided to show the practicability of the approach.

cs.AI↗

arXiv2020

Recognizing Single-Peaked Preferences on an Arbitrary Graph: Complexity and Algorithms

This paper is devoted to a study of single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of a set of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the sense that all the preferences are traversals of G. Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation (ILP) of the problem of minimizing the number of edges in G or the maximum degree of a vertex in G. We prove that both problems are NP-hard in the general case. However, we show that if the optimal number of edges is m-1 (where m is the number of candidates) then any optimal solution of the ILP is integer and thus the integrality constraints can be relaxed. This provides an alternative proof of the polynomial-time complexity of recognizing single-peaked preferences on a tree. We prove the same result for the case of a path (an axis), providing here also an alternative proof of polynomiality of the recognition problem. Furthermore, we provide a polynomial-time procedure to recognize single-peaked preferences on a pseudotree (a connected graph that contains at most one cycle). We also give some experimental results, both on real and synthetic datasets.

cs.GT↗

arXiv2018

Optimizing a Generalized Gini Index in Stable Marriage Problems: NP-Hardness, Approximation and a Polynomial Time Special Case

This paper deals with fairness in stable marriage problems. The idea studied here is to achieve fairness thanks to a Generalized Gini Index (GGI), a well-known criterion in inequality measurement, that includes both the egalitarian and utilitarian criteria as special cases. We show that determining a stable marriage optimizing a GGI criterion of agents' disutilities is an NP-hard problem. We then provide a polynomial time 2-approximation algorithm in the general case, as well as an exact algorithm which is polynomial time in the case of a constant number of non-zero weights parametrizing the GGI criterion.

cs.CC↗

arXiv2017

A double oracle approach for minmax regret optimization problems with interval data

In this paper, we provide a generic anytime lower bounding procedure for minmax regret optimization problems. We show that the lower bound obtained is always at least as accurate as the lower bound recently proposed by Chassein and Goerigk (2015). This lower bound can be viewed as the optimal value of a linear programming relaxation of a mixed integer programming formulation of minmax regret optimization, but the contribution of the paper is to compute this lower bound via a double oracle algorithm (McMahan et al., 2003) that we specify. The double oracle algorithm is designed by relying on a game theoretic view of robust optimization, similar to the one developed by Mastin et al. (2015), and it can be efficiently implemented for any minmax regret optimization problem whose standard version is "easy". We describe how to efficiently embed this lower bound in a branch and bound procedure. Finally, we apply our approach to the robust shortest path problem. Our numerical results show a significant gain in the computation times compared to previous approaches in the literature.

cs.DS↗

arXiv2012

An Axiomatic Approach to Robustness in Search Problems with Multiple Scenarios

This paper is devoted to the search of robust solutions in state space graphs when costs depend on scenarios. We first present axiomatic requirements for preference compatibility with the intuitive idea of robustness.This leads us to propose the Lorenz dominance rule as a basis for robustness analysis. Then, after presenting complexity results about the determination of robust solutions, we propose a new sophistication of A* specially designed to determine the set of robust paths in a state space graph. The behavior of the algorithm is illustrated on a small example. Finally, an axiomatic justification of the refinement of robustness by an OWA criterion is provided.

cs.AI↗

arXiv2009

Exact algorithms for OWA-optimization in multiobjective spanning tree problems

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. In the case where the weights of the OWA are strictly decreasing, we then propose a mixed integer programming formulation, and provide dedicated optimality conditions yielding an important reduction of the size of the program. Next, we present two bounds that can be used to prune subspaces of solutions either in a shaving phase or in a branch and bound procedure. The validity of these bounds does not depend on specific properties of the weights (apart from non-negativity). All these exact resolution algorithms are compared on the basis of numerical experiments, according to their respective validity scopes.

cs.DS↗