arXiv subjects

Nicolas Bousquet

Publications and source records attributed to Nicolas Bousquet.

At least 19 recordsLinked to original sources

arXiv2026

Fair Division of Graphs: Beyond Traceability

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

cs.GT↗

arXiv2026

A Linear Kernel for Independent Set Reconfiguration in Planar Graphs

Fix a positive integer $r$, and a graph $G$ that is $K_{3,r}$-minor-free. Let $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a ``token'' on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated ``token jumping'', removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ into $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono (2014) to be fixed-parameter tractable. That is, the problem can be solved in time $f(k)\cdot Poly(n)$, for some function $f$ and polynomial $Poly(n)$, where $n$ denotes the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a $K_{3,r}$-minor-free graph (for some fixed positive integer $r$) into an equivalent problem on a ($K_{3,r}$-minor-free) graph with order $O(k)$. This answers positively a question of Bousquet, Mouawad, Nishimura, and Siebertz (2024) and improves the recent quadratic kernel of Cranston, Mühlenthaler, and Peyrille (2026). For planar graphs, we further strengthen this upper bound to get a kernel of size at most $39k$.

math.CO↗

arXiv2026

Exact Functional ANOVA Decomposition for Categorical Inputs Models

Functional ANOVA offers a principled framework for interpretability by decomposing a model's prediction into main effects and higher-order interactions. For independent features, this decomposition is well-defined, strongly linked with SHAP values, and serves as a cornerstone of additive explainability. However, the lack of an explicit closed-form expression for general dependent distributions has forced practitioners to rely on costly sampling-based approximations. We completely resolve this limitation for categorical inputs. By bridging functional analysis with the extension of discrete Fourier analysis, we derive a closed-form decomposition without any assumption. Our formulation is computationally very efficient. It seamlessly recovers the classical independent case and extends to arbitrary dependence structures, including distributions with non-rectangular support. Furthermore, leveraging the intrinsic link between SHAP and ANOVA under independence, our framework yields a natural generalization of SHAP values for the general categorical setting.

stat.ML↗

arXiv2026

The local complexity of certifying parity

In this paper, we consider the problem of locally certifying that the size of a network is even, or more generally, congruent to some fixed number. The parity property is one of the simplest global properties, and it plays an intriguing role in local certification. On the one hand, it is one of the simplest properties in cycles because it is equivalent to 2-colorability, and hence can be certified with a single bit. On the other hand, in general graphs, no non-trivial lower bound on the size of the certificates is known, and the known upper bound basically consists in certifying the \emph{exact} value of $n$. In addition, the nature of the problem makes all the known lower bound approaches fail. We uncover a surprising landscape for parity across different models and graph structures: * In general graphs equipped with identifiers, when allowing verification radius 2, parity can be certified with a constant number of bits. * But in the model of anonymous graphs and allowing verification radius only 1, parity requires $Ω(\log \log^*n)$ bits. * Finally, in bounded expansion graph classes (such as bounded-degree graphs and planar graphs), the lower bound does not apply: in the same restricted model we can design a constant-size certification. We introduce several new tools that we expect to be useful in other contexts, in particular ways to \emph{encode a parent at each node with a constant number of bits} (via implicit use of the IDs and conflict-free colorings) and a new lower bound technique, with complex topologies and higher-order Ramsey-type arguments.

cs.DC↗

arXiv2026

Generalized Functional ANOVA in Closed-Form: A Unified View of Additive Explanations

The functional ANOVA, or Hoeffding decomposition, provides a principled framework for interpretability by decomposing a model prediction into main effects and higher-order interactions. For independent inputs, this classical decomposition is explicit. It is closely connected to SHAP values, generalized additive models, and orthogonal polynomial expansions, and therefore constitutes a fundamental tool for additive explainability. In the more general and realistic dependent setting, however, obtaining a tractable representation and estimating the decomposition from data remain challenging. In this work, we address this problem for continuous inputs. By combining Hilbert space methods with the generalized functional ANOVA, we build an explicit decomposition Riesz Basis allowing to easily compute the decomposition. Our formulation recovers the classical independent case and its associated orthogonal decomposition. Building on this representation, we propose a simple but mighty algorithm to estimate the decomposition from a data sample in a model-agnostic setting and we compare it empirically with several state-of-the-art explanation methods, demonstrating the power of the approach.

stat.ML↗

arXiv2026

Fourier Analysis on the Boolean Hypercube via Hoeffding Functional Decomposition

Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an \emph{explicit} decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any \emph{arbitrary} probability measure on the Boolean hypercube. We formulate the computation of the entire functional decomposition as a least squares problem and also provide a method to address the classical \emph{curse of dimensionality} challenge. We provide a comprehensive generalization of Fourier analysis on the Boolean hypercube, enabling the handling of non-uniform configuration spaces inherent to real-world machine learning tasks, \textit{e.g.} when dealing with \emph{one-hot encoded} features. Finally, we demonstrate its practical impact in the field of explainable AI, by conducting comparative studies with feature attribution methods such as SHAP or TreeHFD.

stat.ML↗

arXiv2025

WTNN: Weibull-Tailored Neural Networks for survival analysis

The Weibull distribution is a commonly adopted choice for modeling the survival of systems subject to maintenance over time. When only proxy indicators and censored observations are available, it becomes necessary to express the distribution's parameters as functions of time-dependent covariates. Deep neural networks provide the flexibility needed to learn complex relationships between these covariates and operational lifetime, thereby extending the capabilities of traditional regression-based models. Motivated by the analysis of a fleet of military vehicles operating in highly variable and demanding environments, as well as by the limitations observed in existing methodologies, this paper introduces WTNN, a new neural network-based modeling framework specifically designed for Weibull survival studies. The proposed architecture is specifically designed to incorporate qualitative prior knowledge regarding the most influential covariates, in a manner consistent with the shape and structure of the Weibull distribution. Through numerical experiments, we show that this approach can be reliably trained on proxy and right-censored data, and is capable of producing robust and interpretable survival predictions that can improve existing approaches.

stat.ML↗

arXiv2025

Renaming in distributed certification

Local certification is the area of distributed network computing asking the following question: How to certify to the nodes of a network that a global property holds, if they are limited to a local verification? In this area, it is often essential to have identifiers, that is, unique integers assigned to the nodes. In this short paper, we show how to reduce the range of the identifiers, in three different settings. More precisely, we show how to rename identifiers in the classical local certification setting, when we can (resp.\ cannot) choose the new identifiers, and we show how a global certificate can help to encode very compactly a new identifier assignment that is not injective in general, but still useful in applications. We conclude with a number of applications of these results: For every $\ell$, there are local certification schemes for the properties of having clique number at most $\ell$, having diameter at most $\ell$, and having independence number at most~2, with certificates of size $O(n)$. We also show that there is a global certification scheme for bipartiteness with a certificate of size $O(n)$. All these results are optimal.

cs.DC↗

arXiv2025

On Algorithmic Meta-Theorems for Solution Discovery: Tractability and Barriers

Solution discovery asks whether a given (infeasible) starting configuration to a problem can be transformed into a feasible solution using a limited number of transformation steps. This paper investigates meta-theorems for solution discovery for graph problems definable in monadic second-order logic (MSO$_1$ and MSO$_2$) and first-order logic (FO) where the transformation step is to slide a token to an adjacent vertex, focusing on parameterized complexity and structural graph parameters that do not involve the transformation budget $b$. We present both positive and negative results. On the algorithmic side, we prove that MSO$_2$-Discovery is in XP when parameterized by treewidth and that MSO$_1$-Discovery is fixed-parameter tractable when parameterized by neighborhood diversity. On the hardness side, we establish that FO-Discovery is W[1]-hard when parameterized by modulator to stars, modulator to paths, as well as twin cover, numbers. Additionally, we prove that MSO$_1$-Discovery is W[1]-hard when parameterized by bandwidth. These results complement the straightforward observation that solution discovery for the studied problems is fixed-parameter tractable when the budget $b$ is included in the parameter (in particular, parameterized by cliquewidth$+b$, where the cliquewidth of a graph is at most any of the studied parameters), and provide a near-complete (fixed-parameter tractability) meta-theorems investigation for solution discovery problems for MSO- and FO-definable graph problems and structural parameters larger than cliquewidth.

cs.DS↗

arXiv2025

Shallow brambles

A graph class $\mathcal{C}$ has polynomial expansion if there is a polynomial function $f$ such that for every graph $G\in \mathcal{C}$, each of the depth-$r$ minors of $G$ has average degree at most $f(r)$. In this note, we study bounded-radius variants of some classical graph parameters such as bramble number, linkedness and well-linkedness, and we show that they are pairwise polynomially related. Furthermore, in a monotone graph class with polynomial expansion they are all uniformly bounded by a polynomial in $r$.

math.CO↗

arXiv2025

Induced Minor Models. I. Structural Properties and Algorithmic Consequences

A graph $H$ is said to be an induced minor of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. Equivalently, $H$ is an induced minor of $G$ if there exists an induced minor model of $H$ in $G$, that is, a collection of pairwise disjoint subsets of vertices of $G$ labeled by the vertices of $H$, each inducing a connected subgraph in $G$, such that two vertices of $H$ are adjacent if and only if there is an edge in $G$ between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. It is known that for some graphs $H$, testing whether a given graph $G$ contains $H$ as an induced minor is an NP-complete problem. Nevertheless, as algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if $H$ is the $4$-wheel, the $5$-vertex complete graph minus an edge, or a complete bipartite graph $K_{2,q}$, then there is a polynomial-time algorithm to find in a given graph $G$ an induced minor model of $H$ in $G$, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain $K_{2,3}$ as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains $K_{2,3}$ as an induced minor, using the so-called shortest path detector. It turns out that all these induced subgraphs are Truemper configurations.

math.CO↗

arXiv2025

On the complexity of constrained reconfiguration and motion planning

Coordinating the motion of multiple agents in constrained environments is a fundamental challenge in robotics, motion planning, and scheduling. A motivating example involves $n$ robotic arms, each represented as a line segment. The objective is to rotate each arm to its vertical orientation, one at a time (clockwise or counterclockwise), without collisions nor rotating any arm more than once. This scenario is an example of the more general $k$-Compatible Ordering problem, where $n$ agents, each capable of $k$ state-changing actions, must transition to specific target states under constraints encoded as a set $\mathcal{G}$ of $k$ pairs of directed graphs. We show that $k$-Compatible Ordering is $\mathsf{NP}$-complete, even when $\mathcal{G}$ is planar, degenerate, or acyclic. On the positive side, we provide polynomial-time algorithms for cases such as when $k = 1$ or $\mathcal{G}$ has bounded treewidth. We also introduce generalized variants supporting multiple state-changing actions per agent, broadening the applicability of our framework. These results extend to a wide range of scheduling, reconfiguration, and motion planning applications in constrained environments.

cs.CC↗

arXiv2025

Benchmarking XAI Explanations with Human-Aligned Evaluations

We introduce PASTA (Perceptual Assessment System for explanaTion of Artificial Intelligence), a novel human-centric framework for evaluating eXplainable AI (XAI) techniques in computer vision. Our first contribution is the creation of the PASTA-dataset, the first large-scale benchmark that spans a diverse set of models and both saliency-based and concept-based explanation methods. This dataset enables robust, comparative analysis of XAI techniques based on human judgment. Our second contribution is an automated, data-driven benchmark that predicts human preferences using the PASTA-dataset. This scoring called PASTA-score method offers scalable, reliable, and consistent evaluation aligned with human perception. Additionally, our benchmark allows for comparisons between explanations across different modalities, an aspect previously unaddressed. We then propose to apply our scoring method to probe the interpretability of existing models and to build more human interpretable XAI methods.

cs.CV↗

arXiv2025

Complexity landscape for local certification

An impressive recent line of work has charted the complexity landscape of distributed graph algorithms. For many settings, it has been determined which time complexities exist, and which do not (in the sense that no local problem could have an optimal algorithm with that complexity). In this paper, we initiate the study of the landscape for space complexity of distributed graph algorithms. More precisely, we focus on the local certification setting, where a prover assigns certificates to nodes to certify a property, and where the space complexity is measured by the size of the certificates. Already for anonymous paths and cycles, we unveil a surprising landscape: - There is a gap between complexity $O(1)$ and $Θ(\log \log n)$ in paths. This is the first gap established in local certification. - There exists a property that has complexity $Θ(\log \log n)$ in paths, a regime that was not known to exist for a natural property. - There is a gap between complexity $O(1)$ and $Θ(\log n)$ in cycles, hence a gap that is exponentially larger than for paths. We then generalize our result for paths to the class of trees. Namely, we show that there is a gap between complexity $O(1)$ and $Θ(\log \log d)$ in trees, where $d$ is the diameter. We finally describe some settings where there are no gaps at all. To prove our results we develop a new toolkit, based on various results of automata theory and arithmetic, which is of independent interest.

cs.DC↗

arXiv2025

The tape reconfiguration problem and its consequences for dominating set reconfiguration

A dominating set of a graph $G=(V,E)$ is a set of vertices $D \subseteq V$ whose closed neighborhood is $V$, i.e., $N[D]=V$. We view a dominating set as a collection of tokens placed on the vertices of $D$. In the token sliding variant of the Dominating Set Reconfiguration problem (TS-DSR), we seek to transform a source dominating set into a target dominating set in $G$ by sliding tokens along edges, and while maintaining a dominating set all along the transformation. TS-DSR is known to be PSPACE-complete even restricted to graphs of pathwidth $w$, for some non-explicit constant $w$ and to be XL-complete parameterized by the size $k$ of the solution. The first contribution of this article consists in using a novel approach to provide the first explicit constant for which the TS-DSR problem is PSPACE-complete, a question that was left open in the literature. From a parameterized complexity perspective, the token jumping variant of DSR, i.e., where tokens can jump to arbitrary vertices, is known to be FPT when parameterized by the size of the dominating sets on nowhere dense classes of graphs. But, in contrast, no non-trivial result was known about TS-DSR. We prove that DSR is actually much harder in the sliding model since it is XL-complete when restricted to bounded pathwidth graphs and even when parameterized by $k$ plus the feedback vertex set number of the graph. This gives, for the first time, a difference of behavior between the complexity under token sliding and token jumping for some problem on graphs of bounded treewidth. All our results are obtained using a brand new method, based on the hardness of the so-called Tape Reconfiguration problem, a problem we believe to be of independent interest.

cs.CC↗

arXiv2025

Covariance constraints for stochastic inverse problems of computer models

Stochastic inverse problems considered in this article consist of estimating the probability distributions of intrinsically random inputs of computer models. These estimations are based on observable outputs affected by model noise, and such problems are increasingly examined in parametric Bayesian contexts where the parameters of the targeted input distributions are affected by epistemic uncertainties. With the aim of improving the meaningfulness of solutions found by statistical algorithms -- in the sense that forward simulations based on such solutions must lead to relevant observables -- we derive new prior constraints using the principles of global sensitivity analysis and information theory. Primarily formalized as constraints on covariances in Gaussian linear or linearizable situations, they reflect the idea that the solution should explain most of the observable uncertainty, while the model noise remains a secondary factor of this uncertainty. Simulated experiments highlight that, when injected into stochastic inversion algorithms, these constraints can indeed limit the influence of model noise on the result. They provide hope for future extensions in more general frameworks, for example through the use of linear Gaussian mixtures.

math.ST↗

arXiv2025

A subquadratic certification scheme for P5-free graphs

In local certification, vertices of a $n$-vertex graph perform a local verification to check if a given property is satisfied by the graph. This verification is performed thanks to certificates, which are pieces of information that are given to the vertices. In this work, we focus on the local certification of $P_5$-freeness, and we prove a $O(n^{3/2})$ upper bound on the size of the certificates, which is (to our knowledge) the first subquadratic upper bound for this property.

cs.DC↗

arXiv2024

A note on locating-dominating sets in twin-free graphs

In this short note, we prove that every twin-free graph on $n$ vertices contains a locating-dominating set of size at most $\lceil\frac{5}{8}n\rceil$. This improves the earlier bound of $\lfloor\frac{2}{3}n\rfloor$ due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.

math.CO↗