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Alain Hertz

Publications and source records attributed to Alain Hertz.

At least 19 recordsLinked to original sources

Machine Learning-Enhanced Tabu Search for Tactical Wireless Network Design

Designing high-performance tactical wireless networks under realistic operational constraints gives rise to challenging combinatorial optimization problems, where the evaluation of candidate solutions relies on detailed physical and traffic-aware models. Although classical metaheuristics such as Tabu Search offer effective mechanisms for exploring large search spaces, their computational cost remains high because numerous candidate moves must be evaluated at every iteration. In this paper, we propose a data-driven framework that improves the efficiency of Tabu Search by learning to guide its move selection process. Rather than altering the neighborhood structure, our approach exploits the information contained in the search trajectories generated during the optimization process. At each iteration, we record both improving and non-improving edge-based transformations together with a set of descriptive features capturing the structural, geometric, and performance characteristics of the network. This information is used to train a Graph Neural Network (GNN) that predicts the impact of candidate moves on the objective function. The trained model is then integrated into the Tabu Search algorithm to rank candidate transformations according to their predicted quality, thereby reducing the number of costly objective evaluations while maintaining an effective exploration of the search space. Experimental results on synthetic benchmark instances demonstrate that the proposed learning-assisted Tabu Search notably reduces computation time while consistently producing higher-quality solutions than the standard algorithm. These findings highlight the potential of combining machine learning with metaheuristics by leveraging the implicit knowledge embedded in search trajectories, paving the way for more efficient solution methods for large-scale network design problems.

cs.AI

Sensitivity Analysis of Tactical Wireless Network Design Under Realistic Operational Constraints

The design of tactical wireless networks reflects a complex interplay among structural constraints, technological choices, and underlying modeling assumptions. Although optimization-based approaches have been widely explored, the impact of configuration parameters on network topology quality and overall performance is still not fully understood. This paper presents a comprehensive sensitivity analysis of tactical wireless network design under realistic operational constraints. It systematically investigates three categories of parameters: (i) structural topology rules, including master hub selection; (ii) technological factors such as antenna beamwidth; and (iii) modeling parameters embedded in the objective formulation. Optimized topologies are produced using a Tabu Search metaheuristic, and statistical analyses based on the Friedman and Wilcoxon tests are performed to assess the significance of observed variations across different network sizes. The findings reveal scale-dependent technological transitions and threshold effects in structural constraints. The analysis differentiates parameters that fundamentally reshape network topology from those that primarily influence performance magnitude. Together, these insights provide practical guidance for parameter tuning and topology configuration in mission-critical tactical wireless deployments.

cs.NI

Tabu Search for Tactical Wireless Network Design in Challenging Environments

Tactical wireless networks play a vital role in ensuring reliable connectivity in scenarios where conventional telecommunications infrastructure is unavailable or damaged, such as areas impacted by natural disasters. These networks are designed to operate efficiently in difficult and unpredictable environments by adapting to the unique characteristics of the terrain. This research addresses a real-world challenge from the communications industry: designing tactical wireless networks that meet the specific constraints defined by our industrial partner, with the goal of optimizing signal strength and coverage while minimizing interference. To this end, we propose two tabu search algorithms that incorporate several heuristic subroutines, enabling the efficient generation of high-quality network designs. Results from synthetic tests demonstrate that our approach produces networks rapidly and effectively, offering significant improvements over existing methods.

cs.NI

ChemicHull: an online tool for determining extremal chemical graphs of maximum degree at most 3 for any degree-based topological indices

Topological indices are graph-theoretic descriptors that play a crucial role in mathematical chemistry, capturing the structural characteristics of molecules and enabling the prediction of their physicochemical properties. A widely studied category of topological indices, known as degree-based topological indices, are calculated as the sum of the weights of a graph's edges, where each edge weight is determined by a formula that depends solely on the degrees of its endpoints. This work focuses exclusively on chemical graphs in which no vertex has a degree greater than 3, a model for conjugated systems. Within a polyhedral framework, each chemical graph is mapped to a point in a three-dimensional space, enabling extremal values of any degree-based topological index to be determined through linear optimization over the corresponding polyhedron. Analysis within this framework reveals that extremality is limited to a small subset of chemical graph families, implying that certain chemical graphs can never attain extremality for any degree-based topological index. The main objective of this paper is to present ChemicHull, an online tool we have developed to determine and display extremal chemical graphs for arbitrary degree-based topological indices. To illustrate the power of this tool, we easily recover established results, emphasizing its effectiveness for chemically significant graph classes such as chemical trees and unicyclic chemical graphs. This tool also enabled the identification of a counterexample to a previously published extremal result concerning the Randi\'c index.

cs.DM

Neural Logic Networks for Interpretable Classification

Traditional neural networks have an impressive classification performance, but what they learn cannot be inspected, verified or extracted. Neural Logic Networks on the other hand have an interpretable structure that enables them to learn a logical mechanism relating the inputs and outputs with AND and OR operations. We generalize these networks with NOT operations and biases that take into account unobserved data and develop a rigorous logical and probabilistic modeling in terms of concept combinations to motivate their use. We also propose a novel factorized IF-THEN rule structure for the model as well as a modified learning algorithm. Our method improves the state-of-the-art in Boolean networks discovery and is able to learn relevant, interpretable rules in tabular classification, notably on examples from the medical and industrial fields where interpretability has tangible value.

cs.LG

Complete polyhedral description of chemical graphs of maximum degree at most 3

Chemical graphs are simple undirected connected graphs, where vertices represent atoms in a molecule and edges represent chemical bonds. A degree-based topological index is a molecular descriptor used to study specific physicochemical properties of molecules. Such an index is computed from the sum of the weights of the edges of a chemical graph, each edge having a weight defined by a formula that depends only on the degrees of its endpoints. Given any degree-based topological index and given two integers $n$ and $m$, we are interested in determining chemical graphs of order $n$ and size $m$ that maximize or minimize the index. Focusing on chemical graphs with maximum degree at most 3, we show that this reduces to determining the extreme points of a polytope that contains at most 10 facets. We also show that the number of extreme points is at most 16, which means that for any given $n$ and $m$, there are very few different classes of extremal graphs, independently of the chosen degree-based topological index.

math.CO

Extremal chemical graphs of maximum degree at most 3 for 33 degree-based topological indices

We consider chemical graphs that are defined as connected graphs of maximum degree at most 3. We characterize the extremal graphs, meaning those that maximize or minimize 33 degree-based topological indices. This study shows that five graph families are sufficient to characterize the extremal graphs of 29 of these 33 indices. In other words, the extremal properties of this set of degree-based topological indices vary very little.

math.CO

The average size of maximal matchings in graphs

We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, $\avM(G)$ approaches $\frac{1}{2}$. We propose a general technique to determine the asymptotic behavior of $\avM(G)$ for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of $\avM(G)$ which were typically obtained using generating functions, and we then determine the asymptotic value of $\avM(G)$ for other families of graphs, highlighting the spectrum of possible values of this graph invariant between $\frac{1}{2}$ and $1$.

math.CO

Extremal Chemical Graphs for the Arithmetic-Geometric Index

The arithmetic-geometric index is a newly proposed degree-based graph invariant in mathematical chemistry. We give a sharp upper bound on the value of this invariant for connected chemical graphs of given order and size and characterize the connected chemical graphs that reach the bound. We also prove that the removal of the constraint that extremal chemical graphs must be connected does not allow to increase the upper bound.

math.CO

Addressing the cold start problem in privacy preserving content-based recommender systems using hypercube graphs

The initial interaction of a user with a recommender system is problematic because, in such a so-called cold start situation, the recommender system has very little information about the user, if any. Moreover, in collaborative filtering, users need to share their preferences with the service provider by rating items while in content-based filtering there is no need for such information sharing. We have recently shown that a content-based model that uses hypercube graphs can determine user preferences with a very limited number of ratings while better preserving user privacy. In this paper, we confirm these findings on the basis of experiments with more than 1,000 users in the restaurant and movie domains. We show that the proposed method outperforms standard machine learning algorithms when the number of available ratings is at most 10, which often happens, and is competitive with larger training sets. In addition, training is simple and does not require large computational efforts.

cs.IR

A machine learning framework for neighbor generation in metaheuristic search

This paper presents a methodology for integrating machine learning techniques into metaheuristics for solving combinatorial optimization problems. Namely, we propose a general machine learning framework for neighbor generation in metaheuristic search. We first define an efficient neighborhood structure constructed by applying a transformation to a selected subset of variables from the current solution. Then, the key of the proposed methodology is to generate promising neighbors by selecting a proper subset of variables that contains a descent of the objective in the solution space. To learn a good variable selection strategy, we formulate the problem as a classification task that exploits structural information from the characteristics of the problem and from high-quality solutions. We validate our methodology on two metaheuristic applications: a Tabu Search scheme for solving a Wireless Network Optimization problem and a Large Neighborhood Search heuristic for solving Mixed-Integer Programs. The experimental results show that our approach is able to achieve a satisfactory trade-off between the exploration of a larger solution space and the exploitation of high-quality solution regions on both applications.

math.OC

Using Graph Theory to Derive Inequalities for the Bell Numbers

The Bell numbers count the number of different ways to partition a set of $n$ elements while the graphical Bell numbers count the number of non-equivalent partitions of the vertex set of a graph into stable sets. This relation between graph theory and integer sequences has motivated us to study properties on the average number of colors in the non-equivalent colorings of a graph to discover new non trivial inequalities for the Bell numbers. Example are given to illustrate our approach.

cs.DM

Exploring dual information in distance metric learning for clustering

Distance metric learning algorithms aim to appropriately measure similarities and distances between data points. In the context of clustering, metric learning is typically applied with the assist of side-information provided by experts, most commonly expressed in the form of cannot-link and must-link constraints. In this setting, distance metric learning algorithms move closer pairs of data points involved in must-link constraints, while pairs of points involved in cannot-link constraints are moved away from each other. For these algorithms to be effective, it is important to use a distance metric that matches the expert knowledge, beliefs, and expectations, and the transformations made to stick to the side-information should preserve geometrical properties of the dataset. Also, it is interesting to filter the constraints provided by the experts to keep only the most useful and reject those that can harm the clustering process. To address these issues, we propose to exploit the dual information associated with the pairwise constraints of the semi-supervised clustering problem. Experiments clearly show that distance metric learning algorithms benefit from integrating this dual information.

cs.LG

Upper bounds on the average number of colors in the non-equivalent colorings of a graph

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be the average number of colors in the non-equivalent colorings of a graph $G$. We give a general upper bound on $\mathcal{A}(G)$ that is valid for all graphs $G$ and a more precise one for graphs $G$ of order $n$ and maximum degree $Δ(G)\in \{1,2,n-2\}$.

math.CO

Lower Bounds and properties for the average number of colors in the non-equivalent colorings of a graph

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several lower bounds on $\mathcal{A}(G)$ and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2.

math.CO

On graceful difference labelings of disjoint unions of circuits

A graceful difference labeling (gdl for short) of a directed graph G with vertex set V is a bijection f between V and {1,...,|V|} such that, when each arc uv is assigned the difference label f(v)-f(u), the resulting arc labels are distinct. We conjecture that all disjoint unions of circuits have a gdl, except in two particular cases. We prove partial results which support this conjecture.

math.CO

Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other vertex of $G$. This index is helpful for the prediction of biological activities of diverse nature, a molecule being modeled as a graph where atoms are represented by vertices and chemical bonds by edges. We characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order $n$. Also, given two integers $n$ and $p$ with $p\leq n-1$, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order $n$ with $p$ pending vertices.

cs.DM

Maximum Eccentric Connectivity Index for Graphs with Given Diameter

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. The eccentric connectivity index of a connected graph is the sum over all vertices of the product between eccentricity and degree. Given two integers $n$ and $D$ with $D\leq n-1$, we characterize those graphs which have the largest eccentric connectivity index among all connected graphs of order $n$ and diameter $D$. As a corollary, we also characterize those graphs which have the largest eccentric connectivity index among all connected graphs of a given order $n$.

cs.DM