Description du projet
Des recherches sur les problèmes d’optimisation et de satisfaction de contraintes
Les problèmes d’optimisation et de satisfaction de contraintes (CSP) sont de gros problèmes informatiques généralement classés comme tractables (solubles en temps polynomial) ou intraitables (NP-difficile). Cependant, certains CSP résistent à la classification, notamment les problèmes de la théorie des graphes, de la combinatoire et de la théorie mathématique des jeux. Le projet AUTAR, financé par le CER, vise à combler cette lacune en analysant les algorithmes des candidats qui semblent résoudre toutes les instances et ne sont pas limités à la description d’instances difficiles ou d’hypothèses de dureté. L’approche innovante d’AUTAR s’appuiera sur une découverte récente selon laquelle deux méthodes issues de domaines différents — les jeux de galets d’indiscernabilité de la logique mathématique et les hiérarchies de relaxations convexes de la programmation mathématique — présentent le même potentiel.
Objectif
For a large family of computational problems collectively known as constrained optimization and satisfaction problems (CSPs), four decades of research in algorithms and computational complexity have led to a theory that tries to classify them as algorithmically tractable vs. intractable, i.e. polynomial-time solvable vs. NP-hard. However, there remains an important gap in our knowledge in that many CSPs of interest resist classification by this theory. Some such problems of practical relevance include fundamental partition problems in graph theory, isomorphism problems in combinatorics, and strategy-design problems in mathematical game theory. To tackle this gap in our knowledge, the research of the last decade has been driven either by finding hard instances for algorithms that solve tighter and tighter relaxations of the original problem, or by formulating new hardness-hypotheses that are stronger but admittedly less robust than NP-hardness.
The ultimate goal of this project is closing the gap between the partial progress that these approaches represent and the original classification project into tractable vs. intractable problems. Our thesis is that the field has reached a point where, in many cases of interest, the analysis of the current candidate algorithms that appear to solve all instances could suffice to classify the problem one way or the other, without the need for alternative hardness-hypotheses. The novelty in our approach is a program to develop our recent discovery that, in some cases of interest, two methods from different areas match in strength: indistinguishability pebble games from mathematical logic, and hierarchies of convex relaxations from mathematical programming. Thus, we aim at making significant advances in the status of important algorithmic problems by looking for a general theory that unifies and goes beyond the current understanding of its components.
Champ scientifique
Programme(s)
Régime de financement
ERC-COG - Consolidator GrantInstitution d’accueil
08034 Barcelona
Espagne