Projektbeschreibung
Sekundäre Vermutungen über feinkörnige Komplexität: Aufwertung oder Ablehnung
Die Komplexitätstheorie ist ein Teilgebiet der Mathematik, das zwischen relativ effizienten algorithmischen Lösungen und solchen unterscheidet, die unlösbar sind. Die feinkörnige Komplexitätstheorie versucht, die „Auflösung“ solcher Unterscheidungen zu verbessern, indem sie eine quantitative Bestimmung und möglicherweise einen Einblick in die genaue Komplexität rechnerisch „schwieriger“ Probleme ermöglicht. Es bestehen drei Hauptvermutungen, unter denen viele Härteergebnisse gelten. Es gibt noch zehn weitere sekundäre Vermutungen, die vielleicht stärker sind und uns mehr Härteergebnisse beweisen lassen, aber sie sind auch weniger gut untersucht (und weniger wahrscheinlich). Das ERC-finanzierte Projekt CONJEXITY wird die sekundären Vermutungen auflösen, indem es sie entweder widerlegt oder ihre Gleichwertigkeit mit einer primären Vermutung nachweist.
Ziel
Fine-grained complexity theory identifies a small set of conjectures under which a large number of hardness results hold. The fast-growing list of such conditional hardness results already spans many diverse areas of computer science. Improved algorithms for some of the most central problems in these domains are deemed impossible unless one of the core conjectures turns out to be false, terminating decades-long quests for faster algorithms. Much research is going into closing the remaining gaps, addressing more domains, and achieving beyond-worst-case results.
But should these conjectures, that are the foundation of this entire theory, really be treated as laws of nature? In addition to three primary conjectures, the community has put forth about ten others. These ``secondary conjectures'' are often stronger variants of the primary conjectures, stating that the core problems remain hard despite introducing new assumptions on the input; they let us prove more hardness results but are also less extensively studied (and less likely to be true) compared to the original conjectures.
Stepping away from current research that is hustling towards achieving tight bounds for all important problems under such conjectures, this project aims to investigate the conjectures themselves. Our main objective is to resolve the secondary conjectures; either by falsifying them or by establishing their equivalence to a primary conjecture. Either of these two outcomes would be satisfying: Refuting a conjecture must involve disruptive algorithmic techniques, impacting numerous other problems. Unifying a secondary conjecture with an original (primary) conjecture reinforces the validity of the conjecture and all its implications, solidifying the very foundations of Fine-Grained Complexity. We believe that there is a pressing need for such an investigation of this rapidly growing theory.
Wissenschaftliches Gebiet
Programm/Programme
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Thema/Themen
Finanzierungsplan
ERC - Support for frontier research (ERC)Gastgebende Einrichtung
7610001 Rehovot
Israel