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Investigating the Conjectures of Fine-Grained Complexity

Project description

Fine-grained complexity’s secondary conjectures: upgrade or dismissal

Computational complexity theory is a branch of mathematics that distinguishes between relatively efficient algorithmic solutions and those that are intractable. Fine-grained complexity theory seeks to enhance the ‘resolution’ of such distinctions, providing quantitative determination and potentially insight into the exact complexity of computationally ‘hard’ problems. There are three primary conjectures under which many hardness results hold. Ten other secondary conjectures exist; they may be stronger variants, letting us prove more hardness results, but they are also less extensively studied (and less likely to be true). The ERC-funded CONJEXITY project will resolve the secondary conjectures by either falsifying them or establishing their equivalence to a primary conjecture.


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.


Net EU contribution
€ 1 499 250,00
Herzl street 234
7610001 Rehovot

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Activity type
Higher or Secondary Education Establishments
Other funding
€ 0,00