Periodic Reporting for period 1 - CONJEXITY (Investigating the Conjectures of Fine-Grained Complexity)
Berichtszeitraum: 2022-12-01 bis 2025-11-30
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.
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.
Highlights of the project’s achievements so far include:
- Major speedup in combinatorial algorithms for matrix multiplication, nearly refuting the longstanding BMM conjecture. After decades of research into the problem of designing a combinatorial algorithm that is faster than exhaustive search for multiplying Boolean matrices, the fastest algorithms only achieved a speedup of four logarithmic factors. In our STOC 2024 paper we discovered an algorithm achieving a super-poly-log speedup, and coming tantalizingly close to refuting the conjecture that “truly subcubic” algorithms are not possible. This conjecture is central to our CONJEXITY project, and to the field of fine-grained complexity in general, because refuting it is believed to lead to a refutation some of the conjectures in the field (for not-necessarily combinatorial algorithms).
- Resolving a variant of a secondary conjecture using new techniques based on Additive Combinatorics. Specifically, in our STOC 2023 paper (invited to special issue) we show that the listing variant of the 4-cycle conjecture (a secondary conjecture) holds under the 3-Sum Conjecture (a primary conjecture). This work falls within a recent major interest in techniques from Additive Combinatorics for fine-grained complexity and algorithms.
- Ruling out the use of the popular technique of “expander decompositions” for refuting some of the conjectures in fine-grained complexity. In an ITCS 2023 paper we have introduced the notion of a worst-case to expander-case reduction, which can be used as a tool for discouraging researchers from attempting to refute conjectures via the natural and popular method based on decomposing graphs into expanders.
- Insight into the hard cases of the conjectures. A key step towards resolving a conjecture is to undertand it the hard instances of the conjectured-to-be-hard problem. For problems such as All-Pairs Shortest-Paths (and its secondary variants such as k-Clique) our work has revealed new insights about the hard cases. In a recent submission to STOC 2025, we show that if the number of distinct weights on the edges touching any node is sublinear, then the instance can be solved in truly subcubic time. Therefore, hard instances must use many different weights in every neighborhood. This result also utilizes techniques from Additive Combinatorics. In another project, we are identifying the properties of the graph that make triangle detection hard. For example, we show that a graph without 5-cycles is easy for triangle detection.
- Almost-optimal algorithm for All-Pairs Max-Flow, essentially completing a 60-year-old quest for a linear time algorithm for computing a so-called Gomory-Hu tree of a graph. Recent breakthroughs brought the complexity down from cubic to quadratic, and in our FOCS 2023 paper we bring it down to linear (in the number of edges). While this result does not directly make progress on any of the conjectures in fine-grained complexity, it inspires further research into refuting the All-Pairs Shortest-Paths conjecture and its secondary variants, due to the similarity between shortest paths and maximum flows.
- Major speedup in combinatorial algorithms for matrix multiplication, nearly refuting the longstanding BMM conjecture. After decades of research into the problem of designing a combinatorial algorithm that is faster than exhaustive search for multiplying Boolean matrices, the fastest algorithms only achieved a speedup of four logarithmic factors. In our STOC 2024 paper we discovered an algorithm achieving a super-poly-log speedup, and coming tantalizingly close to refuting the conjecture that “truly subcubic” algorithms are not possible. This conjecture is central to our CONJEXITY project, and to the field of fine-grained complexity in general, because refuting it is believed to lead to a refutation some of the conjectures in the field (for not-necessarily combinatorial algorithms).
- Resolving a variant of a secondary conjecture using new techniques based on Additive Combinatorics. Specifically, in our STOC 2023 paper (invited to special issue) we show that the listing variant of the 4-cycle conjecture (a secondary conjecture) holds under the 3-Sum Conjecture (a primary conjecture). This work falls within a recent major interest in techniques from Additive Combinatorics for fine-grained complexity and algorithms.
- Ruling out the use of the popular technique of “expander decompositions” for refuting some of the conjectures in fine-grained complexity. In an ITCS 2023 paper we have introduced the notion of a worst-case to expander-case reduction, which can be used as a tool for discouraging researchers from attempting to refute conjectures via the natural and popular method based on decomposing graphs into expanders.
- Insight into the hard cases of the conjectures. A key step towards resolving a conjecture is to undertand it the hard instances of the conjectured-to-be-hard problem. For problems such as All-Pairs Shortest-Paths (and its secondary variants such as k-Clique) our work has revealed new insights about the hard cases. In a recent submission to STOC 2025, we show that if the number of distinct weights on the edges touching any node is sublinear, then the instance can be solved in truly subcubic time. Therefore, hard instances must use many different weights in every neighborhood. This result also utilizes techniques from Additive Combinatorics. In another project, we are identifying the properties of the graph that make triangle detection hard. For example, we show that a graph without 5-cycles is easy for triangle detection.
- Almost-optimal algorithm for All-Pairs Max-Flow, essentially completing a 60-year-old quest for a linear time algorithm for computing a so-called Gomory-Hu tree of a graph. Recent breakthroughs brought the complexity down from cubic to quadratic, and in our FOCS 2023 paper we bring it down to linear (in the number of edges). While this result does not directly make progress on any of the conjectures in fine-grained complexity, it inspires further research into refuting the All-Pairs Shortest-Paths conjecture and its secondary variants, due to the similarity between shortest paths and maximum flows.
The progress made by the project puts us in a better-than-ever position to tackle the conjectures of fine-grained complexity and either refute some of them, causing a significant reform in what we believe is possible, or proving them under more central conjectures, gaining confidence in our theory. In either case, this is a high-risk high-gain research so progress cannot be guaranteed, but we are optimistic about the potential. Most importantly:
- Refuting conjectures. Having nearly-refuted the Combinatorial Boolean Matrix Multiplication Conjecture, which in the realm of “combinatorial” conjectures would be considered a primary conjecture, it is now high-time to try to fully-refute its secondary variants such as the (combinatorial) k-Clique and Hyper-Clique conjectures. It is very likely that once this is achieved, we would be able to build on the techniques to refute the corresponding standard (non-combinatorial) conjectures, such as the Min-Weight k-Clique Conjecture which is perhaps the most important secondary conjecture due to its large number of consequences.
- Deeper understanding of the hard instances, and proving conjectures via reductions. Building on the aforementioned progress on identifying the hard instances of the central problems, we would like define families of instances that are hard. For some conjectures such as Strong 3SUM and Triangle, this is not known, since the natural random distributions are easy. A specific important achievement would be to figure out for which graphs H the Triangle problem is easy on H-free graphs. Once the hard instances are well-understood, we would be able to design reductions proving the corresponding conjecture assuming other, more central ones.
- Refuting conjectures. Having nearly-refuted the Combinatorial Boolean Matrix Multiplication Conjecture, which in the realm of “combinatorial” conjectures would be considered a primary conjecture, it is now high-time to try to fully-refute its secondary variants such as the (combinatorial) k-Clique and Hyper-Clique conjectures. It is very likely that once this is achieved, we would be able to build on the techniques to refute the corresponding standard (non-combinatorial) conjectures, such as the Min-Weight k-Clique Conjecture which is perhaps the most important secondary conjecture due to its large number of consequences.
- Deeper understanding of the hard instances, and proving conjectures via reductions. Building on the aforementioned progress on identifying the hard instances of the central problems, we would like define families of instances that are hard. For some conjectures such as Strong 3SUM and Triangle, this is not known, since the natural random distributions are easy. A specific important achievement would be to figure out for which graphs H the Triangle problem is easy on H-free graphs. Once the hard instances are well-understood, we would be able to design reductions proving the corresponding conjecture assuming other, more central ones.