DeepAI

# A New Conjecture on Hardness of Low-Degree 2-CSP's with Implications to Hardness of Densest k-Subgraph and Other Problems

We propose a new conjecture on hardness of low-degree 2-CSP's, and show that new hardness of approximation results for Densest k-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the d-to-1 conjecture, and hardness results for 2-CSP's that can be obtained via standard techniques, such as Parallel Repetition combined with standard 2-prover protocols for the 3SAT problem. We hope that this work will motivate further exploration of hardness of 2-CSP's in the regimes arising from the conjecture. We believe that a positive resolution of the conjecture will provide a good starting point for further hardness of approximation proofs. Another contribution of our work is proving that the problems that we consider are roughly equivalent from the approximation perspective. Some of these problems arose in previous work, from which it appeared that they may be related to each other. We formalize this relationship in this work.

• 14 publications
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08/17/2021

### A Note on the Permuted Puzzles Toy Conjecture

In this note, we show that a "Toy Conjecture" made by (Boyle, Ishai, Pas...
08/16/2021

### A Conjecture Equivalent to the Collatz Conjecture

We present a formulation of the Collatz conjecture that is potentially m...
04/17/2020

### Counterexamples to the Low-Degree Conjecture

A conjecture of Hopkins (2018) posits that for certain high-dimensional ...
11/03/2021

### Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle Union

We revisit the classical problem of determining the largest copy of a si...
10/02/2019

### Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

We consider the problem of outputting succinct encodings of lists of gen...
09/12/2020

### Open Problem: Average-Case Hardness of Hypergraphic Planted Clique Detection

We note the significance of hypergraphic planted clique (HPC) detection ...
09/07/2022

### On the Second Kahn–Kalai Conjecture

For any given graph H, we are interested in p_crit(H), the minimal p suc...