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A New Conjecture on Hardness of 2-CSP’s with Implications to Hardness of Densest k-Subgraph and Other Problems

Authors Julia Chuzhoy, Mina Dalirrooyfard, Vadim Grinberg, Zihan Tan

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Author Details

Julia Chuzhoy
  • Toyota Technological Institute at Chicago, IL, USA
Mina Dalirrooyfard
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Vadim Grinberg
  • Weizmann Institute of Science, Rehovot, Israel
Zihan Tan
  • DIMACS, Rutgers University, New Brunswick, NJ, USA

Funding

  • Chuzhoy, Julia: Supported in part by NSF grant CCF-2006464.
  • Dalirrooyfard, Mina: Part of the work done at Toyota Technological Institute at Chicago.
  • Grinberg, Vadim: Part of the work done at Toyota Technological Institute at Chicago. Supported in part by NSF grant CCF-2006464.
  • Tan, Zihan: Supported by a grant to DIMACS from the Simons Foundation (820931) and NSF grant CCF-2006464.

Acknowledgements

The authors thank Irit Dinur and Uri Feige for insightful and helpful discussions.

Cite AsGet BibTex

Julia Chuzhoy, Mina Dalirrooyfard, Vadim Grinberg, and Zihan Tan. A New Conjecture on Hardness of 2-CSP’s with Implications to Hardness of Densest k-Subgraph and Other Problems. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.38

Abstract

We propose a new conjecture on hardness of 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 other 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
Keywords
  • Hardness of Approximation
  • Densest k-Subgraph

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