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.

💡 Research Summary

The paper introduces a new conjecture concerning the hardness of low‑degree 2‑CSPs (the “Low‑Degree CSP Conjecture”) and shows how this conjecture yields strong conditional hardness results for a family of graph optimization problems. The conjecture occupies a middle ground between the well‑studied d‑to‑1 conjecture and the hardness that can be obtained for 2‑CSPs via standard techniques such as parallel repetition combined with 2‑prover protocols for 3‑SAT. Roughly, it asserts that for any constant ε>0, there exists a family of 2‑CSP instances in which each variable participates in only a constant number of constraints, yet no polynomial‑time algorithm (unless NP⊆BPTIME(n^{O(log n)})) can achieve an approximation factor better than (log n)^ε.

Assuming this conjecture, the authors prove that the Densest k‑Subgraph (DkS) problem is NP‑hard to approximate within a factor of 2·(log n)^ε. The proof proceeds by reducing a low‑degree CSP instance to a “bipartite densest (k₁,k₂)‑subgraph” problem, then regularizing the graph, constructing an assignment graph, and performing further regularization steps that preserve approximation ratios while keeping the instance size polynomial. This chain of reductions yields a DkS instance whose optimum density reflects the satisfiability of the original CSP.

Beyond DkS, the paper studies three additional problems:

1. (r,h)‑Graph Partitioning – Given a graph G and integers r, h, find r vertex‑disjoint subgraphs H₁,…,Hᵣ each containing at most h edges, while maximizing the total number of edges inside the subgraphs. This problem was introduced in recent work on Node‑Disjoint Paths (NDP) as a potential proxy for NDP hardness.

2. Dense k‑Coloring – A newly defined problem that can be viewed as an intermediate between DkS and (r,h)‑Graph Partitioning. It asks for a coloring of the vertices with k colors such that each color class induces a dense subgraph, balancing the number of colors against edge density.

3. Maximum Bounded‑Crossing Subgraph – A variation of the Minimum Crossing Number problem. Given a graph G and a crossing budget L, the goal is to select a subgraph H that admits a planar drawing with at most L crossings while maximizing |E(H)|. The problem is interesting for dense graphs where L may be Θ(n).

The authors establish a network of approximation‑preserving reductions among these four problems. In particular:

• If there exists an α(n)‑approximation algorithm for DkS, then a randomized algorithm achieving O(α(n²)·polylog n)‑approximation for (r,h)‑Graph Partitioning can be built.
• Conversely, an α(n)‑approximation for (r,h)‑Graph Partitioning yields a randomized algorithm for DkS with approximation factor O(α(n)^{O(log n)}·log² n) running in quasi‑polynomial time.
• Dense k‑Coloring reduces both to and from DkS and (r,h)‑Graph Partitioning, showing that all three share essentially the same hardness landscape when approximation factors are large.
• Maximum Bounded‑Crossing Subgraph is shown to be roughly equivalent to (r,h)‑Graph Partitioning via two-way reductions, again preserving approximation up to polylogarithmic factors.

These reductions are constructive and rely on a combination of combinatorial regularization, LP‑relaxation and rounding, and probabilistic sampling. Some steps incur a quasi‑polynomial blow‑up in running time, which the authors acknowledge; however, the reductions are sufficient to transfer any hardness result from one problem to the others.

Putting everything together, the paper’s main conditional hardness theorem can be summarized as follows: under the Low‑Degree CSP Conjecture, there exists a constant ε∈(0,½] such that no polynomial‑time algorithm can achieve a 2·(log n)^ε‑approximation for any of the four problems unless NP⊆BPTIME(n^{O(log n)}). This improves upon the best known hardness for DkS (which previously required stronger average‑case assumptions or yielded only super‑constant factors) and simultaneously lifts the same lower bound to the other three problems.

The authors also discuss the broader significance of their work. By positioning the Low‑Degree CSP Conjecture between the d‑to‑1 conjecture and standard 2‑CSP hardness, they open a new avenue for proving strong approximation lower bounds using a conjecture that is arguably more approachable than the full d‑to‑1 conjecture. Moreover, the equivalence network they build suggests that future hardness results for any one of these problems immediately translate to the others, providing a unified framework for studying dense subgraph‑type optimization tasks.

Finally, the paper outlines several directions for future research: proving or refuting the Low‑Degree CSP Conjecture, tightening the quasi‑polynomial reductions to truly polynomial‑time reductions, and extending the reduction framework to other dense‑graph problems such as community detection or dense subhypergraph extraction. The work thus not only delivers new conditional hardness results but also proposes a methodological template for leveraging low‑degree CSP hardness in the study of a broad class of combinatorial optimization problems.