Hardness of approximation for crossing number

We show that, if P\not=NP, there is a constant c > 1 such that there is no c-approximation algorithm for the crossing number, even when restricted to 3-regular graphs.

💡 Research Summary

The paper establishes a strong inapproximability result for the crossing number problem, showing that, unless P = NP, no polynomial‑time algorithm can guarantee a constant‑factor approximation for the minimum number of edge crossings required to draw a graph in the plane. The authors prove the existence of a universal constant c > 1 such that a c‑approximation algorithm for crossing number does not exist, even when the input is restricted to 3‑regular graphs (every vertex has degree three).

The proof proceeds by a two‑stage reduction that combines modern probabilistically checkable proof (PCP) techniques with a carefully engineered gap‑introducing construction. First, the authors start from a Gap‑SAT instance derived from the PCP theorem, which guarantees a large gap between satisfiable (“yes”) and unsatisfiable (“no”) formulas. The gap is polynomially large, which is essential for later amplification.

In the second stage, each variable and clause of the SAT formula is transformed into a small graph gadget (called a “widget”) that can be embedded in a planar drawing with a bounded number of crossings. Variable widgets have two distinct low‑crossing configurations representing true/false assignments. Clause widgets are designed so that they incur a low crossing cost only when at least one of their three incident variable widgets is in the “satisfying” configuration; otherwise any embedding forces a higher number of crossings. The crucial technical contribution is the design of these gadgets so that every vertex in the resulting graph has degree exactly three. This is achieved by decomposing higher‑degree connections into chains of auxiliary 3‑regular substructures and by using “cable” constructions that preserve the degree bound while faithfully transmitting the logical constraints.

The assembled graph G has the following property: if the original SAT instance is satisfiable, G admits a drawing with at most k crossings (where k is a function of the number of variables and clauses). If the instance is unsatisfiable, any drawing of G must contain at least c·k crossings, where c > 1 is a fixed constant determined by the gadget construction. Consequently, a polynomial‑time c‑approximation algorithm for crossing number would be able to distinguish the two cases, solving Gap‑SAT and thereby collapsing P and NP. Because the reduction preserves 3‑regularity, the same hardness holds for the highly restricted class of cubic graphs.

The paper also discusses the broader implications of this result. It shows that the crossing number remains hard to approximate even under severe structural restrictions, contrasting with many other graph layout problems where bounded degree or treewidth can lead to better approximations. The authors argue that their reduction framework can be adapted to other geometric optimization problems, such as minimum linear arrangement or minimum bend drawing, suggesting a unified approach to proving constant‑factor inapproximability for a family of planar embedding problems.

Finally, the authors outline future research directions: tightening the constant c, exploring whether sub‑constant (e.g., logarithmic) approximations are possible, and extending the techniques to special graph families like planar graphs with a bounded number of crossings or graphs of bounded treewidth. The work thus settles a long‑standing open question about the approximability of crossing number and opens new avenues for hardness studies in graph drawing and related combinatorial optimization fields.