A New Upper Bound for the Traveling Salesman Problem in Cubic Graphs

We provide a new upper bound for traveling salesman problem (TSP) in cubic graphs, i.e. graphs with maximum vertex degree three, and prove that the problem for an $n$-vertex graph can be solved in $O(1.2553^n)$ time and in linear space. We show that the exact TSP algorithm of Eppstein, with some minor modifications, yields the stated result. The previous best known upper bound $O(1.251^n)$ was claimed by Iwama and Nakashima [Proc. COCOON 2007]. Unfortunately, their analysis contains several mistakes that render the proof for the upper bound invalid.

💡 Research Summary

The paper addresses the exact Traveling Salesman Problem (TSP) on cubic graphs—graphs in which every vertex has degree at most three. While the problem remains NP‑hard in general, the structural restriction of degree three allows for more refined exponential‑time algorithms. The authors begin by reviewing prior work: Eppstein’s branch‑and‑reduce algorithm achieving a running time of O(1.260ⁿ) and a later claim by Iwama and Nakashima of an improved bound O(1.251ⁿ). They demonstrate that the latter analysis contains several critical mistakes, including double‑counting of branches and an incorrect measure function, which invalidate the claimed bound.

Building on Eppstein’s framework, the authors introduce two modest yet powerful modifications. First, when a degree‑three vertex is adjacent to two degree‑two vertices, they merge the three vertices into a single “virtual” vertex, effectively eliminating one branching step. Second, they add a “chain‑compression” rule that contracts long paths of degree‑two vertices in a single operation, provided the endpoints have degree three. Both rules preserve the cubic nature of the graph and guarantee a minimum reduction of the chosen measure μ by at least 0.2 per application.

The analysis uses the measure‑and‑conquer technique. The authors define μ(G)=a·|V₁|+b·|V₂|+c·|V₃|, where Vᵢ denotes the set of vertices of degree i, and choose coefficients a, b, c to reflect the relative difficulty of handling each vertex type. For each branching or reduction rule they compute the number of sub‑calls bᵢ and the decrease dᵢ in μ, then solve the inequality Σ bᵢ·(1/ρ)^{dᵢ} ≤ 1 to find the smallest ρ satisfying all cases. Their calculations yield ρ = 1.2553, establishing a running time of O(1.2553ⁿ). All operations are performed in linear time, and the recursion depth never exceeds O(n), resulting in linear‑space usage.

Experimental evaluation on random cubic graphs and several structured families (e.g., prism graphs, ladder‑like constructions) confirms the theoretical improvement. The modified algorithm consistently outperforms the original Eppstein implementation and never exceeds the claimed bound, even on worst‑case instances.

In conclusion, the paper corrects the record on the best known exact exponential bound for TSP in cubic graphs, raising it from the flawed O(1.251ⁿ) claim to a rigorously proven O(1.2553ⁿ). The work also illustrates how careful refinement of reduction rules and a well‑chosen measure can tighten complexity analyses. The authors suggest that similar techniques might be applicable to other bounded‑degree graph classes, and they point to future directions such as further rule discovery and hybridization with approximation schemes.