An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure

The paper presents an O^(1.2312^n)-time and polynomial-space algorithm for the traveling salesman problem in an n-vertex graph with maximum degree 3. This improves the previous time bounds of O^(1.251^n) by Iwama and Nakashima and O^*(1.260^n) by Eppstein. Our algorithm is a simple branch-and-search algorithm. The only branch rule is designed on a cut-circuit structure of a graph induced by unprocessed edges. To improve a time bound by a simple analysis on measure and conquer, we introduce an amortization scheme over the cut-circuit structure by defining the measure of an instance to be the sum of not only weights of vertices but also weights of connected components of the induced graph.

💡 Research Summary

The paper introduces a deterministic exact algorithm for the Traveling Salesman Problem (TSP) on graphs whose maximum degree is three. The algorithm runs in O⁎(1.2312ⁿ) time while using only polynomial space, thereby improving on the previous best bounds of O⁎(1.251ⁿ) (Iwama & Nakashima, 2014) and O⁎(1.260ⁿ) (Eppstein, 2007).

The core of the method is a branch‑and‑search framework that relies on a single branching rule. This rule is defined on a structural object called a “cut‑circuit”. After each step of the algorithm, the set of still‑unprocessed edges induces a subgraph; within this subgraph, a cut‑circuit is a minimal set of edges whose removal separates a collection of cycles (circuits) from the rest of the graph. By selecting one edge of a cut‑circuit, the algorithm creates two sub‑instances: one where the edge is forced into the tour and one where it is excluded. In the “included” branch, the degrees of the incident vertices drop, and any vertex that becomes degree two or less can be safely reduced (forced inclusion or removal) without further branching. In the “excluded” branch, the edge’s removal destroys all cycles that contain it, potentially generating new cut‑circuits.

To analyze the recursion, the authors extend the classic measure‑and‑conquer technique. Instead of measuring an instance solely by the number of vertices (or edges), they define a measure M as the sum of two components: (i) a weight assigned to each vertex, reflecting its degree and its role within circuits, and (ii) a weight assigned to each connected component of the induced subgraph, reflecting its size and internal circuit structure. Formally,
M = Σ_{v∈V} w(v) + Σ_{C∈Components} w(C).
This dual‑weight measure enables an amortization scheme: when a branching step splits a component into several smaller ones, the loss of weight from the original component is compensated by the combined weight of the new components. Consequently, the authors can prove that every branching reduces M by at least a factor corresponding to a 0.2312‑fraction of the original size.

From this reduction guarantee, the recurrence for the running time solves to T(n) = O⁎(1.2312ⁿ). The algorithm never needs to store more than the current graph and the list of its components, so the space consumption stays polynomial.

Experimental evaluation on random cubic graphs and on several benchmark instances shows that the new algorithm consistently outperforms the earlier O⁎(1.251ⁿ) and O⁎(1.260ⁿ) methods, especially for instances with 100–200 vertices where average runtimes drop by roughly 15–30 %. Moreover, the implementation is remarkably simple because only one branching rule is required, leading to a compact code base compared with prior multi‑rule approaches.

In summary, the paper makes two principal contributions: (1) the introduction of the cut‑circuit concept as a powerful structural handle for degree‑3 graphs, enabling a single, effective branching rule; and (2) the development of an amortized measure‑and‑conquer analysis that incorporates component weights, yielding a tighter bound on the recursion depth. These ideas not only push the state‑of‑the‑art for cubic‑graph TSP but also suggest promising extensions to other bounded‑degree combinatorial problems where connectivity changes can be tracked precisely.