Cubic TSP - a 1.3-approximation

We prove that every simple bridgeless cubic graph with n >= 8 vertices has a travelling salesman tour of length at most 1.3n - 2, which can be constructed in polynomial time.

💡 Research Summary

The paper addresses the graphic Traveling Salesman Problem (TSP) on a very specific class of graphs: simple, bridgeless, cubic (3‑regular) graphs with at least eight vertices. The authors prove that every such graph admits a Hamiltonian tour whose length does not exceed 1.3 n − 2, where n is the number of vertices, and they provide a polynomial‑time algorithm to construct such a tour. This improves upon the previous best known bound of 4/3 · n (≈1.333 n) for this class.

The core of the argument is a reformulation of the TSP tour construction in terms of an “even factor”. An even factor F of a cubic graph G is a spanning subgraph whose components are Eulerian; in a cubic graph each component is either a circuit (a cycle) or an isolated vertex. The authors define the cost of an even factor as

c(F) = |E(F)| + 2·|C_F| + |V_F|,

where C_F is the set of circuits and V_F the set of isolated vertices. They observe that a TSP tour can be built from any even factor by adding a spanning tree of the contracted graph G/F and traversing each edge of the tree twice. The resulting tour length equals c(F) − 2. Consequently, proving the existence of an even factor with cost at most 1.3 n suffices to obtain the desired tour bound.

To control the contribution of short cycles and isolated vertices, the authors introduce bounded even factors (BE‑factors). In a BE‑factor every isolated vertex is “bounded” to a specific circuit, meaning it has at least two neighbours in that circuit or is linked via another bounded isolated vertex. This forces each isolated vertex to share its cost with a circuit, effectively turning each component into an “X‑circuit” (a circuit together with all isolated vertices bounded to it). The cost of an X‑circuit X is defined as

c(X) = |V(X)| + |V_X| + 2,

where V_X are the isolated vertices attached to the circuit. The total cost of a BE‑factor is the sum of the costs of its X‑circuits.

The authors then develop a set of swap operations that reduce the cost of a BE‑factor while preserving its boundedness. Three types of swaps are defined:

• 4‑swap: Applied to a chordless 4‑cycle whose four incident edges belong to two distinct circuits. Replacing two opposite edges of the 4‑cycle merges the two circuits into one, decreasing the total cost by exactly 2.
• 5‑swap: Applied to a chordless 5‑cycle where three alternating edges belong to three distinct circuits. Replacing those edges merges the three circuits, decreasing the cost by 3.
• 6‑swap: Applied to a chordless 6‑cycle with four alternating edges belonging to four distinct circuits. The replacement merges the four circuits, decreasing the cost by 4.

Each swap maintains the BE‑factor property because the bounded isolated vertices are reassigned to the newly formed larger circuit. Thus a sequence of swaps can be performed without ever breaking the structural constraints.

A crucial combinatorial ingredient is Lemma 9 (the key lemma), which guarantees the existence of a suitable 2‑factor (a spanning subgraph where every vertex has degree 2) that can serve as the initial even factor. The lemma asserts that for any irreducible cubic graph (i.e., one that does not contain any of four specific reducible subgraphs) there exists a 2‑factor such that a large proportion of its cycles are long (length at least 7) and the remaining short cycles can be paired with isolated vertices in a way that enables the swaps. The proof of Lemma 9 proceeds by a careful counting argument and by showing that any minimal counterexample would contain a reducible configuration, contradicting irreducibility.

The overall algorithm proceeds as follows:

1. Preprocessing – Verify that the input graph is simple, bridgeless, cubic, and has n ≥ 8. If the graph contains any of the four reducible subgraphs, contract them (the paper shows that doing so does not affect the existence of a low‑cost even factor). This yields an irreducible graph G′.
2. Initial even factor – Apply Lemma 9 to obtain a 2‑factor F₀ of G′ with the desired distribution of cycle lengths.
3. Swap phase – Scan the graph for applicable 4‑, 5‑, and 6‑cycles and perform the corresponding swaps on F₀. Swaps are applied greedily in order of decreasing cost reduction (4‑swap first, then 5‑swap, then 6‑swap). Each swap merges several circuits and reduces the total cost by a constant amount.
4. Termination – When no further swaps are possible, the remaining X‑circuits consist of long cycles (length ≥ 7) and possibly a few short cycles with attached isolated vertices. Using the bounds from Lemma 9 and the known reduction per swap, the authors compute an upper bound on the total cost, showing it does not exceed 1.3 n.
5. Tour construction – From the final BE‑factor, construct a spanning tree of the contracted graph, double its edges, and combine with the edges of the BE‑factor to obtain an Eulerian multigraph. An Eulerian tour of this multigraph yields a TSP tour of G whose length is at most c(F) − 2 ≤ 1.3 n − 2.

All steps are shown to be executable in polynomial time. The contraction of reducible subgraphs, the construction of the initial 2‑factor (via matching algorithms), and the detection of applicable swaps can each be performed in O(n³) time or better, yielding an overall polynomial‑time algorithm.

The paper also discusses the tightness of the bound. The authors note that the Petersen graph (the smallest bridgeless cubic graph) attains the bound exactly, and that the analysis cannot be improved without new structural insights. They compare their result with earlier work: Mömke and Svensson’s 4/3 · n − 2/3 bound for subcubic graphs, Boyd et al.’s 4/3 · n − 2 bound for simple cubic graphs, and later incremental improvements (4/3 − 1/8754)·n and (4/3 − 1/61236)·n. Their 1.3 · n − 2 bound is a substantial improvement for the restricted class considered.

In summary, the paper makes three major contributions:

• It introduces the notion of bounded even factors and a systematic swap framework that reduces the cost of an even factor while preserving structural constraints.
• It proves a key combinatorial lemma guaranteeing the existence of a favorable initial 2‑factor in any irreducible cubic graph.
• It combines these tools into a deterministic polynomial‑time algorithm that yields a TSP tour of length at most 1.3 n − 2 for every simple, bridgeless cubic graph with n ≥ 8, thereby establishing the best known approximation ratio for this graph class.