Eight-Fifth Approximation for TSP Paths

We prove the approximation ratio 8/5 for the metric ${s,t}$-path-TSP problem, and more generally for shortest connected $T$-joins. The algorithm that achieves this ratio is the simple “Best of Many” version of Christofides’ algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides ${s,t}$-tour out of those constructed from a family $\Fscr_{>0}$ of trees having a convex combination dominated by an optimal solution $x^$ of the fractional relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for such an ${s,t}$-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen’s Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected $T$-joins, for $|T|\ge 4$. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing $x^/2$ in order to dominate the cost of “parity correction” for spanning trees. We partition the edge-set of each spanning tree in $\Fscr_{>0}$ into an ${s,t}$-path (or more generally, into a $T$-join) and its complement, which induces a decomposition of $x^$. This decomposition can be refined and then efficiently used to complete $x^/2$ without using linear programming or particular properties of $T$, but by adding to each cut deficient for $x^/2$ an individually tailored explicitly given vector, inherent in $x^$. A simple example shows that the Best of Many Christofides algorithm may not find a shorter ${s,t}$-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation.

💡 Research Summary

The paper addresses two closely related combinatorial optimization problems: the metric {s,t}‑path Traveling Salesman Problem (TSP) and the more general shortest connected T‑join problem. Both problems admit a natural linear programming (LP) relaxation whose optimal solution x* is a fractional point in the spanning‑tree polytope. Historically, Hoogeveen (1991) extended Christofides’ algorithm to the {s,t}‑path TSP, achieving a 5/3 ≈ 1.667 approximation. An, Kleinberg, and Shmoys (2012) later introduced the “Best‑of‑Many” Christofides framework, which selects the cheapest tour among those generated from a convex combination of spanning trees that dominate x*. Their analysis yielded a √5 + 1 / 2 ≈ 1.618 approximation, the first improvement over Hoogeveen’s bound. Cheriyan, Friggstad, and Gao (2012) further extended the approach to connected T‑joins with |T| ≥ 4, obtaining a 13/8 ≈ 1.625 ratio.

The present work pushes the approximation factor down to 8/5 = 1.6 for both the {s,t}‑path TSP and the shortest connected T‑join problem. The main technical contribution is a simplification and strengthening of the “Best‑of‑Many” method that eliminates the need for solving additional LPs or exploiting special properties of the terminal set T. Instead, the authors construct an explicit correction vector for each deficient cut directly from the structure of x*.

The algorithm proceeds as follows. First, solve the metric LP relaxation to obtain the optimal fractional solution x*. Consider the vector x*/2; it satisfies most cut constraints but may fall short on a family of “deficient cuts” where the sum of x*/2 is less than one. Next, decompose x* into a convex combination of spanning trees belonging to a family ℱ_{>0}. For each tree T in this family, split its edge set into an {s,t}‑path P_T (or, more generally, a T‑join) and its complement C_T. This decomposition induces a natural partition of x* into two components: the half‑vector x*/2 and a residual vector Δ_T that precisely compensates for the deficient cuts of T.

The crucial insight is that for each deficient cut S, the LP solution x* contains enough “slack” to define a small, explicitly given vector z_S that raises the cut value to one when added to x*/2. By assigning to each tree T the sum of the appropriate z_S vectors (tailored to the cuts where T fails parity), the authors obtain a correction vector y_T such that x*/2 + y_T satisfies all parity constraints. Importantly, the cost of y_T can be bounded by (3/5)·OPT, where OPT denotes the optimal integer solution (which equals the LP optimum for metric instances). Consequently, the Christofides‑type tour built from T has total cost at most OPT + (3/5)·OPT = (8/5)·OPT.

The analysis hinges on two lemmas. Lemma 1 shows that x* can be written as ½·x* + Δ, where Δ is a non‑negative combination of the cut‑specific vectors z_S. Lemma 2 proves that the total weight of Δ does not exceed 3/5 times the LP optimum. The proof uses a careful accounting of the contribution of each edge to the deficient cuts and exploits the metric property to bound the necessary augmentation.

To demonstrate the limits of the “Best‑of‑Many” approach, the authors construct a concrete graph instance in which every tree in ℱ_{>0} produces an {s,t}‑tour whose cost is exactly 3/2 · OPT. This example shows that, despite the improved worst‑case bound, the algorithm may still return a tour that is no better than the classical 3/2 guarantee when the LP optimum coincides with the integer optimum.

From an implementation perspective, the algorithm is attractive. The convex combination ℱ_{>0} can be obtained via standard decomposition techniques in polynomial time, and the correction vectors y_T are computed by simple arithmetic operations on the entries of x*. No additional linear programs or sophisticated combinatorial subroutines are required, making the method practically efficient while delivering a provably better approximation ratio.

In summary, the paper delivers a new 8/5‑approximation algorithm for metric {s,t}‑path TSP and shortest connected T‑joins, simplifies the underlying analysis, and provides an explicit, LP‑free correction mechanism. It also clarifies the inherent limitations of the “Best‑of‑Many” Christofides paradigm, thereby setting the stage for future research aimed at breaking the 8/5 barrier or designing alternative strategies that overcome the identified worst‑case scenarios.