Improved Inapproximability for TSP

The Traveling Salesman Problem is one of the most studied problems in computational complexity and its approximability has been a long standing open question. Currently, the best known inapproximability threshold known is 220/219 due to Papadimitriou and Vempala. Here, using an essentially different construction and also relying on the work of Berman and Karpinski on bounded occurrence CSPs, we give an alternative and simpler inapproximability proof which improves the bound to 185/184.

💡 Research Summary

The paper presents a new hardness of approximation result for the Traveling Salesman Problem (TSP), improving the previously best known inapproximability ratio of 220/219 (Papadimitriou‑Vempala) to 185/184. The authors achieve this by constructing a three‑stage reduction that is conceptually simpler than the earlier work while preserving, and slightly strengthening, the hardness parameters.

The starting point is the well‑known MAX‑E3‑LIN2 problem, for which Håstad proved optimal inapproximability: distinguishing between instances where almost all equations can be satisfied and those where at most half plus a small ε can be satisfied is NP‑hard. Papadimitriou and Vempala reduced directly from this CSP to TSP, but their reduction required intricate gadgets to avoid introducing extra variables and to keep the number of occurrences of each variable low.

The new approach first applies the bounded‑occurrence amplification technique of Berman and Karpinski. By embedding each original variable into a “cloud” of copies and auxiliary variables linked through a bipartite expander graph, the authors obtain an equivalent CSP (call it I₂) in which every variable appears exactly five times. This bounded‑occurrence property is crucial for the later construction because it guarantees that each variable can be represented by a small, fixed number of gadgets in the TSP instance.

Next, the authors transform I₂ into a MAX‑1‑in‑3‑SAT instance (I₃). Two‑variable equations become simple clauses of the form (x ∨ y). Three‑variable equations are replaced by a cluster of three clauses that involve two new auxiliary variables per original equation. The cluster has the property that if the original equation is satisfied, all three clauses can be satisfied simultaneously; otherwise at most two clauses can be satisfied, and exactly which clause is left unsatisfied can be controlled by the assignment to the auxiliary variables. Consequently, the number of unsatisfied clauses in I₃ is linearly related to the number of unsatisfied equations in the original CSP.

The core of the paper is the reduction from I₃ to a TSP instance G. For each variable (whether a main variable, a checker variable, or an auxiliary variable) the construction creates two terminal vertices, denoted x_L and x_R, and connects each terminal to a central hub vertex s via a “forced edge”. Forced edges are implemented by subdividing a single edge into a long path of many tiny‑weight edges; any feasible tour must traverse these paths, effectively forcing the inclusion of the corresponding edges in the tour. Between the two terminals of the same variable the construction adds two parallel non‑forced edges of weight 1; a tour must pick exactly one of them, thereby encoding a true/false assignment for that variable.

Clause gadgets are then attached to the appropriate terminals. A 2‑literal clause (x ∨ y) is represented by two forced edges of weight 3/2 linking the terminals of x and y, together with the routing of the true‑edge of each variable through this gadget. A 3‑literal cluster is modeled by a more elaborate structure involving six auxiliary terminals and several forced edges of weight 5/4 and 1, forming small triangles that enforce the “exactly one true” condition of the 1‑in‑3 predicate. If a clause is satisfied, the tour can pass through the gadget using only forced edges; if it is unsatisfied, the tour is forced to use at least one extra non‑forced edge, incurring an additional cost of exactly 1.

The authors compute the baseline cost L of any feasible tour as the sum of the weights of all forced edges, which equals 91.8 · m where m is the number of original equations. They then prove Lemma 1: if an assignment leaves k clauses unsatisfied in I₃, there exists a tour of total cost L + k, and conversely any tour of cost L + k yields an assignment with at most k unsatisfied clauses. Combining this with the hardness of distinguishing between (1 − ε)·m satisfied equations and (½ + ε)·m satisfied equations in the original MAX‑E3‑LIN2 instance, they obtain the final theorem: for every ε > 0, achieving a (92.3 − ε)/(91.8 − ε)‑approximation for TSP would imply P = NP. Numerically this ratio simplifies to 185/184, improving the previous bound.

The paper’s contributions are twofold. First, it provides a modest but concrete improvement in the inapproximability ratio for TSP, showing that the gap between known algorithms (≈1.4) and hardness remains large but can be narrowed further. Second, it introduces a modular reduction framework: bounded‑occurrence CSP → 1‑in‑3‑SAT → TSP, each step accompanied by clean, self‑contained gadgets. This modularity makes the proof more accessible and suggests that further refinements—perhaps using tighter bounded‑occurrence constructions or more efficient clause gadgets—could push the ratio even higher.

The authors acknowledge that the constant 185/184 is unlikely to be optimal and that their techniques apply to the general weighted (non‑metric) TSP; extending the approach to the metric case or to related problems such as Graphic TSP remains an open direction. Nonetheless, the work demonstrates that even after a decade of stagnation, careful exploitation of bounded‑occurrence CSP tools can still yield progress on one of the most classic problems in combinatorial optimization.