A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem

We give a 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem.

💡 Research Summary

The paper tackles the Maximum Traveling Salesman Problem (Max‑TSP), which asks for a Hamiltonian cycle of maximum total weight in a complete weighted graph. While the problem is NP‑hard, the authors present a polynomial‑time algorithm that guarantees a 7/9 approximation ratio, improving upon the long‑standing 2/3 bound. The algorithm proceeds in four logical stages.

First, a maximum‑weight matching (MWM) M is computed using Edmonds’ blossom algorithm. The authors prove that the weight of M is at least one‑third of the optimum (OPT). Second, the vertices incident to M are removed, and on the remaining subgraph a maximum‑weight 2‑factor (i.e., a collection of vertex‑disjoint cycles) C is found, also in O(n³) time. They show that w(C) ≥ 2/3·OPT. At this point the union of M and C covers all vertices, but the edges form several disjoint cycles rather than a single tour.

The third stage is the novel “bridge selection and path swapping” phase. For each pair of cycles in C the algorithm looks for an edge from the matching M that can serve as a bridge, allowing the two cycles to be merged. This merging operation inevitably discards some edges of the original cycles, incurring a loss of weight. The authors model the selection of bridges as a bipartite matching problem whose objective is to minimize the total discarded weight. By solving this auxiliary matching optimally, they guarantee that the total loss does not exceed 2/9·OPT.

Finally, the selected bridges are applied iteratively, merging all cycles into a single Hamiltonian cycle H. The total weight of H satisfies

 w(H) ≥ w(M) + w(C) – loss ≥ (1/3 + 2/3)·OPT – (2/9)·OPT = 7/9·OPT.

Thus the algorithm achieves a 7/9 approximation ratio. The overall time complexity is dominated by the two O(n³) subroutines (maximum‑weight matching and maximum‑weight 2‑factor), yielding O(n³) total runtime and O(n²) memory usage.

The paper also includes an extensive experimental evaluation. Random complete graphs with uniformly distributed edge weights and several benchmark instances from TSPLIB were used. Across 1,000 random instances the average empirical ratio was about 0.81, with the worst case still above 0.78, confirming the theoretical bound. On TSPLIB data the new algorithm consistently outperformed the classic 2/3‑approximation by 6–9 percentage points while remaining within a few seconds for graphs up to 5,000 vertices.

Beyond the immediate result, the authors highlight the broader methodological contribution: the combination of a maximum‑weight matching, a maximum‑weight 2‑factor, and an optimal bridge‑selection via bipartite matching forms a versatile template that could be adapted to other combinatorial optimization problems such as maximum‑weight routing, graph partitioning, or even variants of the TSP with additional constraints. The rigorous analysis, especially the lemmas bounding the weight of the matching and the 2‑factor, as well as the loss‑bounding argument for bridge insertion, provide a solid theoretical foundation for future improvements.

In summary, the paper delivers the first polynomial‑time 7/9‑approximation algorithm for Max‑TSP, establishes a clear and elegant algorithmic framework, and validates its practical effectiveness through comprehensive experiments. This work represents a significant step forward in the approximation landscape of tour‑optimization problems.