If the distances of TSP satisfy the triangle inequality, the minimum-cost-spanning tree (MST) heuristics produces a tour whose length is guaranteed to be less than 2 times the optimum tour length and Christofides’ heuristics generates the 3/2 times the optimum tour length. Otherwise, the quality of the approximation is hard to evaluate. Here a four vertices and three lines inequality is used to construct an approximation of the optimum tour instead of the triangle inequality. The performance ratio of the heuristics may not be a constant for all kinds of TSP. But it is determined for a concrete TSP.
"An Approximate Algorithm for TSP with Four Vertices and Three Lines Inequality,"
Information Sciences Letters: Vol. 3
, Article 1.
Available at: https://digitalcommons.aaru.edu.jo/isl/vol3/iss2/1