Average-case approximation ratio of the 2-opt algorithm for the TSP

The traveling salesman problem (TSP) is one of the most important problems in combinatorial optimization: Given a complete graph with edge weights, the goal is to find a Hamiltonian cycle (also called a tour) of minimum weight. 2-opt is probably the most widely used local search heuristic for the TSP. It incrementally improves an initial tour by exchanging two edges of the tour with two other edges, until a local optimum is reached. More formally: Let w be the edge weights. If {a, b} and {c, d} are two edges of the current cycle such that a, b, c, d appear in that order in the cycle, then we can improve the tour by replacing {a, b} and {c, d} by {a, c} and {b, d}, provided that w({a, c})+w({b, d}) < w({a, b})+w({c, d}). On randomly generated instances, 2-opt comes within a small percentage of the global optimum [3]. Chandra et al. [1] analyzed 2-opt’s worst-case approximation ratio: On instances that fulfil the triangle inequality it is O( √ n), where n is the number of nodes. This means that the worst local optimum is within a factor of O( √ n) of the global optimum. For Euclidean instances, 2-opt’s worst-case approximation ratio is O(log n). Englert et al. [2] showed that the expected approximation ratio of O( d √ φ) for d-dimensional Euclidean instances that are drawn according to density functions bounded by φ. To explain the good performance of subtour patching for TSP, Karp [4] analyzed its approximation performance in a simple probabilistic setting: all edge weights are drawn uniformly and independently at random from the interval [0, 1]. In this setting, the triangle inequality is usually not fulfilled. In the worst-case, TSP cannot be approximated at all without triangle inequality, and also 2-opt cannot provide any approximation guarantee. We use Karp’s probabilistic model [4] to analyze the approximation performance of 2-opt. Let WLOn be the weight of the worst, i.e., heaviest, locally optimal tour of a graph of n