Average optimal cost for the Euclidean TSP in one dimension

Abstract

The traveling-salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We study the traveling salesman problem when the positions of the cities are chosen at random in the unit interval and the cost associated to the travel between two cities is their distance elevated to an arbitrary power p ∈ R. We characterize the optimal cycle and compute the average optimal cost for every number of cities when the measure used to choose the position of the cities is flat and asymptotically for large number of cities in the other cases. We also show that the optimal cost is a self-averaging quantity, and we test our analytical results with extensive simulations.Submitted to: J. Phys. A: Math. Gen.

Plastic number and possible optimal solutions for an Euclidean 2-matching in one dimension

“…Similar conclusions can be derived in the case of the complete graph K N since, as we have said, both the analytical solution for the matching [21] and the TSP [24] are known. Let us order the points in increasing order, i.e.…”

“…Indeed the calculation we perform below can be done also for general p > 1, but it is much more involved. Then we will examine the complete graph case, where we have obtained a very simple expression of the average optimal cost for every N, and for every p > 1 [24].…”

“…We restrict here to a particular case in which the parameter p appearing in the definition of the cost (5) is such that p > 1, that is the weight associated to an edge is a convex and increasing function of the Euclidean distance between its two vertices. In such a case we know exactly, for every number of points, the optimal solution of the matching problem both on the bipartite [19][20][21] and the complete graph [21], and of the TSP problem, again on both its bipartite [22] and complete graph version [24]. The knowledge of the optimal configuration of those problems permits one to write down several properties of the solution of the 2-matching.…”

Selberg integrals in 1D random Euclidean optimization problems

“…The possibility of dealing with correlations as a perturbation has been investigated [12] and is eective for very large dimensions of the Euclidean space. On the contrary, detailed results, at least in the limit of an asymptotic number of points, have been recently obtained in one dimension [13][14][15][16][17][18] and in two dimensions [19][20][21][22][23]. Note that related problems in dimensions higher than one have also been rigorously studied [24][25][26].…”

Exact value for the average optimal cost of the bipartite traveling salesman and two-factor problems in two dimensions

“…In the bipartite version of the problem the set of 2N points is partitioned in two subsets each with N points and steps are allowed only from points in one subset to points in the other subset, in the monopartite version all the points can be reached from any other point. Interestingly enough in d = 1, when p > 1, that is when the cost function is convex and increasing, the search for the optimal tour can be exactly solved both in the bipartite [16], as well as in the monopartite [17], version of the problem. There have been, recently, what we consider three relevant progresses in the field: i) for other optimization problems similar to the TSP, the monopartite and bipartite versions have different optimal cost properties.…”