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

In this work we consider the problem of finding the minimum-weight loop cover of an undirected graph. This combinatorial optimization problem is called 2-matching and can be seen as a relaxation of the traveling salesman problem since one does not have the unique loop condition. We consider this problem both on the complete bipartite and complete graph embedded in a one dimensional interval, the weights being chosen as a convex function of the Euclidean distance between each couple of points. Randomness is introduced throwing independently and uniformly the points in space. We derive the average optimal cost in the limit of large number of points. We prove that the possible solutions are characterized by the presence of "shoelace" loops containing 2 or 3 points of each type in the complete bipartite case, and 3, 4 or 5 points in the complete one. This gives rise to an exponential number of possible solutions scaling as p^N , where p is the plastic constant. This is at variance to what happens in the previously studied one-dimensional models such as the matching and the traveling salesman problem, where for every instance of the disorder there is only one possible solution.