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In this work we consider the problem of finding the minimum-weight loop coverof an undirected graph. This combinatorial optimization problem is called2-matching and can be seen as a relaxation of the traveling salesman problemsince one does not have the unique loop condition. We consider this problemboth on the complete bipartite and complete graph embedded in a one dimensionalinterval, the weights being chosen as a convex function of the Euclideandistance between each couple of points. Randomness is introduced throwingindependently and uniformly the points in space. We derive the average optimalcost in the limit of large number of points. We prove that the possiblesolutions are characterized by the presence of "shoelace" loops containing 2 or3 points of each type in the complete bipartite case, and 3, 4 or 5 points inthe complete one. This gives rise to an exponential number of possiblesolutions scaling as p^N , where p is the plastic constant. This is at varianceto what happens in the previously studied one-dimensional models such as thematching and the traveling salesman problem, where for every instance of thedisorder there is only one possible solution.

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