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

TL;DR

This paper analyzes the Euclidean 2-matching problem in one dimension, deriving the average optimal cost and characterizing the structure of solutions, revealing an exponential number of solutions related to the plastic constant.

Contribution

It provides the first detailed analysis of the 2-matching problem in 1D Euclidean space, including solution characterization and cost scaling, extending understanding beyond classical matching and TSP models.

Findings

Average optimal cost derived for large N

Solutions characterized by 'shoelace' loops with 2-5 points

Exponential number of solutions scaling as p^N, with p as the plastic constant

Abstract

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.