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With applications to many disciplines, the traveling salesman problem (TSP) is a classical computer science optimization problem with applications to industrial engineering, theoretical computer science, bioinformatics, and several other…
In this paper, we study the optimal transport problem induced by separable cost functions. In this framework, transportation can be expressed as the composition of two lower-dimensional movements. Through this reformulation, we prove that…
Although many well-known algorithms can solve each bipartite matching problem instance efficiently, it remains an open question how one could estimate the expected optimal matching distance for arbitrary numbers of randomly distributed…
A disordered medium is often constructed by $N$ points independently and identically distributed in a $d$-dimensional hyperspace. Characteristics related to the statistics of this system is known as the random point problem. As $d \to…
Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We…
A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based…
We propose a learning algorithm for solving the traveling salesman problem based on a simple strategy of trial and adaptation: i) A tour is selected by choosing cities probabilistically according to the ``synaptic'' strengths between…
The $k$-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces $k$ edges of the tour by $k$ other edges, as long as this yields a shorter tour. We…
Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of…
We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant.…
We consider combinatorial optimization problems defined over random ensembles, and study how solution cost increases when the optimal solution undergoes a small perturbation delta. For the minimum spanning tree, the increase in cost scales…
In this work, we solve a discrete optimal transport problem in a nonuniform environment. To solve the optimal transport problem, we build the cost matrix and then use classical solvers for discrete optimal transport. The challenge is to…
We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $\Omega$ of unit area. It is known that the average cost scales as $E_{\Omega}(N)\sim\frac{1}{2\pi}\ln N$ with a correction that…
The 2-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces two edges of the tour by two other edges, as long as this yields a shorter tour. We will…
We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set,…
We establish the validity of asymptotic limits for the general transportation problem between random i.i.d. points and their common distribution, with respect to the squared Euclidean distance cost, in any dimension larger than three.…
We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and…
This study provides new results about the probabilistic behaviour of a class of Euclidean algorithms: the asymptotic distribution of a whole class of cost-parameters associated to these algorithms is normal. For the cost corresponding to…
We consider Monge-Kantorovich optimal transport problems on $\mathbb{R}^d$, $d\ge 1$, with a convex cost function given by the cumulant generating function of a probability measure. Examples include the Wasserstein-2 transport whose cost…
The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms…