Local matching indicators for transport with concave costs

In this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of $N$ demands and $N$ supplies in $\mathbf{R}$ in the case where the cost function is concave. The computational cost of these indicators is small and independent of $N$. A hierarchical use of them enables to obtain an efficient algorithm

Planar diagrams from optimization

We propose a new toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model the sequential intervals between neighboring monomers along a chain are considered as quenched random variables. Using the optimization procedure for a special class of concave--type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA--like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale--free) and demonstrate the existence of a topological crossover from sequential to essentially embedded (nested) configurations of paired links.Comment: 10 pages, 10 figures, the proof is added. arXiv admin note: text overlap with arXiv:1102.155

Minimum-weight perfect matching for non-intrinsic distances on the line

Consider a real line equipped with a (not necessarily intrinsic) distance. We deal with the minimum-weight perfect matching problem for a complete graph whose points are located on the line and whose edges have weights equal to distances along the line. This problem is closely related to one-dimensional Monge-Kantorovich trasnport optimization. The main result of the present note is a "bottom-up" recursion relation for weights of partial minimum-weight matchings.Comment: 13 pages, figures in TiKZ, uses xcolor package; introduction and the concluding section have been expande

Local matching indicators for concave transport costs

International audienceIn this note, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of $N$ demands and $N$ supplies in $\mathbf{R}$ in the case where the cost function is concave. The computational cost of these indicators is small and independent of $N$. A hierarchical use of them enables to obtain an efficient algorithm

Local matching indicators for transport problems with concave costs

In this paper, we introduce a class of indicators that enable to compute efficiently optimal transport plans associated to arbitrary distributions of N demands and M supplies in R in the case where the cost function is concave. The computational cost of these indicators is small and independent of N. A hierarchical use of them enables to obtain an efficient algorithm

Fast transport optimization for Monge costs on the circle

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of $N$ real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.Comment: Added affiliation for the third author in arXiv metadata; no change in the source. AMS-LaTeX, 20 pages, 5 figures (pgf/TiKZ and embedded PostScript). Article accepted to SIAM J. Applied Mat

Batch Testing, Adaptive Algorithms, and Heuristic Applications for Stable Marriage Problems

In this dissertation we focus on different variations of the stable matching (marriage) problem, initially posed by Gale and Shapley in 1962. In this problem, preference lists are used to match n men with n women in such a way that no (man, woman) pair exists that would both prefer each other over their current partners. These two would be considered a blocking pair, preventing a matching from being considered stable. In our research, we study three different versions of this problem. First, we consider batch testing of stable marriage solutions. Gusfield and Irving presented an open problem in their 1989 book The Stable Marriage Problem: Structure and Algorithms\u3c\italic\u3e on whether, given a reasonable amount of preprocessing time, stable matching solutions could be verified in less than O(n^2) time. We answer this question affirmatively, showing an algorithm that will verify k different matchings in O((m + kn) log^2 n) time. Second, we show how the concept of an adaptive algorithm can be used to speed up running time in certain cases of the stable marriage problem where the disorder present in preference lists is limited. While a problem with identical lists can be solved in a trivial O(n) running time, we present an O(n+k) time algorithm where the women have identical preference lists, and the men have preference lists that differ in k positions from a set of identical lists. We also show a visualization program for better understanding the effects of changes in preference lists. Finally, we look at preference list based matching as a heuristic for cost based matching problems. In theory, this method can lead to arbitrarily bad solutions, but through empirical testing on different types of random sources of data, we show how to obtain reasonable results in practice using methods for generating preference lists “asymmetrically” that account for long-term ramifications of short-term decisions. We also discuss several ways to measure the stability of a solution and how this might be used for bicriteria optimization approaches based on both cost and stability