Quadratically-Regularized Optimal Transport on Graphs

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport over a graph. We theoretically analyze the behavior of quadratically-regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily-implemented Newton-type optimization algorithm.Comment: 27 page

Some results on heuristical algorithms for shortest path problems in large road networks

This thesis studies the shortest path problem in large road networks. The classical algorithm for networks with non-negative edge weights is due to Dijkstra and has a worst-case performance of O ( |E |+ |V |log |V |) using a simple priority queue as data structure for temporarily labeled nodes. We present a new, so-called tree heuristic, which is based on the similarity of shortest path trees and which can be used to speed up the shortest path search especially in practical applications like microscopic simulation of traffic or route guidance systems. Instead of searching a path in the original network, the tree heuristic partitions the network into classes of about equal size and constructs a special searchgraph for each class. On a test road network of about one million nodes the tree heuristic outperforms Dijkstra\'s algorithm by a factor of more than three with respect to runtime and about seven with respect to permanently labeled nodes where the found paths can be expected to have a relative error below 1%, if the starting and end node are not too close to each other. We also analyze the A -algorithm with overdo-factor, originally devised for Euclidean networks and derive an interval [1.... 27......,5] from which an optimal overdo-factor should be chosen in practical applications. Finally we give an algorithm which calculates edge tolerances for a shortest path and which can be used to generate reasonable alternative routes to the exact shortest path

Advanced analysis of branch and bound algorithms

Als de code van een cijferslot zoek is, kan het alleen geopend worden door alle cijfercom­binaties langs te gaan. In het slechtste geval is de laatste combinatie de juiste. Echter, als de code uit tien cijfers bestaat, moeten tien miljard mogelijkheden bekeken worden. De zogenaamde 'NP-lastige' problemen in het proefschrift van Marcel Turkensteen zijn vergelijkbaar met het 'cijferslotprobleem'. Ook bij deze problemen is het aantal mogelijkheden buitensporig groot. De kunst is derhalve om de zoekruimte op een slimme manier af te tasten. Bij de Branch and Bound (BnB) methode wordt dit gedaan door de zoekruimte op te splitsen in kleinere deelgebieden. Turkensteen past de BnB methode onder andere toe bij het handelsreizigersprobleem, waarbij een kortste route door een verzameling plaatsen bepaald moet worden. Dit probleem is in algemene vorm nog steeds niet opgelost. De economische gevolgen kunnen groot zijn: zo staat nog steeds niet vast of bijvoorbeeld een routeplanner vrachtwagens optimaal laat rondrijden. De huidige BnB-methoden worden in dit proefschrift met name verbeterd door niet naar de kosten van een verbinding te kijken, maar naar de kostentoename als een verbinding niet gebruikt wordt: de boventolerantie.

An investigation of shortest paths algorithms

In this work, we classify the shortest path problems, review all source algorithms and analyse the different implementations of single source algorithms using various list structures and labelling techniques. Furthermore, we study the Sensitivity Analysis of one-to-all problems and present an algorithm, Senet, for their Post Optimality Analysis. Senet determines all the critical values for the weight of an arc (which could be optimal, non-optimal or non-existant) at which the optimal solution changes. Senet also provides the updated optimal solution for every range formed by two successive critical values

Robustness of Mission Plans for Unmanned Aircraft.

This thesis studies the robustness of optimal mission plans for unmanned aircraft. Mission planning typically involves tactical planning and path planning. Tactical planning refers to task scheduling and in multi aircraft scenarios also includes establishing a communication topology. Path planning refers to computing a feasible and collision-free trajectory. For a prototypical mission planning problem, the traveling salesman problem on a weighted graph, the robustness of an optimal tour is analyzed with respect to changes to the edge costs. Specifically, the stability region of an optimal tour is obtained, i.e., the set of all edge cost perturbations for which that tour is optimal. The exact stability region of solutions to variants of the traveling salesman problems is obtained from a linear programming relaxation of an auxiliary problem. Edge cost tolerances and edge criticalities are derived from the stability region. For Euclidean traveling salesman problems, robustness with respect to perturbations to vertex locations is considered and safe radii and vertex criticalities are introduced. For weighted-sum multi-objective problems, stability regions with respect to changes in the objectives, weights, and simultaneous changes are given. Most critical weight perturbations are derived. Computing exact stability regions is intractable for large instances. Therefore, tractable approximations are desirable. The stability region of solutions to relaxations of the traveling salesman problem give under approximations and sets of tours give over approximations. The application of these results to the two-neighborhood and the minimum 1-tree relaxation are discussed. Bounds on edge cost tolerances and approximate criticalities are obtainable likewise. A minimum spanning tree is an optimal communication topology for minimizing the cumulative transmission power in multi aircraft missions. The stability region of a minimum spanning tree is given and tolerances, stability balls, and criticalities are derived. This analysis is extended to Euclidean minimum spanning trees. This thesis aims at enabling increased mission performance by providing means of assessing the robustness and optimality of a mission and methods for identifying critical elements. Examples of the application to mission planning in contested environments, cargo aircraft mission planning, multi-objective mission planning, and planning optimal communication topologies for teams of unmanned aircraft are given.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120837/1/mniendo_1.pd

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

A Multi-Objective Decision Support Model for the Turkish Armed Forces Personnel Assignment System

The Turkish Armed Forces (TAF) assign more than 25,000 active duty personnel annually. TAF wants to obtain maximum utilization of its personnel by assigning the right person to the right job at the right time, To accomplish this task, decision-makers and personnel assignment staff should consider conflicting multiple objectives that create the widely known problem called personnel assignment problem . To assist in this complicated task from a quantitative perspective, a preemptive goal programming approach was used to develop an integer programming (IP) model to capture the multiple objectives flexibly and interactively. A realistic size IP problem with random data was tested for computational efficiency and analysis. The mean solution time for different instances of the problem was reasonably small. An application of the methodology in an actual assignment decision support system of any large-scale government or non-government organization has a potential to help decision-makers make better use of their personnel