Multipoint-to-multipoint network communication

We have formulated an exact ILP model for the problem of communicating on a virtual network. While this ILP model was successful in solving small problems, it is not recommended to handle larger instances, due to the fact that the number of variables in the model grows exponentially as the graph size grows. However, this ILP model can provide a benchmark for heuristic algorithms developed for this problem. We have also described a heuristic approach, and explored several variants of the algorithm. We found a solution that seems to perform well with reasonable computation time. The heuristic is able to find solutions that respect the degree constraints, but show a small number of violations of the desired time constraints. Tests on small problems show that heuristic is not always able to find feasible solutions, even though the exact method has shown they exist. It would be interesting in the future to look at whether insights gained by looking at exact solutions can be used to improve the heuristic

Optimal Trees

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Improving the Asymmetric TSP by Considering Graph Structure

Recent works on cost based relaxations have improved Constraint Programming (CP) models for the Traveling Salesman Problem (TSP). We provide a short survey over solving asymmetric TSP with CP. Then, we suggest new implied propagators based on general graph properties. We experimentally show that such implied propagators bring robustness to pathological instances and highlight the fact that graph structure can significantly improve search heuristics behavior. Finally, we show that our approach outperforms current state of the art results.Comment: Technical repor

On a general implementation of hh- and pp-adaptive curl-conforming finite elements

Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, specially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR. We cover into detail how to implement arbitrary order (i.e., $p$-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation

Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP

Abstract — We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2 − ɛ)-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier. 1

Cloud-Based Centralized/Decentralized Multi-Agent Optimization with Communication Delays

We present and analyze a computational hybrid architecture for performing multi-agent optimization. The optimization problems under consideration have convex objective and constraint functions with mild smoothness conditions imposed on them. For such problems, we provide a primal-dual algorithm implemented in the hybrid architecture, which consists of a decentralized network of agents into which centralized information is occasionally injected, and we establish its convergence properties. To accomplish this, a central cloud computer aggregates global information, carries out computations of the dual variables based on this information, and then distributes the updated dual variables to the agents. The agents update their (primal) state variables and also communicate among themselves with each agent sharing and receiving state information with some number of its neighbors. Throughout, communications with the cloud are not assumed to be synchronous or instantaneous, and communication delays are explicitly accounted for in the modeling and analysis of the system. Experimental results are presented to support the theoretical developments made.Comment: 8 pages, 4 figure

Combinatorial Path Planning for a System of Multiple Unmanned Vehicles

In this dissertation, the problem of planning the motion of m Unmanned Vehicles (UVs) (or simply vehicles) through n points in a plane is considered. A motion plan for a vehicle is given by the sequence of points and the corresponding angles at which each point must be visited by the vehicle. We require that each vehicle return to the same initial location(depot) at the same heading after visiting the points. The objective of the motion planning problem is to choose at most q(≤ m) UVs and find their motion plans so that all the points are visited and the total cost of the tours of the chosen vehicles is a minimum amongst all the possible choices of vehicles and their tours. This problem is a generalization of the wellknown Traveling Salesman Problem (TSP) in many ways: (1) each UV takes the role of salesman (2) motion constraints of the UVs play an important role in determining the cost of travel between any two locations; in fact, the cost of the travel between any two locations depends on direction of travel along with the heading at the origin and destination, and (3) there is an additional combinatorial complexity stemming from the need to partition the points to be visited by each UV and the set of UVs that must be employed by the mission. In this dissertation, a sub-optimal, two-step approach to motion planning is presented to solve this problem:(1) the combinatorial problem of choosing the vehicles and their associated tours is based on Euclidean distances between points and (2) once the sequence of points to be visited is specified, the heading at each point is determined based on a Dynamic Programming scheme. The solution to the first step is based on a generalization of Held-Karp’s method. We modify the Lagrangian heuristics for finding a close sub-optimal solution. In the later chapters of the dissertation, we relax the assumption that all vehicles are homogenous. The motivation of heterogenous variant of Multi-depot, Multiple Traveling Salesmen Problem (MDMTSP) derives form applications involving Unmanned Aerial Vehicles (UAVs) or ground robots requiring multiple vehicles with different capabilities to visit a set of locations