38,710 research outputs found
A rigorous analysis of the cavity equations for the minimum spanning tree
We analyze a new general representation for the Minimum Weight Steiner Tree
(MST) problem which translates the topological connectivity constraint into a
set of local conditions which can be analyzed by the so called cavity equations
techniques. For the limit case of the Spanning tree we prove that the fixed
point of the algorithm arising from the cavity equations leads to the global
optimum.Comment: 5 pages, 1 figur
A rigorous analysis of the cavity equations for the minimum spanning tree
We analyze a new general representation for the Minimum Weight Steiner Tree
(MST) problem which translates the topological connectivity constraint into a
set of local conditions which can be analyzed by the so called cavity equations
techniques. For the limit case of the Spanning tree we prove that the fixed
point of the algorithm arising from the cavity equations leads to the global
optimum.Comment: 5 pages, 1 figur
Essential Constraints of Edge-Constrained Proximity Graphs
Given a plane forest of points, we find the minimum
set of edges such that the edge-constrained minimum spanning
tree over the set of vertices and the set of constraints contains .
We present an -time algorithm that solves this problem. We
generalize this to other proximity graphs in the constraint setting, such as
the relative neighbourhood graph, Gabriel graph, -skeleton and Delaunay
triangulation. We present an algorithm that identifies the minimum set
of edges of a given plane graph such that for , where is the
constraint -skeleton over the set of vertices and the set of
constraints. The running time of our algorithm is , provided that the
constrained Delaunay triangulation of is given.Comment: 24 pages, 22 figures. A preliminary version of this paper appeared in
the Proceedings of 27th International Workshop, IWOCA 2016, Helsinki,
Finland. It was published by Springer in the Lecture Notes in Computer
Science (LNCS) serie
Prize-Collecting TSP with a Budget Constraint
We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint
Deep Q-Learning-based Distribution Network Reconfiguration for Reliability Improvement
Distribution network reconfiguration (DNR) has proved to be an economical and
effective way to improve the reliability of distribution systems. As optimal
network configuration depends on system operating states (e.g., loads at each
node), existing analytical and population-based approaches need to repeat the
entire analysis and computation to find the optimal network configuration with
a change in system operating states. Contrary to this, if properly trained,
deep reinforcement learning (DRL)-based DNR can determine optimal or
near-optimal configuration quickly even with changes in system states. In this
paper, a Deep Q Learning-based framework is proposed for the optimal DNR to
improve reliability of the system. An optimization problem is formulated with
an objective function that minimizes the average curtailed power. Constraints
of the optimization problem are radial topology constraint and all nodes
traversing constraint. The distribution network is modeled as a graph and the
optimal network configuration is determined by searching for an optimal
spanning tree. The optimal spanning tree is the spanning tree with the minimum
value of the average curtailed power. The effectiveness of the proposed
framework is demonstrated through several case studies on 33-node and 69-node
distribution test systems
Spanning trees with generalized degree constraints arising in the design of wireless networks
In this paper we describe a minimum spanning tree problem with generalized degree constraints which arises in the design of wireless networks. The signal strength on the receiver side of a wireless link decreases with the distance between transmitter and receiver. In order to work properly, the interference on the receiving part of the link must be under a given threshold. In order to guarantee this constraint, for each node we impose a degree constraint that depends on the ”length” of the links adjacent to the corresponding node, more precisely, nodes adjacent to long links must have a smaller degree and vice-versa. The problem is complicated by considering different signal strengths for each link. Increasing the strength in a link increases the cost of the link. However, it also reduces the maximum allowed degree on its end nodes. We create two models using adequate sets of variables, one may be considered an extended version of the other, and relate, from a theoretical perspective, the corresponding linear programming relaxations.FCT - POCTI-ISFL-1-152FCT - PTDC/EIA/64772/200
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Improving spanning trees by upgrading nodes
We study budget constrained optimal network upgrading problems. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. A general problem in this setting is the following. We are given an edge weighted graph G = (V, E) where nodes represent processors and edges represent bidirectional communication links. The processor at a node v {element_of} V can be upgraded at a cost of c(v). Such an upgrade reduces the delay of each link emanating from v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has the best performance with respect to some measure. We consider the problem under two measures, namely, the weight of a minimum spanning tree and the bottleneck weight of a minimum bottleneck spanning tree. We present approximation and hardness results for the problem. Our results are tight to within constant factors. We also show that these approximation algorithms can be used to construct good approximation algorithms for the dual versions of the problems where there is a budget constraint on the upgrading cost and the objectives are minimum weight spanning tree and minimum bottleneck weight spanning tree respectively
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