Length 3 Edge-Disjoint Paths Is NP-Hard

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and in this note we show that this problem is NP-hard. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard

Military route planning in battlefield simulation: effectiveness problems and potential solutions, Journal of Telecommunications and Information Technology, 2003, nr 4

Path searching is challenging problem in many domains such as simulation war games, robotics, military mission planning, computer generated forces (CGF), etc. Effectiveness problems in military route planning are related both with terrain modelling and path planning algorithms. These problems may be considered from the point of view of many criterions. It seems that two criterions are the most important: quality of terrain reflection in the terrain model and computational complexity of the on(off)-line path planning algorithm. The paper deals with two above indicated problems of route planning effectiveness. Comparison of approaches used in route planning is presented. The hybrid, terrain merging-based and partial path planning, approach for route planning in dynamically changed environment during simulation is described. It significantly increase effectiveness of route planning process. The computational complexity of the method is given and some discussion for using the method in the battlefield simulation is conducted. In order to estimate how many times faster we can compute problem for finding shortest path in network with n big squares (b-nodes) with relation to problem for finding shortest path in the network with V small squares (s-nodes) acceleration function is defined and optimized

Journal of Telecommunications and Information Technology, 2003, nr 4

kwartalni

path-constrained network flows

This thesis focuses on approximation algorithms and complexity assessments concerning network flows. It deals with various network flow problems with path restrictions. These restrictions cover the number of paths that are used to route commodities as well as the amount of flow that is routed along a single path or the path's length. Concerning the first restriction we study the unsplittable flow problem-a generalization of the NP-hard edge-disjoint paths problem. Given a network with commodities that must be routed from their sources to their sinks, the unsplittable flow problem forbids each commodity to use more than one path. For this problem we prove a new lower bound on the performance guarantee of randomized rounding which so far belongs to the best approximation algorithms known for this problem. Further, we present an interesting relation between unsplittable flows and classical (splittable) multicommodity flows in the case that all commodities share a common source: Each single source multicommodity flow can be represented as a convex combination of unsplittable flows of congestion at most 2. Further, we combine different path restrictions from the ones mentioned above. In the k-splittable flow problem with path capacities, we study the NP-hard problem that each commodity may be sent along a limited number of paths while the flow value of each path is bounded. This yields a generalization of the unsplittable flow problem, but we show how one can obtain the same asymptotic approximation ratios. For the length-bounded k-splittable flow problem, we consider the single commodity case and develop a constant factor approximation algorithm. A crucial characteristic of network flows occurring in real-world applications is flow variation over time and the fact that flow does not travel instantaneously through a network but requires a certain amount of time to travel through each arc. Both characteristics are captured by "flows over time" which specify a flow rate for each arc and each point in time. We consider the quickest single commodity k-splittable flow problem and give a constant factor approximation algorithm for it. So far only results for k-splittable flows as well as for length-bounded flows and flows over time have been known, but nothing was known for combinations of them. Bounding the flow value of each path is also interesting in the classical maximum s-t-flow problem. We study the case that each path may carry at most one unit of flow and prove that this restriction makes the maximum s-t-flow problem strongly NP-hard. In contrast to the classical maximum s-t-flow problem, the fractional and the integral problem diverge strongly with the new restriction. For the integral problem, we even prove APX-hardness. We develop an FPTAS for the fractional problem and an O(log m)-approximation algorithm for the integral one. (Here, m is the number of arcs in the network under consideration.) Similar results emerge for the multicommodity case. For the objective to find a maximum integral multicommodity flow our asymptotic approximation ratio of O(m^{0.5}) is proven to be best possible, unless P = NP

Modelos de programação linear inteira para problemas de dimensionamento e engenharia de tráfego de redes de telecomunicações

Tese de doutoramento, Estatística e Investigação Operacional (Optimização), 2009, Universidade de Lisboa, Faculdade de CiênciasIn this thesis, we develop Integer Linear Programming models to address network dimensioning and traffic engineering problems in telecommunications networks. Given the ever increasing need for reliable and delay sensitive services, our main focus is on survivability and delay constraints. We study the theoretical properties of the proposed models and we test their efficiency. In particular: 1. We analyse different classes of (sub)models which guarantee the existence of D paths between two nodes; 2. Based on those submodels, we compare the different formulations thus obtained for a MPLS network design problem; 3. In the context of a MPLS over WDM network dimensioning problem, we analyse the extent to which network costs are determined by different survivability mechanisms. We also develop a heuristic approach to this problem; 4. We study two classes of models disaggregated and aggregated to address two traffic engineering problems defined over the network solutions mentioned in 3. The theoretical and computational results discussed in this thesis clearly demonstrate, we believe, the value of good problem formulations using Integer Linear Programming.Fundação para a Ciência e Tecnologia (SFRH/ BD/ 28837/ 2006