Packing trees in communication networks

Given an undirected edge-capacitated graph and given (possibly) different subsets of vertices, we consider the problem of selecting a maximum (weighted) set of Steiner trees, each tree spanning a subset of vertices, without violating the capacity constraints. This problem is motivated by applications in multicast communication networks. We give an integer linear programming (ILP) formulation for the problem, and observe that its linear programming (LP) relaxation is a fractional packing problem with exponentially many variables and a block (sub-)problem that cannot be solved in polynomial time. To this end, we take an r-approximate block solver (a weak block solver) to develop a (1−ε)/r-approximation algorithm for the LP relaxation. The algorithm has a polynomial coordination complexity for any ε∈(0,1). To the best of our knowledge, this is the first approximation result for fractional packing problems with only weak block solvers (with arbitrarily large approximation ratio) and a coordination complexity that is polynomial in the input size. This leads also to an approximation algorithm for the underlying tree packing problem. Finally, we extend our results to an important multicast routing and wavelength assignment problem in optical networks, where each Steiner tree is to be assigned one of a limited set of given wavelengths, so that trees crossing the same fiber are assigned different wavelengths. A preliminary version of this paper appeared in the Proceedings of the 1st Workshop on Internet and Network Economics (WINE 2005), LNCS, vol. 3828, pp. 688–697. Research supported by a MITACS grant for all the authors, an NSERC post doctoral fellowship for the first author, the NSERC Discovery Grant #5-48923 for the second and fourth author, NSERC Discovery Grant #15296 for the third author, the Canada Research Chair Program for the second author, and an NSERC industrial and development fellowship for the fourth author.     

An Improved Approximation Algorithm for Multicast <Emphasis Type="Italic">k</Emphasis>-Tree Routing

An improved approximation algorithm is presented in this paper for the multicast k-tree routing problem. The algorithm has a worst case performance ratio of (2.4 + ρ), where ρ is the best approximation ratio for the metric Steiner tree problem (and is about 1.55 so far). The previous best approximation algorithm for the multicast k-tree routing problem has a performance ratio of 4. Two techniques, weight averaging and tree partitioning, are developed to facilitate the algorithm design and analysis.Research supported by AICML, CFI, NSERC, PENCE, a Startup Grant from the University of Alberta, and NNSF Grant 60373012.     

Approximation algorithms for multicast routing in ad hoc wireless networks

Energy efficient multicast problem is one of important issues in ad hoc networks. In this paper, we address the energy efficient multicast problem for discrete power levels in ad hoc wireless networks. The problem of our concern is: given n nodes deployed over 2-D plane and each node v has l(v) transmission power levels and a multicast request (s,D) (clearly, when D is V∖{s}, the multicast request is a broadcast request), how to find a multicast tree rooted at s and spanning all destinations in D such that the total energy cost of the multicast tree is minimized. We first prove that this problem is NP-hard and it is unlikely to have an approximation algorithm with performance ratio ρlnn(ρ<1). Then, we propose a general algorithm for the multicast/broadcast tree problem. And based on the general algorithm, we propose an approximation algorithm and a heuristics for multicast tree problem. Especially, we also propose an efficient heuristic for broadcast tree problem. Simulations ensure our algorithms are efficient.     

Reliability evaluation of a multicast over coded packet networks

Network coding is a generalization of conventional routing methods that allows a network node to code information flows before forwarding them. While it has been theoretically proved that network coding can achieve maximum network throughput, theoretical results usually do not consider the stochastic nature in information processing and transmission, especially when the capacity of each arc becomes stochastic due to failure, attacks, or maintenance. Hence, the reliability measurement of network coding becomes an important issue to evaluate the performance of the network under various system settings. In this paper, we present analytical expressions to measure the reliability of multicast communications in coded networks, where network coding is most promising. We define the probability that a multicast rate can be transmitted through a coded packet network under a total transmission cost constraint as the reliability metric. To do this, we first introduce an exact mathematical formulation to construct multicast connections over coded packet networks under a limited transmission cost. We then propose an algorithm based on minimal paths to calculate the reliability measurement of multicast connections and analyze the complexity of the algorithm. Our results show that the reliability of multicast routing with network coding improved significantly compared to the case of multicast routing without network coding.     

Minimum failure explanations for path vector routing changes

Path vector protocols in routing networks convey entire path information to each destination. When links fail, affected paths are replaced by new paths, and by observing the entire path information, one might hope to infer the failed links that caused these changes. However, inferring the exact topological changes behind observed routing changes may not be possible due to limited information, and the same changes may be explained by more than one set of candidate failures. In this paper, using a simple path vector routing model, we present the problem of finding the candidate set with minimum number of failures to explain observed route changes. We call this problem the minimum e-set problem and present algorithms for solving it optimally for certain cases. We also show that the minimum e-set problem is NP-complete in the general case. This material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No N66001-04-1-8926 and by National Science Fundation(NSF) under Contract No ANI-0221453. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the DARPA or NSF. Part of the work was done when Akash Nanavati was at DA-IICT, India     

Inapproximability and approximability of maximal tree routing and coloring

Let G be a undirected connected graph. Given g groups each being a subset of V(G) and a number of colors, we consider how to find a subgroup of subsets such that there exists a tree interconnecting all vertices in each subset and all trees can be colored properly with given colors (no two trees sharing a common edge receive the same color); the objective is to maximize the number of subsets in the subgroup. This problem arises from the application of multicast communication in all optical networks. In this paper, we first obtain an explicit lower bound on the approximability of this problem and prove Ω(g^1−ε)-inapproximability even when G is a mesh. We then propose a simple greedy algorithm that achieves performance ratio O√|E(G)|, which matches the theoretical bounds. Supported in part by the NSF of China under Grant No. 70221001 and 60373012.     

Broadcast Routing with Minimum Wavelength Conversion in WDM Optical Networks

We consider dynamic routing of broadcast connections in WDM optical networks. Given the current network state, we want to find a minimum set of network nodes S such that a broadcast routing using only the nodes in S as wavelength conversion nodes can be found. This ensures that the average conversion delay from the source to all destinations is minimized. We refer to the problem as Broadcast Conversion Node Selection (BCNS) problem. We prove that BCNS has no polynomial-time approximation with performance ratio ln n for < 1 unless NPDTIME(n^{O(log log n)}) where n is the number of vertices in the input graph. We present a greedy approximation algorithm for BCNS that achieves approximation ratio 2+ln n. Simulation results show that the algorithm performs very well in practice, obtaining optimal solutions in most of the cases.     

Solution Structure of Some Inverse Combinatorial Optimization Problems

In this paper we consider some inverse combinatorial optimization problems, that is, for a given set of feasible solutions of an optimization problem, we wish to know: under what weight vectors (or capacity vectors) will these feasible solutions become optimal solutions? We analysed shortest path problem, minimum cut problem, minimum spanning tree problem and maximum-weight matching problem, and found that in each of these cases, the solution set of the inverse problem is charaterized by solving another combinatorial optimization problem. The main tool in our approach is Fulkerson's theory of blocking and anti-blocking polyhedra with some necessary revisions.     

A Genetic Algorithm for the Weight Setting Problem in OSPF Routing

With the growth of the Internet, Internet Service Providers (ISPs) try to meet the increasing traffic demand with new technology and improved utilization of existing resources. Routing of data packets can affect network utilization. Packets are sent along network paths from source to destination following a protocol. Open Shortest Path First (OSPF) is the most commonly used intra-domain Internet routing protocol (IRP). Traffic flow is routed along shortest paths, splitting flow at nodes with several outgoing links on a shortest path to the destination IP address. Link weights are assigned by the network operator. A path length is the sum of the weights of the links in the path. The OSPF weight setting (OSPFWS) problem seeks a set of weights that optimizes network performance. We study the problem of optimizing OSPF weights, given a set of projected demands, with the objective of minimizing network congestion. The weight assignment problem is NP-hard. We present a genetic algorithm (GA) to solve the OSPFWS problem. We compare our results with the best known and commonly used heuristics for OSPF weight setting, as well as with a lower bound of the optimal multi-commodity flow routing, which is a linear programming relaxation of the OSPFWS problem. Computational experiments are made on the AT&T Worldnet backbone with projected demands, and on twelve instances of synthetic networks.     

An Approximation for Minimum Multicast Route in Optical Networks with Nonsplitting Nodes

Consider the problem of computing the minimum-weight multicast route in an optical network with both nonsplitting and splitting nodes. This problem can be reduced to the minimum Hamiltonian path problem when all nodes are nonsplitting, and the Steiner minimum tree problem when all nodes are splitting. Therefore, the problem is NP-hard. Previously, the best known polynomial-time approximation has the performance ratio 3. In this paper, we present a new polynomial-time approximation with performance ratio of 1+ρ, where ρ is the best known approximation performance ratio for the Steiner minimum tree in graph and it has been known that ρ < 1.55. Support in part by National Science Foundation under grants CCF-0514796 and CNS-0524429     

Reliable Ad Hoc Routing Based on Mobility Prediction

Reliability is a very important issue in Mobile Ad hoc NETworks (MANETs). Shortest paths are usually used to route packets in MANETs. However, a shortest path may fail quickly, because some of the wireless links along a shortest path may be broken shortly after the path is established due to mobility of mobile nodes. Rediscovering routes may result in substantial data loss and message exchange overhead. In this paper, we study reliable ad hoc routing in the urban environment. Specifically, we formulate and study two optimization problems. In the minimum Cost Duration-bounded Path (CDP) routing problem, we seek a minimum cost source to destination path with duration no less than a given threshold. In the maximum Duration Cost-bounded Path (DCP) routing problem, we seek a maximum duration source to destination path with cost no greater than a given threshold. We use a waypoint graph to model the working area of a MANET and present an offline algorithm to compute a duration prediction table for the given waypoint graph. An entry in the duration prediction table contains the guaranteed worst-case duration of the corresponding wireless link. We then present an efficient algorithm which computes a minimum cost duration-bounded path, using the information provided in the duration prediction table. We also present a heuristic algorithm for the DCP routing problem. In addition, we show that the proposed prediction and routing schemes can be easily applied for designing reliable ad hoc routing protocols. Simulation results show that our mobility prediction based routing algorithms lead to higher network throughput and longer average path duration, compared with the shortest path routing. This research was supported in part by ARO grant W911NF-04-1-0385 and NSF grant CCF-0431167. The information reported here does not reflect the position or the policy of the federal government.     

Exact and heuristic algorithms for solving the generalized minimum filter placement problem

We consider a problem of placing route-based filters in a communication network to limit the number of forged address attacks to a prescribed level. Nodes in the network communicate by exchanging packets along arcs, and the originating node embeds the origin and destination addresses within each packet that it sends. In the absence of a validation mechanism, one node can send packets to another node using a forged origin address to launch an attack against that node. Route-based filters can be established at various nodes on the communication network to protect against these attacks. A route-based filter examines each packet arriving at a node, and determines whether or not the origin address could be legitimate, based on the arc on which the packet arrives, the routing information, and possibly the destination. The problem we consider seeks to find a minimum cardinality subset of nodes to filter so that the prescribed level of security is achieved. We formulate a mixed-integer programming model for the problem and derive valid inequalities for this model by identifying polynomially-solvable subgraphs of the communication network. We also present three heuristics for solving the filter placement problem and evaluate their performance against the optimal solution provided by the mixed-integer programming model. The authors gratefully acknowledge the comments of two anonymous referees, whose input led to an improved version of this paper. Dr. Smith gratefully acknowledges the support of the Office of Naval Research under Grant #N00014-03-1-0510 and the Defense Advanced Research Projects Agency under Grant #N66001-01-1-8925.     

Directed Steiner trees with diffusion costs

Given a directed arc-weighted graph G with n nodes, a root r and k terminals, the directed steiner tree problem (DST) consists in finding a minimum-weight tree rooted at r and spanning all the terminals. If this problem has several applications in multicast routing in packet switching networks, the modeling is not adapted anymore in networks based upon the circuit switching principle in which some nodes, called non diffusing nodes, are unable to duplicate packets. We define a more general problem, namely the directed steiner tree with a limited number of diffusing nodes (DSTLD), that enables us to model multicast in a network containing at most d diffusing nodes. We show that DSTLD is XP with respect to d, and use this result to build a \(\left\lceil \frac{k-1}{d} \right\rceil \)-approximation algorithm for DST that is XP in d. We deduce from that result a strong inapproximability property. In particular, we prove that, under the assumption that NP \(\not \subseteq \) ZTIME \([n^{\log ^{O(1)}n}]\), there is no polynomial-time approximation algorithm for DSTLD with ratio \(\varOmega \left( \frac{k}{d}\right) \). We finally give an evaluation of performances of an exact algorithm dedicated to the case \(d \le 3\).     

Improving robustness of next-hop routing

A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple \(\frac{1}{2}\)-approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.     

Traffic Partition in WDM/SONET Rings to Minimize SONET ADMs

SONET (Synchronous Optical NETworks) add-drop multiplexers (ADMs) are the dominant cost factor in the WDM(Wavelength Division Multiplexing)/SONET rings. The number of SONET ADMs required by a set of traffic streams is determined by the routing and wavelength assignment of the traffic streams. Previous works took as input the traffic streams with routings given a priori and developed various heuristics for wavelength assignment to minimize the SONET ADM costs. However, little was known about the performance guarantees of these heuristics. This paper contributes mainly in two aspects. First, in addition to the traffic streams with pre-specified routing, this paper also studies minimizing the ADM requirement by traffic streams without given routings, a problem which is shown to be NP-hard. Several heuristics for integrated routing and wavelength assignment are proposed to minimize the SONET ADM costs. Second, the approximation ratios of those heuristics for wavelength assignment only and those heuristics for integrated routing and wavelength assignment are analyzed. The new Preprocessed Iterative Matching heuristic has the best approximation ratio: at most 3/2.     

Modeling the problem of locating collection areas for urban waste management. An application to the metropolitan area of Barcelona

Reverse logistics problems arising in municipal waste management are both wide-ranging and varied. The usual collection system in UE countries is composed of two phases. First, citizens leave their refuse at special collection areas where different types of waste (glass, paper, plastic, organic material) are stored in special refuse bins. Subsequently, each type of waste is collected separately and moved to its final destination (a recycling plant or refuse dump). The present study focuses on the problem of locating these collection areas. We establish the relationship between the problem, the set covering problem and the MAX-SAT problem and then go on to develop a genetic algorithm and a GRASP heuristic to, respectively, solve each formulation. Finally, the quality of the algorithms is tested in a computational experience with real instances from the metropolitan area of Barcelona, as well as a reduced set of set covering instances from the literature.     

Maximizing Profits of Routing in WDM Networks

Let G = (V, E) be a ring (or chain) network representing an optical wavelength division multiplexing (WDM) network with k channels, where each edge e_j has an integer capacity c_j. A request s_i,t_i is a pair of two nodes in G. Given m requests s_i,t_i, i = 1, 2, ..., m, each with a profit value p_i, we would like to design/route a k-colorable set of paths for some (may not be all) of the m requests such that each edge e_j in G is used at most c_j times and the total profit of the set of designed paths is maximized. Here two paths cannot have the same color (channel) if they share some common edge(s).This problem arises in optical communication networks. In this paper, we present a polynomial-time algorithm to solve the problem when G is a chain. When G is a ring, however, the optimization problem is NP-hard (Wan and Liu, 1998), we present a 2-approximation algorithm based on our solution to the chain network. Similarly, some results in a bidirected chain and a bidirected ring are obtained.     

动态需求下车辆路径问题的周期性优化模型及求解

针对客户点不断更新的动态需求车辆路径问题,依据滚动时域对配送中心工作时间进行划分,提出基于延迟服务的周期性客户点实时重置策略,策略中延迟服务机制能结合车辆启动延迟系数对照当前时域的时间进行检验,满足所有客户点的服务需求,保证车辆满足中心时间窗约束。设计多阶段求解的混合变邻域人工蜂群算法对各时间片内子问题进行连续迭代优化,算法中子路径动态转变的设计能较好平衡原有客户点和新客户点对路径更新和车辆实时信息匹配的要求。算例验证及对比分析表明本文策略和算法在求解动态问题时的有效性和可行性。     

Multi-depot vehicle routing problem with time windows under shared depot resources

A new variant of multi-depot vehicle routing problem with time windows is studied. In the new variant, the depot where the vehicle ends is flexible, namely, it is not entirely the same as the depot that it starts from. An integer programming model is formulated with the minimum total traveling cost under the constrains of time window, capacity and route duration of the vehicle, the fleet size and the number of parking spaces of each depot. As the problem is an NP-Hard problem, a hybrid genetic algorithm with adaptive local search is proposed to solve it. Finally, the computational results show that the proposed method is competitive in terms of solution quality. Compared with the classic MDVRPTW, allowing flexible choice of the stop depot can further reduce total traveling cost.     

A multi-objective vehicle routing and scheduling problem with uncertainty in customers’ request and priority

In this paper, a multi-objective vehicle routing and scheduling problem with uncertainty in priority and request of customers is presented. In the proposed model, a set of dynamic requests is received over time, and the planner does not have any information regarding their location and size until they arrive. Moreover, the routing model aims to satisfy different customers according to their specific time windows which were predefined by an expert as (being very important, important, casual or unimportant). This paper uses the proposed model as a multi-objective problem where the total required number of vehicles, the total distance travelled and the waiting time imposed on vehicles are minimized, and the total customers’ satisfaction for service is maximized. An efficient framework for solving this model is designed and its performance is evaluated in different steps for various test problems generalized from Solomon’s VRPTW benchmark problems. The various heuristics and improvement concepts incorporate local exploitation in the evolutionary search, and the concept of Pareto optimality for the multi-objective optimization is used in the proposed procedure. The computational experiments on data sets illustrate the efficiency and effectiveness of the proposed approach.