On the Hardness of Equal Shortest Path Routing

In telecommunication networks packets are carried from a source s to a destination t on a path that is determined by the underlying routing protocol. Most routing protocols belong to the class of shortest path routing protocols. In such protocols, the network operator assigns a length to each link. A packet going from s to t follows a shortest path according to these lengths. For better protection and efficiency, one wishes to use multiple (shortest) paths between two nodes. Therefore the routing protocol must determine how the traffic from s to t is distributed among the shortest paths. In the protocol called OSPF-ECMP (for Open Shortest Path First-Equal Cost Multiple Path) the traffic incoming at every node is uniformly balanced on all outgoing links that are on shortest paths. In that context, the operator task is to determine the “best” link lengths, toward a goal such as maximizing the network throughput for given link capacities.In this work, we show that the problem of maximizing even a single commodity flow for the OSPF-ECMP protocol cannot be approximated within any constant factor ratio. Besides this main theorem, we derive some positive results which include polynomial-time approximations and an exponential-time exact algorithm. We also prove that despite their weakness, our approximation and exact algorithms are, in a sense, the best possible.     

Increasing Internet Capacity Using Local Search

Open Shortest Path First (OSPF) is one of the most commonly used intra-domain internet routing protocol. Traffic flow is routed along shortest paths, splitting flow evenly at nodes where several outgoing links are on shortest paths to the destination. The weights of the links, and thereby the shortest path routes, can be changed by the network operator. The weights could be set proportional to the physical lengths of the links, but often the main goal is to avoid congestion, i.e. overloading of links, and the standard heuristic recommended by Cisco (a major router vendor) is to make the weight of a link inversely proportional to its capacity.We study the problem of optimizing OSPF weights for a given a set of projected demands so as to avoid congestion. We show this problem is NP-hard, even for approximation, and propose a local search heuristic to solve it. We also provide worst-case results about the performance of OSPF routing vs. an optimal multi-commodity flow routing. Our numerical experiments compare the results obtained with our local search heuristic to the optimal multi-commodity flow routing, as well as simple and commonly used heuristics for setting the weights. Experiments were done with a proposed next-generation AT&T WorldNet backbone as well as synthetic internetworks.     

Intra-domain traffic engineering with shortest path routing protocols

Throughout the last decade, extensive deployment of popular intra-domain routing protocols such as open shortest path first and intermediate system–intermediate system, has drawn an ever increasing attention to Internet traffic engineering. This paper reviews optimization techniques that have been deployed for managing intra-domain routing in networks operated with shortest path routing protocols, and the state-of-the-art research that has been carried out in this direction.     

Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints, such as capacity or time windows, are solved via column generation and branch-and-price, it is common that the pricing subproblem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the vertices. A common solution technique for this problem is dynamic programming. In this paper we illustrate how the basic dynamic programming algorithm can be improved by bounded bi-directional search and we experimentally evaluate the effectiveness of the enhancement proposed. We consider as benchmark problems the elementary shortest path problems arising as pricing subproblems in branch-and-price algorithms for the capacitated vehicle routing problem, the vehicle routing problem with distribution and collection and the capacitated vehicle routing problem with time windows.     

Intra-domain traffic engineering with shortest path routing protocols

Throughout the last decade, extensive deployment of popular intra-domain routing protocols such as open shortest path first (OSPF) and intermediate system–intermediate system (IS-IS), has drawn an ever increasing attention to internet traffic engineering. This paper reviews optimization techniques that have been deployed for managing intra-domain routing in networks operated with shortest path routing protocols, and the state-of-the-art research that has been carried out in this direction.     

满足路径约束的最优路算法

满足路径约束的最优路问题已被证明是NP-hard问题。本针对源点到宿点满足两个QoS(服务质量)度量的路由问题，给出一种保证时延的最小费用路由启发式算法。这个算法的优点是计算较简单、占用内存小、时间短。算法的复杂度是多项式的，表明算法是有效的。     

A note on the separation of subtour elimination constraints in elementary shortest path problems

This note proposes an alternative procedure for identifying violated subtour elimination constraints (SECs) in branch-and-cut algorithms for elementary shortest path problems. The procedure is also applicable to other routing problems, such as variants of travelling salesman or shortest Hamiltonian path problems, on directed graphs. The proposed procedure is based on computing the strong components of the support graph. The procedure possesses a better worst-case time complexity than the standard way of separating SECs, which uses maximum flow algorithms, and is easier to implement.     

On single-path network routing subject to max-min fair flow allocation

Fair allocation of flows in multicommodity networks has been attracting a growing attention. In Max-Min Fair (MMF) flow allocation, not only the flow of the commodity with the smallest allocation is maximized but also, in turn, the second smallest, the third smallest, and so on. Since the MMF paradigm allows to approximate the TCP flow allocation when the routing paths are given and the flows are elastic, we address the network routing problem where, given a graph with arc capacities and a set of origin-destination pairs with unknown demands, we must route each commodity over a single path so as to maximize the throughput, subject to the constraint that the flows are allocated according to the MMF principle. After discussing two properties of the problem, we describe a column generation based heuristic and report some computational results.     

Path problems in generalized stars, complete graphs, and brick wall graphs

Path problems such as the maximum edge-disjoint paths problem, the path coloring problem, and the maximum path coloring problem are relevant for resource allocation in communication networks, in particular all-optical networks. In this paper, it is shown that maximum path coloring can be solved optimally in polynomial time for bidirected generalized stars, even in the weighted case. Furthermore, the maximum edge-disjoint paths problem is proved NP-hard for complete graphs (undirected or bidirected), a constant-factor off-line approximation algorithm is presented for the weighted case, and an on-line algorithm with constant competitive ratio is given for the unweighted case. Finally, an open problem concerning the existence of routings that simultaneously minimize the maximum load and the number of colors is solved: an example for a graph and a set of requests is given such that any routing that minimizes the maximum load requires strictly more colors for path coloring than a routing that minimizes the number of colors.     

Reliability evaluation according to a routing scheme for multi-state computer networks under assured accuracy rate

In many real-time networks such as computer networks, each arc has stochastic capacity, lead time, and accuracy rate. Such a network is named a multi-state computer network (MSCN). Under the strict assumption that the capacity of each arc is deterministic, the quickest path (QP) problem is to find a path that sends a specific amount of data with minimum transmission time. From the viewpoint of internet quality, the transmission accuracy rate is one of critical performance indicators to assess internet network for system administrators and customers. Under both assured accuracy rate and time constraint, this paper extends the QP problem to discuss the flow assignment for a MSCN. An efficient algorithm is proposed to find the minimal capacity vector meeting such requirements. The system reliability, the probability to send \(d\) units of data through multiple minimal paths under both assured accuracy rate and time constraint, can subsequently be computed. Furthermore, two routing schemes with spare minimal paths are adopted to reinforce the system reliability. The enhanced system reliability according to the routing scheme is calculated as well. The computational complexity in both the worst case and average case are analyzed.     

Bottleneck routing with elastic demands

Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing her fixed demand. The disutility of a user only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria.     

An origin-based model for unique shortest path routing

Link weights are the main parameters of shortest path routing protocols, the most commonly used protocols for IP networks. The problem of optimally setting link weights for unique shortest path routing is addressed. Due to the complexity of the constraints involved, there exist challenges to formulate the problem in such a way based on which a more efficient solution algorithm than the existing ones may be developed. In this paper, an exact formulation is first introduced and then mathematically proved correct. It is further illustrated that the formulation has advantages over a prior one in terms of both constraint structure and model size for a proposed decomposition method to solve the problem.     

Sensible decisions based on QoS

Network Quality of Service (QoS) criteria of interest include conventional metrics such as throughput, delay, loss, and jitter, as well as new QoS criteria based on power utilization, reliability and security. Variable and adaptive routing have again become of interest in networking because of the increasing importance of mobile ad-hoc networks. In this paper we develop a probability model of adaptive routing algorithms which use the expected QoS to select paths in the network. Our objective is not to analyze QoS, but rather to design randomized routing policies which can improve QoS. We define QoS metrics as non-negative random variables associated with network paths which satisfy a sub-additivity condition along each path. We define the QoS of a path, under some routing policy, as the expected value of a non-decreasing measurable function of the QoS metric. We discuss sensitive and insensitive QoS metrics, the latter being dependent on the routing policy which is used. We describe routing policies simply as probabilistic choices among all possible paths from some source to some given destination. Incremental routing policies are defined as those which can be derived from independent decisions taken at certain points (or nodes) along paths. Sensible routing policies are then introduced: they take decisions based simply on the QoS of each available path. Sensible policies, which make decisions based on the QoS of the paths, are introduced. We prove that the routing probability of a sensible policy can always be uniquely obtained. A hierarchy of m-sensible probabilistic routing policies is then introduced. A 0-sensible policy is simply a random choice of routes with equal probability, while a 1-sensible policy selects a path with a probability which is inversely proportional to the (expected) QoS of the path. We prove that an m + 1-sensible policy provides better QoS on the average than an m-sensible policy, if the QoS metric is insensitive. We also show that under certain conditions, the same result also holds for sensitive QoS metrics.Accepted: May 2003, This work was supported by the U.S. Army and Navy under contracts N611339-00-K-0002, N61339-02-C0050, N61339-02-C0080, N61339-02-C0117, and by NSF grants EIA0086251, EIA0203446, ECS0216381.     

A New Multiobjective Dynamic Routing Method for Multiservice Networks: Modelling and Performance

There are potential advantages in formulating the routing problems in modern multiservice networks as multiple objective problems. This paper presents a novel hierarchical bi-level multiobjective dynamic routing model for multiservice networks. It is based on a bi-objective shortest path algorithm, with dynamically adapted soft-constraints, to compute alternative paths for each node pair and on a heuristic to synchronously select alternative routing plans for the network in a dynamic alternative routing context. It is a routing method which periodically changes alternative paths as a function of periodic updates of certain QoS related parameters obtained from real-time measurements. The performance of the proposed routing method is compared with two reference dynamic routing methods namely RTNR and DAR by means of a discrete-event simulator.A previous short version of this work was presented at INOC’03 (International Network Optimisation Conference). Work partially supported by programme POSI of the III EC programme cosponsored by FEDER and national funds.     

A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

Given an undirected graph with nonnegative edge lengths and nonnegative vertex weights, the routing requirement of a pair of vertices is assumed to be the product of their weights. The routing cost for a pair of vertices on a given spanning tree is defined as the length of the path between them multiplied by their routing requirement. The optimal product-requirement communication spanning tree is the spanning tree with minimum total routing cost summed over all pairs of vertices. This problem arises in network design and computational biology. For the special case that all vertex weights are identical, it has been shown that the problem is NP-hard and that there is a polynomial time approximation scheme for it. In this paper we show that the generalized problem also admits a polynomial time approximation scheme.     

Path relinking for the vehicle routing problem

This paper describes a tabu search heuristic with path relinking for the vehicle routing problem. Tabu search is a local search method that explores the solution space more thoroughly than other local search based methods by overcoming local optima. Path relinking is a method to integrate intensification and diversification in the search. It explores paths that connect previously found elite solutions. Computational results show that tabu search with path relinking is superior to pure tabu search on the vehicle routing problem.     

基于聚类和动态规划的组合路径策略

为提高电商仓库的拣货作业效率,本文提出了基于聚类和动态规划的组合路径策略,实现了生成路径消耗时间和路径长度之间的平衡,并将这一策略成功地应用到多区型仓库。该策略分四步:首先,根据待拣储位分布特征,运用聚类分析法对其进行分类;然后,以各类的首末储位作为节点,运用动态规划法对已得分类进行排序,得到相应的类序;其次,得到各类内部路径;最后,依次拣取待拣商品,并返回出发点完成拣货作业。在提出新的路径策略后,通过仿真方法将新策略与三种传统路径策略(穿越策略、最大间隙策略和混合策略)和一种优质算法(蚁群算法)进行了对比分析,结果表明:该策略具备良好的适用性和实用性。     

On a bi-dimensional dynamic alternative routing method

The analysis of a bi-dimensional dynamic routing model for alternative routing telecommunication networks led to the identification of an instability problem in the synchronous path selection associated with the complex interdependencies among the coefficients of the objective functions and the computed paths for every node pair. In this paper an analytical model enabling to make explicit this problem and evaluate its effects in terms of two global network criteria, is presented. Also a heuristic procedure dedicated to overcome this instability problem and select “good” compromise solutions in terms of network performance is developed. Finally the performance of the proposed routing method using the heuristic is compared by recurring to discrete-event simulation with a reference dynamic routing method (Real Time Network Routing) for some test networks.     

An optimal greedy routing algorithm for triangulated polygons

In this paper, we prove that a triangulated polygon G admits a greedy embedding into an appropriate semi-metric space such that using an appropriate distance definition, for any two vertices u and w in G, a most virtual distance decreasing path is always a minimum-edge path between u and w. Therefore, our greedy routing algorithm is optimal. The greedy embedding of G can be obtained in linear time. To the best of our knowledge, this is the first optimal greedy routing algorithm for a nontrivial subcategory of graphs.     

Multiobjective routing problems

Summary Many network routing problems, particularly where the transportation of hazardous materials is involved, are multiobjective in nature; that is, it is desired to optimise not only physical path length but other features as well. Several such problems are defined here and a general framework for multiobjective routing problems is proposed. The notion of “efficient solution” is defined and it is demonstrated, by means of an example, that a problem may have very many solutions which are efficient. Next, potentially useful solution methods for multiobjective routing problems are discussed with emphasis being placed on the use of shortest/k-shortest path techniques. Finally, some directions for possible further research are indicated. Invited by B. Pelegrin