Genetic local search for multicast routing with pre-processing by logarithmic simulated annealing

Abstract

Over the past few years, several local search algorithms have been proposed for various problems related to multicast routing in the off-line mode. We describe a population-based search algorithm for cost minimisation of multicast routing. The algorithm utilises the partially mixed crossover operation (PMX) under the elitist model: for each element of the current population, the local search is based upon the results of a landscape analysis that is executed only once in a pre-processing step; the best solution found so far is always part of the population. The aim of the landscape analysis is to estimate the depth of the deepest local minima in the landscape generated by the routing tasks and the objective function. The analysis employs simulated annealing with a logarithmic cooling schedule (logarithmic simulated annealing—LSA). The local search then performs alternating sequences of descending and ascending steps for each individual of the population, where the length of a sequence with uniform direction is controlled by the estimated value of the maximum depth of local minima. We present results from computational experiments on three different routing tasks, and we provide experimental evidence that our genetic local search procedure that combines LSA and PMX performs better than algorithms using either LSA or PMX only.

Introduction

Multicast routing has become an important topic in combinatorial optimisation. A recent overview on multicast routing and associated optimisation algorithms has been presented by Oliveira and Pardalos [1]. The focus of this overview, as in most papers on multicast routing, are on-line algorithms [2], [3]. An early summary of problems and technical solutions related to multicast communication was given by Diot et al. [4]. Great effort has been undertaken to incorporate quality of service (QoS) into data communication networks such as ATM and IP networks [5], [6], [7], [8], [9], [10], [11]. Many multicast applications, such as video conferencing, distance-learning, and multimedia broadcasting are QoS-sensitive in nature and thus they should benefit from the QoS improvement in the underlying networks.

Designing multicast routing algorithms is a complex and challenging task. Among the various issues involved are the design of optimal routes taking into consideration different cost functions, the minimisation of network load and the avoidance of loops and traffic congestion, and the provision of basic support for reliable transmission. In the present paper, we focus on the design of optimal routes in the off-line mode, as discussed, e.g., in the survey [9] (see Section 2 therein) and [12], [13].

The problem of minimising the tree costs of single requests under the constraint that all path capacities are within a user-specified capacity bound, i.e. the requests are executed simultaneously, is referred to as the capacity constrained multicast routing problem (CCMRP) [1], [4], [14], [15].

The CCMRP can be formalised as a constrained Steiner tree problem, which is known to be NP-complete [16]. We note that in applications like video conferencing, multimedia broadcasting, and distance-learning the routing procedure is updated only from time to time, e.g. when new customers register to use one of the services. In such cases, off-line routing algorithms are an appropriate way to solve the routing problem. Since we are dealing with an NP-complete problem, local search methods are a natural choice to tackle the problem; see [9], [12], [13].

In [17], [18], [19], [20], [21], [22], [23], [24], algorithms utilising genetic algorithms (GA) or tabu search are presented. For an overview of search methods, in particular, GA applied to various problem settings in multicast routing, we refer the reader to [25, cf. p. 20–21 therein]. Here, we discuss only a few of the issues raised on this topic. We note that most of the papers are dealing with single trees, but not with routing multiple requests (trees) simultaneously.

The GA proposed in [17], [18] assume that several messages all have to be transferred from several sources to multiple destinations, and this has to be executed simultaneously without any order or priority for certain messages. The GA uses a population of chromosomes, where each chromosome is a permutation of the numbers that are assigned to the requests. The algorithms start with a subset of $k$ out of $n$ requests. By using a Steiner tree algorithm, the $k$ requests are routed in the order they appear in the chromosome (partial permutation). Then, to pairs of chromosomes the partially mixed crossover (PMX) operation and the new population (of the same fixed size) is generated by roulette wheel selection, where a sector of a “roulette wheel” is assigned to each offspring whose size is proportional to the fitness measure. The algorithm runs a fixed number of steps, and then $k$ is increased by one in order to check whether $(k+1)$ requests can be scheduled conflict-free. The same procedure is repeated for $(k+1)$ until either $k=n$, or repeated attempts to schedule simultaneously $k$ requests are unsuccessful. The search-based methods from [17], [18] are, in part, incorporated into our approach and are discussed in more detail in Section 4.1.

The paper by Ericsson et al. [19] demonstrates a variety of routing problems that can be tackled by GA. The authors apply GA to a routing problem where the link weights are assigned by the network operator, i.e. the problem setting is somewhat different from ours. Then the weight setting problem seeks a set of weights that optimises network performance. Given a set of projected demands, the weight assignment problem, with the objective of minimising network congestion, is NP-hard. The individuals of the population are weight vectors, where the range of components is from $1$ to $2^{16}-1$. The crossover operator acts on one elite and one non-elite parent and selects each component of the resulting weight vector according to independently chosen random numbers from $(0,1)$. The evaluation of the fitness function is rather complicated, since it involves the whole process of routing and the computation of arc loads. The method was successfully tested on the AT&T Worldnet backbone with projected demands, and on several synthetic networks.

Barolli et al. [20] focus on creating a robust path finding solution for mobile ad hoc networks (MANTETs). Since the nodes are mobile, the creation of routing paths is affected by the addition and deletion of nodes, i.e. the topology of the network may change rapidly and unexpectedly. Therefore, QoS is only guaranteed as long as a signal to the node actually exists. The authors propose a genetic algorithm for mobile ad hoc networks (GAMAN) where the network and, respectively, the individuals of the population are represented by trees. The GAMAN algorithm uses the single point crossover and a mutation operation where the “tree junctions” are chosen randomly in the range from zero up to $1/\ell$, for $\ell =\text{length of individuals}$. The algorithm employs the elitist model, where the individual with the highest fitness value in a population is left unchanged in the next generation. The simulation results show that the algorithm is reasonably fast on small to medium size networks.

Yang [21] devised a tabu search algorithm for finding a single, feasible multicast tree efficiently that satisfies a number of QoS constraints. The method is tested on randomly generated networks with $100$ nodes (and on $8\times 8$ meshes). A similar setting (Steiner tree computation under certain constraints by tabu search) has been investigated by Skorin-Kapov and Kos [22]. The tests on a large number of benchmark problems have shown that the tabu search heuristic from [22] is superior in quality for medium sized problems.

Wang et al. [23] discuss the same problem as in [21], [22]. The authors propose an efficient algorithm based on tabu search for delay constrained, low cost multicast trees (TSDLMRA). To evaluate the efficiency of TSDLMRA, the authors utilise a random link generator, which yields networks with an average node degree of 4–6. The link delay function is defined as the propagation delay of the link. The TSDLMRA algorithm is shown to be of low time complexity, with the ability to find multicast trees if such solutions exist.

Yang and Wen [12] apply tabu search to the problem of pre-planning delay-constrained backup paths for multicast trees to minimise the total cost of all the backup paths. The neighbourhood structure of the search algorithm is based upon the random selection of a single link in the current solution for backup paths. The computational experiments were carried out on networks with 30–50 nodes.

Apart from GA and tabu search, simulated annealing-based search [26], [27], [28], [29], [30] has been utilised recently for multicast routing, in particular, under QoS considerations [13], [31], [32], [33], [34], [35]. In [31], the QoS issue is reduced to a path constraint problem (multiple requests are not considered), where along a path from source to destination each link has to obey a vector of weight restrictions. The constraint vector is transformed into an energy function by a max-operation over the component-wise ratio of link weights and capacity constraints. The search for appropriate paths is then executed by simulated annealing. The paper [32] demonstrates how different QoS requirements, like available CPU resources, buffer resources, error rates, queuing delay and sending delay at each node as well as available bandwidth, transmission delay and error rate at each link, can be incorporated into a single energy function for a given potential multicast routing solution (see Section 3.2 for more details). Simulated annealing is then applied to this energy function (the experiments are executed on small networks), where the underlying model is a homogeneous Markov chain (cf. Section 3.1). Since the specific QoS requirements considered in [32] do not affect the general methodology, we have chosen only two QoS parameters for the calculation of the energy function (cf. Sections 2 and 3.2). In [13] (see also [9], Section 2), an overlay multicast network infrastructure is proposed which forms a multicast data delivery backbone. The overlay topology is continuously adapted (off-line) with changes in the distribution of the clients as well as changes in network conditions. The performance optimisation is executed by a simulated annealing-based algorithm defined by homogeneous Markov chains. The paper [35] employs a QoS setup similar to [32] and investigates three different search methods to calculate routing trees: simulated annealing, tabu search, and GA. The three methods are tested on small (14 nodes, 21 edges) and relatively large ($100$ nodes, $800$ edges; randomly generated) networks. The findings suggest that simulated annealing can solve multicastrouting problems efficiently with high-quality solutions, tabu search-based algorithms show a good time performance when the group size is large, and that GA genetic-based methods slightly outperform SA in terms of the solution quality.

In the present paper, we introduce an new search method that combines landscape analysis with a genetic local search procedure. The notion of landscape analysis was first mentioned in [36] and has become a major topic in combinatorial optimisation in recent years [37], [38], [39]. Our tool for landscape analysis is logarithmic simulated annealing (LSA) [28], [29], [40], i.e. in our approach we employ simulated annealing based on inhomogeneous Markov chains. The annealing procedure allows us to estimate the depth of the deepest local minima. Recently, simulated annealing algorithms, in particular variants based on inhomogeneous Markov chains, have been used to investigate problems from Computational Biology [41], [42]. Another motivation for choosing simulated annealing is based upon recent advances in GA research [43], where the convergence of GA to optimum solutions has been ensured by employing simulated annealing-based selection in a variety of ways (see Section 10 in [43] for a summary of results). Since we focus on algorithmic aspects of multicast routing in off-line mode, we reduce the QoS requirements to bandwidth and delay constraints (see Section 3.2). The present paper is an extension of the short conference presentation [44]. The competitiveness of LSA in relation to GA and tabu search was demonstrated in [45] for the job shop scheduling problem, which is one of the hardest NP-complete problems.

We performed computational experiments on three instances of the OR library [46] (steinb10, steinb11, steinb18), which were modified for multicast routing. The results provide evidence that our genetic local search heuristic performs better than “pure” LSA.

The paper is structured as follows: in Section 2, we provide a formal definition of the multicast routing problem, along with explanations about parameters and cost functions. In Section 3, we describe LSA pre-processing as a tool for landscape analysis. In Section 4, we introduce our genetic local search heuristic, and in Section 5 we present the results from computational experiments on the modified instances no. steinb10, steinb11, and steinb18, including the results from the landscape analysis which is performed in a pre-processing step.