ILP formulation of the exact solution of multi-constrained minimum cost multicast

Abstract

Multimedia applications such as videoconferencing and collaborative applications require the satisfaction of several Quality of Service constraints (QoS). The routing with respect to QoS constraints was proposed in order to satisfy the user requirement and guarantee a certain level of performance to a data flow. As the communication architecture of these applications is often multicasting, the problem of finding a multicast route satisfying the QoS constraints proves to be challenging. In this paper we propose an Integer Linear Program (ILP) for finding the multicast route respecting a set of QoS constraints with minimum cost. Since the problem is NP-hard, we propose an efficient pretreatment algorithm (ArcReduce) to accelerate the resolution time. The pretreatment process can even answer in polynomial time, whether the problem has a solution or not, before starting the resolution process. The computation of the exact solution also allows for comparison of the heuristic solutions to the exact solution. We conduct an analysis of the ILP and the ArcReduce with various sizes of input data regarding the execution time, the success rate and the quality of the generated multicast route.

Introduction

Multicast routing with Quality of Service (QoS) in networks is considered as an important field of research worldwide. With the appearance of next-generation multimedia Internet applications, much of work has been done on this subject to meet the requirements of users and to improve the communication in networks. The significant increase of connected users in Internet involves accessing large volume of data, often with QoS requirement which made these tasks more challenging [1], [14].

Recently, with the advance in technology, multimedia applications are widely used, especially real-time multicast applications like videoconferencing, Video on Demand, Game on Demand, and Internet Protocol Television. In these applications the data are distributed from a source to several users. In Video on Demand, for instance, a single server delivers a high quality video to a large number of customers. Selecting a tree for routing, which is rooted at the source and contains all the destinations with minimal cost, is called the Steiner problem, which is NP-hard [2]. Exact algorithms as well as heuristics for solving the Steiner problem were proposed in the literature [13]. However, multimedia applications must respect several QoS constraints to operate properly (delay, delay jitter, bandwidth, packet loss...). Consequently, the basic algorithms designed for the Steiner problem are no longer adequate.

Therefore, several multicast routing algorithms are proposed not only for finding the multicast route which minimizes the cost, but also for satisfying the QoS constraints [3], [4], [6], [7], [8], [14]. The problem of constructing a multi-constrained multicast route with minimal cost, the so-called Multi-Constrained Minimum Cost Multicast (MCMCM) problem is NP-hard.

In fact, there is a plethora of algorithms treating the MCMCM problem using different ways, but the goal is the same which is finding a multicast tree that respects a set of End-to-end constraints. Important works have adopted metaheuristic techniques based on genetic algorithm [16], [17], ant colony optimization [18], [19] and tabu search algorithm [20].

The majority of MCMCM algorithms focusing on finding a multicast tree [8], while some other solve special cases of the problem (e.g. the Delay-Constrained Minimum Cost Multicast Routing Problem (DCMCM)) [7].

To solve MCMCM Kuipers et al. [6] proposed an efficient heuristic named MAMCRA. This algorithm computes a route which is not always a tree. First, MAMCRA calculates the set of optimal paths with minimum non-linear length¹ from the source to each destination using an exact multi-constrained path algorithm SAMCRA [11]. Since this set may contain some overlaps, MAMCRA uses in the second step a greedy algorithm to eliminate some of these overlaps without violating the constraints. The final structure is neither a tree nor an optimal solution.

In [8] Raayatpanah proposes a new QoS multicast solution following two steps. In the first step, data envelopment analysis (DEA) technique is used to evaluate the arc efficiency in the network. Afterward, each arc in the network can be considered as a decision making unit by replacing its weights to be minimized and maximized with inputs and outputs respectively. In the second step, the problem is formulated as an integer linear programming based on the relative efficiency scores of arcs to determine the multicast tree.

Actually the problem of QoS multicast routing is a hot topic in wireless networks and related applications. Since the problem is NP-hard, Lu and Zhu proposed in [23] a genetic algorithm based and energy efficient heuristic for QoS multicasting in MANETs.

Wei et al. [21] designed the (EA-ACA-QMRA) algorithm for MCMCM problem which is the combination of an evolutionary algorithm (EA) and an ant colony algorithm (ACA). The key potential of EA-ACA-QMR is the rapid global search capability of EAs with the pheromone feedback factor of ACAs while accounting for multiple constraints. In short, the algorithm combines EA and ACA to describe a strategy for satisfying multiple QoS constraints on a multicast tree.

An interesting algorithm has proposed in [24] to optimize the multi constrained multicast tree using Teaching Learning Based Optimization method (TLBO). TLBO is a population-based method, the population here is considered as a class of learners. The TLBO process is divided into two parts. The first part is the Teacher phase which means learning from teacher and the second part is Learning phase which means learning by the interaction between learners. The TLBO method works on the effect of influence of a teacher on learners in a class. So, output is considered in term of results or grades. Naik et al. have used this concept of algorithm for optimization MCMCM problem. According to the authors, in term of optimal solution quality TLBO performs better than the existing MCMCM algorithms.

Ajay et al. [22] proposed a new method based on fuzzy logic. In this mechanism all the metrics of the routs are combined into the same metric (i.e. fuzzy metric). After though, the minimum fuzzy cost solution will be considered as the optimal solution. The proposed fuzzy logic is very interesting when the goal is to optimize globally the network. In other words, when there are no defined constraints to respect. Unfortunately, it is not the case with some recent applications having strict requirements (e.g. the end-to-end delay must be less than 30 ms, etc.). Therefore, using the combination of metrics does not guarantee that each path respects the end-to-end constraints from the root to any node.

It will be interesting to manage the problem of QoS routing with a general framework with resource allocation [25]. The crux of the method is the formulation of QoS routing, which incorporates the hardware/software implementation and its relation to the allocated resources into a single framework. Our biggest concerns in this paper is that of finding a multicast QoS route that satisfies the end-to-end constraints without resource allocation.

As mentioned in [12] the optimal solution of MCMCM is not a tree (not a sub-graph) but a hierarchy (cf. Section 2). To the best of our knowledge, the exact ILP formulation has not yet been published. The first part of the paper presents the proposed Integer Linear Program (ILP) that finds optimal hierarchies for MCMCM. It is based on a multi-flow method. The ILP gives the optimal route, if it exists. This ILP can also be used to evaluate the efficiency of earlier proposed heuristics. In this paper we propose tests for MAMCRA, which is one of the most efficient heuristics to date. The second part of the paper presents, a new pretreatment algorithm designed to reduce the search area. The pretreatment step consists of eliminating the arcs that cannot be part of the feasible QoS multicast route. Therefore, the pretreatment algorithm accelerates the process and improve significantly the scalability of the ILP.

The rest of the paper is organized as follows. Section 2 presents the problem formulation based on the hierarchies. We demonstrate in the same section some properties of hierarchies and the MCMCM problem. In Section 3 we illustrate the limitation of using standard ILP formulations. Section 4 describes the ILP and explains why we propose a multi-flow method. The efficiency of our ILP is demonstrated in Section 5. Section 6 presents the pretreatment algorithm called ArcReduce, while Section 7 is the experimental part of ArcReduce. Finally, Section 8 concludes this paper.