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Intl. Trans. in Op. Res. 26 (2019) 1339–1364DOI: 10.1111/itor.12605

INTERNATIONALTRANSACTIONS

IN OPERATIONALRESEARCH

Multicast routing under quality of service constraintsfor vehicular ad hoc networks: mathematical formulation

and a relax-and-fix heuristic

Celso C. Ribeiroa, Tiago de A. Santosa,b,† and Cid C. de Souzac,∗aInstitute of Computing, Universidade Federal Fluminense, Niteroi, RJ 24210-346, Brazil

bInstituto Federal de Educacao, Ciencia e Tecnologia Fluminense, Campos dos Goytacazes, RJ 28030-130, BrazilcInstitute of Computing, University of Campinas (UNICAMP), Campinas, SP, Brazil

E-mail: [email protected] [Ribeiro]; [email protected] [de Souza]

Received 20 July 2018; received in revised form 30 October 2018; accepted 31 October 2018

Abstract

In this paper, we investigate a multicast routing problem with quality of service constraints on ad hocvehicular networks. An integer programming formulation for the problem is proposed that forms the basis ofa relax-and-fix heuristic designed with the goal of producing feasible solutions of good quality. In addition,preprocessing procedures relying on simple and constrained shortest paths are developed that reduce themodel size to the point of making it viable to compute. Computational experiments on benchmark instancesgenerated to mimic realistic settings are reported. The results highlight the effectiveness of the relax-and-fixheuristic and the importance of the preprocessing routines for the computability of the proposed mathematicalmodel.

Keywords: multicast routing; quality of service; vehicular networks; ad hoc networks; integer programming; relax-and-fix;heuristics

1. Introduction

Advances in wireless technologies have contributed to the emergence of mobile ad hoc networks(MANETs). A MANET is an autoconfigured network consisting of a collection of wireless mobilenodes independently linked one to another without the need of any infrastructure. Usually thesenodes are personal computer devices, small mobile devices, sensors, and cell phones, among others.

With the expansion of mobile technologies, the design of intelligent transport has attracted theattention of many researchers and industries that aim to provide such types of technologies. Among

∗Corresponding author.†Tiago de A. Santos passed away on 5 June 2018 at the time of writing of this article.

C© 2018 The Authors.International Transactions in Operational Research C© 2018 International Federation of Operational Research SocietiesPublished by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,USA.

1340 C. C. Ribeiro et al. / Intl. Trans. in Op. Res. 26 (2019) 1339–1364

the most important intelligent transport systems are ad hoc vehicular networks or VANETs (Bitamand Mellouk, 2013). A VANET is a type of MANET where nodes are vehicles. There are two typesof communication in VANETs: between vehicles (V2V—vehicle to vehicle) and between vehicles andinfrastructured nodes (V2I—vehicle to infrastructure), also called RSU (roadside unit). Karim (2008)showed that VANETs are more suitable for vehicular communication because they have lower costsof implementation and communication even in places where there is no infrastructure. In addition,VANETs also have lower latency in data delivery when compared to other technologies such as3G/4G networks and infostations.

One possible application of VANETs consists in assisting vehicles to communicate and coordinatewith each other, with the aim of avoiding any type of critical situation such as road accidents, trafficjams, and poor roads, as well as other applications such as speed control, detailed information onaccidents for rescue teams, free passage of vehicles, and invisible obstacles. Another category ofapplications is information and entertainment for passengers and drivers. These are called applica-tions for the comfort of drivers and passengers, such as Internet access, chats and interactive gamesbetween cars next to each other, free places for parking, and detailed information on fuel price,among others.

In view of their high node mobility, the protocols of routing in vehicular ad hoc networks mustbe properly adapted and offer optimized routing. According to Peterson and Davie (2012), ina communication network there are three fundamental methods for data transmission: unicast,broadcast, and multicast. The multicast routing strategy reduces the communication cost andstrongly explores the bandwidth and network resources, since the data packets can be transmittedto all destinations (members of the multicast group) by a single transmission, while with the unicastrouting the source node sequentially transmits the same packet several times to different destinationsand with the broadcast routing the source node transmits to all nodes in the network, whether theyare interested or not in the message.

In a VANET, as well as in other types of networks, many applications have special requirementsin terms of network resources. The use of such special requirements is directly related to the qualityof service (QoS). According to Oliveira and Pardalos (2011), in many applications it is extremelyimportant to maintain high levels of service quality from the Web. For example, security-relatedapplications, such as crash warning messages, accidents, or poor track conditions, are sensitive todelayed delivery of messages, as well as multimedia applications are sensitive to the bandwidth.Among the most important requirements, we find the end-to-end delay, the jitter (i.e., the variationin delay between messages with the same destination), the variation in delay between differentdestinations, the bandwidth, the number of hops between the source and the destination, and theduration of the connection.

In this work, we introduce an integer programming (IP) formulation for the multicast routingproblem with QoS constraints on VANETs and present a relax-and-fix heuristic for solving theproblem. Realistic test problems are generated and computational experiments are reported. Thisarticle is organized as follows. In the next section, we present a review of the literature on theproblem of multicast routing with QoS constraints applied to VANETs. Section 3 introduces aninteger programming model for the multicast routing problem with QoS constraints (MRPQOS) basedon a multicommodity flow formulation. We also describe how instances were generated to validatethe formulation, and report on preliminary results obtained by a mathematical solver that computesthe model. Section 4 details preprocessing strategies for model reduction, i.e., procedures capable

C© 2018 The Authors.International Transactions in Operational Research C© 2018 International Federation of Operational Research Societies

C. C. Ribeiro et al. / Intl. Trans. in Op. Res. 26 (2019) 1339–1364 1341

of eliminating variables and constraints of the model to make it more manageable by a linearprogramming solver, as well as the computational results obtained with their application. Section 5presents a heuristic for the multicast routing problem with QoS constraints based on the relax-and-fix approach. Finally, we conclude and discuss some perspectives for future work in the last section.

2. Related work

According to Oliveira and Pardalos (2005), the minimum Steiner tree problem (SMT) closelyresembles the multicast routing problem and is useful for its formulation and the representationof its solutions. The minimum Steiner tree problem is a classic NP-hard problem (Garey andJohnson, 1990) and an extensive literature exists on the subject, including the surveys of Maculan(1987), Winter (1987), Hwang and Richards (1992), and Du et al. (2009).

Exact methods for solving the SMT appear in Wong (1984), Beasley (1989), and Koch and Martin(1998), among others. In addition to these, there are also implementations of algorithms based onmetaheuristics. Examples are the works of Gendreau et al. (1999), Martins et al. (2000), Ribeiroand De Souza (2000), and Ribeiro et al. (2002), among many others.

The SMT variant with delay constraints is also attractive to represent solutions for the multicastrouting problem with QoS constraints. According to Oliveira and Pardalos (2005), the most studiedversion of the minimum Steiner tree problem applied to multicast routing with QoS constraint isthat considering delays (Kompella et al., 1993, 1996; Jia, 1998; Sriram et al., 1998; Raghavan et al.,1999; Leggieri et al., 2014). Combinations with other QoS constraints in other applications lead toeven harder to solve problems.

Bitam and Mellouk (2013) proposed a bee swarm optimization algorithm for solving the multicastrouting problem with constraints of QoS in VANETs. Given the origin node of the multicast treeand a set of destinations, the objective is to find a tree minimizing the sum of four functions:tree total cost, total delay, total jitter, and bandwidth. The weight assigned to each of the functionscorresponds to the importance assigned to each of them to obtain the QoS. The bee swarm algorithmwas compared to others in the literature in a scenario containing 20 vehicular nodes in an area of1200 m × 1200 m during a period of 120 seconds. This information is passed to a simulator, wherevehicles travel at random. The location of the nodes is extracted at the 85th second, together withthe value of the four criteria in all network links. The results showed that the proposed algorithmpresented improvements with respect to the others involved in the comparison, including fasterconvergence to the best solution.

Sebastian et al. (2010) proposed a change in the routing scheme to make the dissemination ofsecurity messages more efficient in VANETs. The multicast routing problem was formulated as theminimum Steiner tree problem with delay constraints. The routing scheme detects the vehicles thatare threatened in the event of accidents and place them in the set of recipient of alert messages.There is also a feature that aims to increase the efficiency of the communication channels of thewireless network, since there is a selection of the nodes that will receive such emergency alerts,thereby reducing the number of messages sent.

Fazio et al. (2013) developed a new protocol for VANETs that has at its core an optimizedalgorithm for route construction. This algorithm takes into account three constraints: end-to-enddelay, co-channel interference (two or more entities—nodes or antennas—in the same geographic



area transmitting in the same frequency), and the likelihood of link duration. The goal of thework is to find a minimum generating tree considering the QoS constraints. The proposed protocolwas revealed to be superior with respect to the evaluation of the following criteria: bandwidth,packet delivery rate, and order-to-order delay. However, the authors noticed a small overload ofprocessing. In addition to the protocol simulation, the authors also tested the algorithm to evaluateits performance in constructing the routes according to the QoS constraints.

Souza (2012) developed the MAV-AODV routing protocol for VANETs based on ant colonies,which is an extension of a known routing protocol called MAODV (multicast ad hoc on-demanddistance vector routing protocol) and uses vehicle mobility information to increase multicast routingstability. The author proposed an algorithm based on ant colonies to optimize the construction andmaintenance of multicast trees. The proposed protocol was evaluated in a discrete event simulationenvironment for computer networks in a scenario of 1600 m × 1500 m, with simulation times of150 seconds and the number of vehicles varying between 25 and 100. The protocol developed wascompared with two protocols that are used in MANETs, MAODV and PUMA (protocol for uni-fied multicasting through announcement). Five criteria were evaluated: maximum end-to-end delay,variation of end-to-end delay, routing overhead, delivery rate of delivered packets, and packet redun-dancy. The results showed that the proposed protocol was superior to MAODV for several criteria.However, its performance was lower than that of PUMA, except with respect to packet redundancy.

Correia et al. (2011) proposed a routing protocol to VANETS, based on the DYMO reactiveprotocol modified by an ant colony optimization strategy. The optimization makes use of informa-tion such as the speed and the position of vehicles to aid in the choice of routes. The protocol wasvalidated in a software for network simulation in a scenario of 1600 m × 1500 m, with a varyingnumber of vehicles (25, 50, 75, 100, and 150). Three measures of QoS were considered to evaluatethe results: average number of packets, average end-to-end delay, and routing overhead. The resultsshowed that the proposed protocol was superior to the native DYMO protocol with respect to theaverage end-to-end criterion, while protocol DYMO performed better for the other two criteria.

3. A mathematical formulation for multicast routing with QoS constraints

In this section, we develop an integer programming model for the multicast routing problem withQoS constraints. This model is based on a multicommodity flow formulation, where several productsmust be shipped in a network from their source to their destination vertices. We also describe howinstances were generated to validate this formulation, and report on preliminary results obtainedby a mathematical solver that computes the model.

3.1. Integer programming model

The multicast routing problem (MRP) was investigated earlier in the literature (Wang and Hou,2000; Koch et al., 2001; Oliveira and Pardalos, 2011). Oliveira and Pardalos (2011) state that themulticast routing problem generalizes the classic minimum Steiner tree problem in graphs, which isknown to be NP-hard. Our IP formulation for the multicast routing problem with QoS constraintsexplores this fact insofar, as it includes several features of the best formulations available for theSteiner problem.



In that vein, the multicast routing problem with QoS constraints is modeled through a directedgraph G = (V, A) representing a network, where V denotes its set of vertices, which can be vehiclesor fixed nodes containing some of the network infrastructure, and A a set of arcs (links). A sourcevertex s is defined, as well as a subset D of destination vertices or terminals, with D ⊆ V \{s}. Theremaining vertices of G, i.e., those in V \(D ∪ {s}), constitute the set S, and are called optional orSteiner vertices.

Four real nonnegative values are associated to each arc (i, j) ∈ A: λi j , ξi j , αi j , and ωi j . These valuesrepresent, respectively, the end-to-end delay, the jitter, an estimation of the connection duration,and the bandwidth in the corresponding link, which are the coefficients of the QoS constraintsmentioned in Section 1. Two facts are worth mentioning at this point. First, since the links in thecommunication network are bidirectional, if arc (i, j) belongs to A, then the reverse arc ( j, i) is alsoin A and is associated with the same four values just described. Second, when the bandwidth ωi j oflink (i, j) falls below the threshold required for the QoS, the corresponding arcs are not added to Aas they can never be part of a feasible solution to MRPQOS.

A multicast arborescence T of G has the source vertex s as its root and spans all the destinationvertices in D and, possibly, some vertices of S, but not necessarily all of them. To be a feasiblesolution for MRPQOS, T must be such that the total delay and the total jitter in the path from s toevery destination vertex cannot exceed the maximum values allowed for these measures. In addition,the difference between the maximum and the minimum delays of such paths is also constrained bya given upper bound.

Although the graph described above allows for a correct modeling of the problem, we slightly alterit, to ensure that any feasible solution corresponds to a spanning arborescence. This is convenientbecause strong IP formulations are known for arborecences (Magnanti and Wolsey, 1995). Thechanges in the graph are the following. First, we add a new vertex s′ and a new arc from s to s′.Then, an extra arc from s′ to each optional vertex q ∈ S is also created. The resulting graph isG′ = (V ′, A′), where V ′ = V ∪ {s′} and A′ = A ∪ {(s, s′)} ∪ {(s′, q) : q ∈ S}. As said before, feasiblesolutions in this graph are given by arborescences rooted at s that span all the vertices in V ′.However, some additional constraints need to be met. The arborescence must include the arc (s, s′)and, whenever an arc (s′, q) is present for some q ∈ S, the vertex q is a leaf. The purpose here is tohave in the subtree rooted at s′ all the optional vertices that are not in the solution of the originalgraph G. Figure 1 illustrates the situation. The optimal multicast tree corresponds to the subtreerooted at s that is obtained after the removal of the arc (s, s′). The other subtree, rooted at s′, is astar with the leaves corresponding to the unused Steiner vertices.

Prior to describing the formulation of MRPQOS, we list the parameters used in the equations:

� �d : maximum delay allowed in any path;� �v: maximum delay variation between any two paths;� � j : maximum jitter allowed in any path;� θc: unitary cost per link used;� θp: unitary cost for delay;� θd : unitary cost for the estimated duration of a link;� λi j : end-to-end delay of the link between vertices i and j;� αi j : estimation of the duration of the connection between vertices i and j; and� ξi j : jitter of the link between vertices i and j.



s′

s

Fig. 1. An arborescence representing a solution to MRPQOS. Squares denote Steiner vertices and trianglesdenote the terminal vertices.

The variables of the model are defined as follows:

� xki j : binary variable indicating if the arc (i, j) belongs to the path from source s to k ∈ D or,

alternatively, if the flow from s to k goes through arc (i, j);� xq

i j : binary variable indicating if the arc (i, j) belongs to the path from source s to q ∈ S or,alternatively, if the flow from s to q goes through arc (i, j); and

� yi j : binary variable indicating if the arc (i, j) belongs to the optimal arborescence.

The integer programming model for the MRPQOS now reads:

(MIP) min∑

(i, j)∈A′μi jyi j (1)

subject to∑

(s, j)∈A′xk

s j −∑

( j,s)∈A′xk

js = 1 ∀ k ∈ D ∪ S (2)

∑(i, j)∈A′

xki j −

∑( j,i)∈A′

xkji = 0 ∀ j ∈ V ′ \ {s, k}, ∀ k ∈ D ∪ S (3)

∑(k, j)∈A′

xkk j −

∑( j,k)∈A′

xkjk = −1 ∀ k ∈ D ∪ S (4)

xki j ≤ yi j ∀ (i, j) ∈ A, ∀ k ∈ D (5)



∑(i, j)∈A′

λi jxki j ≤ �d ∀ k ∈ D (6)

∑(i, j)∈A′

ξi jxki j ≤ � j ∀ k ∈ D (7)

∑(i, j)∈A′

λi j(xki j − xl

i j ) ≤ �v ∀ k, l ∈ D, ∀ k = l (8)

xkss′ = 0 ∀ k ∈ D (9)

xes′q = 0 ∀ q ∈ S, ∀ e ∈ D ∪ S\{q} (10)

yqi + yiq + xqs′q ≤ 1 ∀ q ∈ S, ∀ i = s′, ∀ i ∈ D ∪ S (11)

∑(i, j)∈A′

yi j = |V | (12)

xqi j ≤ yi j ∀ (i, j) ∈ A′, ∀ q ∈ S (13)

xki j ∈ {0, 1} ∀ (i, j) ∈ A′, ∀ k ∈ D (14)

xqi j ∈ {0, 1} ∀ (i, j) ∈ A′, ∀ q ∈ S (15)

yi j ∈ {0, 1} ∀ (i, j) ∈ A′. (16)

The objective function (1) minimizes the weighted total cost of the solution. The coefficient μi jassociated to an arc (i, j) ∈ A in the objective function is given by μi j = θc + θpλi j − θdαi j , whichencompasses the weighted cost of using the link and paying for its delay, as well as being rewardedfor the duration of its connection. For arcs in A′ \ A, that is, those having s′ as head or tail, the costcoefficient is null. Constraints (2)–(4) impose the flow conservation at each vertex, while ensuringthat only one unit of flow goes from the source s to each destination k ∈ D ∪ S. Constraints (5)guarantee that there can be no flow passing through arc (i, j) and headed to a vertex k ∈ D, unlessthe arc belongs to the optimal arborescence. Constraint (6) limits the maximum end-to-end delayfor each path from source s to a destination k ∈ D to �d . Constraint (7) certifies that the maximumjitter of a path that connects the source s to any destination is at most � j . Constraint (8) ensuresthat the difference between the delays from the source and any pair of terminals does not exceeda given limit �v. Constraint (9) enforces that the flow destined to any vertex k ∈ D cannot passthrough s′. Constraint (10) imposes that, for every optional vertex q ∈ S, no flow can pass througharc (s′, q) unless it is destined to q itself. Constraint (11) ensures that, if s′ is in the path from s toq ∈ S, no arc entering or leaving q and coming from or going to any node i ∈ D ∪ S belongs to theoptimal arborescence. Constraint (12) enforces that the solution has |V ′| = |D| + |S| + 1 arcs and,therefore, spans the graph G′. Constraints (13) are similar to (5) in that they forbid the flow headingto a vertex q ∈ S to circulate in an arc that is not in the arborescence. Constraints (9)–(13) enforce



the connection of all optional vertices that are not in paths from the source s to any destinationvertex. They are leaves of the arborescence and can be reached from the source s via vertex s′.Finally, constraints (14)–(16) define the variable domains.

Notice that there are no inequalities in the formulation forcing variable yi j to be set to zerowhen none of the paths from the source s to a terminal uses it. This is done because, in real-world instances, its coefficient μi j in the objective function is positive and MRPQOS is naturally aminimization problem.

3.2. Instance generation

There is no consensus among authors of previously published papers on the routing problem inVANETs with regard to the instances to be used for testing. Each author creates his or her ownbenchmark for the validation of the proposed algorithm or protocol. This appears in the worksof Bitam and Mellouk (2013), Souza (2012), Souza et al. (2013), and Fazio et al. (2013). Moreimportant, the set of QoS metrics that are considered in these articles differ from that consideredhere and, therefore, the instances there miss relevant data for MRPQOS. As a consequence, thesebenchmarks are of no use for us. In light of this, we built our own benchmark to evaluate themathematical model of Section 3.1.

The first step in generating instances for the MRPQOS involves the choice of the roads and highwaysto form the street network. For this task, we use the maps available at the Open Street Map(OSM; Haklay and Weber, 2008), which is a collaborative mapping project to create a free andeditable map of the world that has been in development since 2004.

After selecting the desired information from the OSM, i.e., the part of the map of interest, thedata are processed, converted, and then imported into SUMO (Simulation of Urban Mobility), anopen-source platform for simulating urban mobility (Krajzewicz et al., 2012).

The next step to start the vehicle simulation process in SUMO is the creation of the mobilitymodel through the insertion of vehicles, traffic lights, and other components inherent to car trafficon the map. Once this task is completed, the vehicle traffic simulation process begins, exporting thedata of vehicle movement, which are necessary for the next step.

The data exported from SUMO are fed into network simulator NS-2, which is a discrete eventsimulator for telecommunication network research, initially developed by the University of Berke-ley and broadly used by the scientific community (Issariyakul and Hossain, 2011). The networksimulation process is then initiated and, at any moment, one can extract from NS-2 the informationrelative to the current topology of the network and build the graph corresponding to the presentstatus of the communication network, with its nodes (vehicles) and links (connections betweenthe vehicles). Such a graph serves for evaluating our mathematical model as, among the gatheredinformation, we also have the metrics that are needed to describe an instance of MRPQOS. Despite thelimitations inherent to the simulation process, this instance is believed to reflect real situations veryclosely.

Figure 2 depicts a simplified flowchart of the instance generation process. It begins with the mapextraction in OSM, goes through the simulation in SUMO, accompanied by the definition of themobility model, and ends with the simulation process in NS-2, from where the instance is finallybuilt.



Fig. 2. Simplified flowchart of the instance generation process.

The tests were carried out in three groups totaling 39 instances created from parts of the realcity maps of Washington DC, Berlin, and Geneva. The number of vertices of these instances variesbetween 9 and 500, the number of terminal vertices between 3 and 125, and the number of arcsbetween 26 and 2430.

3.3. Experiments with the MIP model

All the programs implemented in this work were coded in C++ and compiled with GNU G++6.3.1. To compute the IP models we used IBM CPLEX 12.6.1 with the runtime limit set to 2400seconds for each run of each instance. The experiments were conducted on a computer with a 3.40GHz Intel Core i7 processor, 24 GB of RAM, and running under the Linux (Fedora 25) operatingsystem.

Table 1 exhibits the relevant data of the instances of our benchmark along with the CPLEX resultswhen receiving the MIP model as input. All results consider the following weights for the cost metrics:θc = 27, θp = 65, and θd = 8.

The first column of Table 1 identifies each instance, whereas the second, third, and fourth columnsindicate their numbers of vertices, arcs, and terminals, respectively. The next two columns presentthe number of constraints and variables in the model. The seventh column displays the time inseconds needed to compute the model. The acronym TLE stands for “time limit exceeded,” meaningthat the solver was interrupted prematurely without reaching the optimum. The eighth columncorresponds to the number of nodes explored in the enumeration tree at the time the computationstopped. Finally, the last two columns show, respectively, the upper (primal) and lower (dual) boundscomputed at this point.

Of the 39 instances in the benchmark, only four had their optimality proven. The respectiverows are highlighted in dark gray ( ) in the table. The rows in light gray ( ) correspond to the



Table 1Results of the MIP model obtained by CPLEX

Instance |V | |A| |D| Constraints VariablesTime(seconds) Nodes

Upperbound

Lowerbound

Washington-9-3 9 26 3 375 288 0.13 0 2886.87 2886.87Washington-20-3 20 276 3 6147 5780 1.93 77 1417.89 1417.89Washington-20-5 20 276 5 6017 5660 7.36 508 1712.36 1712.36Washington-20-10 20 276 10 5935 5560 TLE 70,576 2918.79 2736.62Washington-40-7 40 690 7 30,026 28,520 TLE 10,577 3785.69 3549.49Washington-40-10 40 690 10 30,226 28,720 TLE 24,069 3821.84 3614.03Washington-40-20 40 690 20 24,774 23,040 TLE 9483 8142.29 7071.59Berlin-100-5 100 1118 5 129,434 119,500 TLE 309 − 3967.68Berlin-100-10 100 1118 10 123,365 113,400 TLE 767 8523.66 6520.31Berlin-100-17 100 1118 17 128,415 118,300 TLE 228 − 8473.32Berlin-100-25 100 1118 25 122,898 112,500 TLE 276 − 10,110.36Berlin-125-5 125 1234 5 181,072 165,500 TLE 67 − 4042.34Berlin-125-10 125 1234 10 189,844 152,625 TLE 318 7665.14 6993.93Berlin-125-21 125 1234 21 181,123 165,250 TLE 87 − 10,571.22Berlin-125-31 125 1234 31 182,307 166,000 TLE 3 − 13,244.27Berlin-150-5 150 1690 5 294,414 271,950 TLE 1 − 3923.99Berlin-150-10 150 1690 10 281,209 258,750 TLE 34 − 5452.83Berlin-150-25 150 1690 25 270,256 247,350 TLE 0 − 12,547.45Berlin-150-38 150 1690 38 282,459 258,900 TLE 0 − 15,789.23Berlin-sparse-100-5 100 470 5 63,705 53,700 2116.12 5116 5130.09 5130.09Berlin-sparse-100-10 100 470 10 61,659 51,600 TLE 2361 − 7923.45Berlin-sparse-100-17 100 470 17 64,932 54,700 TLE 1129 − 9406.31Berlin-sparse-100-25 100 470 25 62,248 51,700 TLE 1589 − 11,289.58Berlin-sparse-125-5 125 600 5 105,622 90,000 TLE 413 − 6084.16Berlin-sparse-125-10 125 600 10 104,811 89,125 TLE 395 − 7479.33Berlin-sparse-125-21 125 600 21 102,742 86,750 TLE 235 − 12,530.05Berlin-sparse-125-31 125 600 31 103,248 86,750 TLE 56 − 17,287.80Berlin-sparse-150-5 150 720 5 142,950 120,450 TLE 314 − 6022.92Berlin-sparse-150-10 150 720 10 142,266 119,700 TLE 316 − 8040.22Berlin-sparse-150-25 150 720 25 137,819 114,750 TLE 26 − 16,670.60Berlin-sparse-150-38 150 720 38 145,036 121,200 TLE 19 − 20,592.92Geneva-sparse-400-5 400 2200 5 1,158,787 998,800 TLE 0 − −Geneva-sparse-400-10 400 2200 10 1,183,250 1,023,200 TLE 0 − −Geneva-400-67 400 2200 67 1,119,033 954,800 TLE 0 − −Geneva-sparse-400-100 400 2200 100 1,155,233 985,600 TLE 0 − 1687.61Geneva-sparse-500-5 500 2430 5 1,708,496 1,458,500 TLE 0 − −Geneva-sparse-500-10 500 2430 10 1,707,053 1,457,000 TLE 0 − −Geneva-sparse-500-83 500 2430 83 1,664,109 1,407,500 TLE 0 − 7722.70Geneva-sparse-500-125 500 2430 125 1,661,763 1,396,500 TLE 0 − 17,362.47

six additional instances for which a feasible solution was found, but the duality gap could not beclosed. For the remaining 29 instances (74%), no feasible solution was found and, for five of them,not even a dual bound was computed, meaning that CPLEX could not handle the linear relaxationof the problem. The latter is an indication that the MIP model is maybe too big to be solved. In such



case, it would be handy to design procedures capable of eliminating variables and constraints of themodel to make it more manageable. This strategy is explored next.

4. Preprocessing for model reduction

A common technique applied to reduce the size of IP models is to look for variables that can havetheir values fixed without sacrificing optimality (Uchoa et al., 2002). Specifically, in the case ofMRPQOS, given a terminal vertex k and an arc (i, j), we try to establish that arc (i, j) cannot takepart in a feasible path leading from the source vertex s to a terminal k. If so, variable xk

i j can besafely set to zero or, in other words, it can be eliminated from the model, reducing its size. To checkthe viability of using arc (i, j) in a path from the source s to a terminal k, we can compute boundsfor both the minimum delay and the minimum jitter needed for building such path.

4.1. Preprocessing MRPQOS

The general idea goes as follows. Let r be a resource associated to the arcs of the network G so that,for arc (i, j), ri j is the amount of resource r consumed when this arc is traversed. For a path P fromthe source s to a terminal k to be feasible, the sum of the r values for all arcs in P cannot exceed agiven quantity �r. Now, let LBri be a lower bound on the amount of resource r that is needed to gofrom s to i and, analogously, let LBjk be a lower bound on the resource r that is needed to go fromj to k. Then, if

LBsi + ri j + LBjk > �r, (17)

we know that no feasible path going from the source s to terminal k can use arc (i, j). As aconsequence, variable xk

i j can be set to zero or, equivalently, can be dropped from the MIP model.For MRPQOS, there are two resources to consider: delay and jitter.

In view of the above arguments, we devise three procedures intended to eliminate variables fromour formulation. Essentially, they differ in the way the bounds in Equation (17) are obtained.Nevertheless, these bounds are all based on shortest path computations as explained next.

Consider an instance of MRPQOS defined by the directed graph G and the functions λ and ξ thatassign, respectively, delay and jitter values to each of its arcs. The following notations and definitionsare used:

� Gλ: directed graph G with arc costs given by the function λ and arc resources given by function ξ ;� Gξ : same as above, but with the roles of the functions λ and ξ interchanged;� GT

r : the transpose of graph Gr for r ∈ {λ, ξ} (i.e., with arc directions reversed);� SP(r, u, v): the shortest path from u to v in graph Gr, where r is in {λ, ξ} and u and v is any pair

of distinct vertices in Gr;� CSP(r, u, v, �t ): the shortest path from u to v in Gr constrained to use at most �t units of resource

t, where r and t are in {λ, ξ}, r = t and u and v is any pair of distinct vertices in Gr; and� CSPT (r, u, v, �t ): same as above with Gr replaced by GT

r (the transpose graph).



We now describe the tests we devised to verify if an arc variable xki j can be excluded from the

MIP model. These tests are classified as weak, moderate, and strong, depending on how many timesshortest and constrained shortest path computations are used to obtain the bounds in Equation (17).If both bounds correspond to the length of a shortest path, the test is considered weak. On theopposite side, a test is said to be strong if the two bounds are based on constrained shortest paths.Finally, in a moderate test, one bound is given by a shortest path and the other by a constrainedshortest path. The tests are the following:

1. Weak vertex elimination (WVE). For a given vertex i ∈ S and a resource r ∈ {λ, ξ}, if

SP(r, s, i) + SP(r, i, k) > �r, for all k ∈ D, (18)

then vertex i can be removed from the graph.Explanation and variable fixing. The least amount of resource r necessary to go from source s toterminal k through vertex i exceeds the allowable limit. Hence, vertex i cannot be in any feasiblepath and, therefore, it can be removed from the graph together with all arcs incident to it. Thismeans that all variables of the form xk

i j and xkji for all k ∈ D are eliminated from the model.

2. Moderate vertex elimination (MVE). For a given vertex i ∈ S and resources r and t in {λ, ξ} withr = t, if

CSP(r, s, i, �t ) + SP(r, i, k) > �r, for all k ∈ D, (19)

then vertex i can be removed from the graph.Explanation and variable fixing. The argument is analogous to the previous test. The differencehere is that the bound in the first term is strengthened by using a shortest path constrained to donot overuse resource t.

3. Weak arc elimination (WAE). For a given arc (i, j) ∈ A, a terminal k ∈ D and a resourcer ∈ {λ, ξ}, if

SP(r, s, i) + ri j + SP(r, j, k) > �r, (20)

then the arc (i, j) cannot appear in a feasible path from s to k.Explanation and variable fixing. The minimum amount of resource r necessary to go from sources to i, traverse arc (i, j), and then go from j to terminal k surpasses the permitted limit. So, thevariable xk

i, j can be eliminated.4. Moderate arc elimination (MAE). For a given arc (i, j) ∈ A, a terminal k ∈ D and resources r

and t in {λ, ξ} with r = t, if

CSP(r, s, i, �t ) + ri j + SP(r, j, k) > �r, (21)

then the arc (i, j) cannot be in a feasible path from s to k.Explanation and variable fixing. Similar to the previous test, except that the first bound wasstrengthened by the computation of a constrained shortest path in the same way as in Equa-tion (19).



5. Strong arc elimination (SAE). For a given arc (i, j) ∈ A, a terminal k ∈ D and resources r and tin {λ, ξ} with r = t, consider again Equation (17). The lower bound LBsi can be computed by

LBsi = CSP(r, s, i, �t − min�∈D,� =i

{SP(t, i, �)}). (22)

This is because, if the flow goes from the source s to terminal k and traverses arc (i, j), then thequantity of resource t consumed to go from i to k is limited from below by min{SP(t, i, �) : � ∈D, � = i}. Thus, one is left with at most �t minus this amount of resource t to go from s to i.Consequently, the length of the constrained shortest path given by this formula is a lower boundto the amount of resource r that is spent when going from the source s to i. Now, going back toEquation (17), the second lower bound is obtained by the equation

LBjk = CSPT (r, k, j, �t − CSP(t, s, j, �r)). (23)

Initially, let us analyze the meaning of the term �t − CSP(t, s, j, �r) in this expression. Itcorresponds to an upper bound on the amount of resource t that is available to go from j to k,as it assumes we have spent the least possible amount of this resource to reach j coming from s.It is clear then that the outer constrained shortest path in the formula gives a valid lower boundto the amount of resource r needed to go from j to k. The fact that this constrained shortestpath is computed for the transpose graph has to do with the computational complexity that willbe discussed later. From the above, the SAE test is derived: if

CSP(

r, s, i, �t − min�∈D,� =i

{SP(t, i, �)})

+ ri j

+CSPT (r, k, j, �t − CSP(t, s, j, �r)) > �r,

(24)

then the arc (i, j) cannot appear in a feasible path from the source s to terminal k.Explanation and variable fixing. Once the validity of the lower bounds have already been estab-lished above, the arguments from the previous test can be repeated, leading to the elimination ofthe variable xk

i j .

A few words about the computational complexity of the tests above are worth adding. All thesecomputations are done on the graph G = (V, A) or its transpose. Let n = |V |, m = |A| and d = |D|,i.e., d is the number of terminal vertices in V . Since the running time of the tests depends on thecomputation of shortest and constrained shortest paths required in each case, let #SP and #CSP be,respectively, the complexities of computing, respectively, shortest and constrained shortest pathsin G or GT . It is well known that #SP is O(n2) for single source and O(n3) for all-pair shortestpaths computations (see, e.g., Cormen et al., 2009). On the other hand, the constrained shortestpath problem is NP-hard (see Handler and Zang, 1980, Minoux and Ribeiro, 1986, and Ribeiroand Minoux, 1985). This said, the complexities of our tests are given by (a) WVE: O(n2 · #SP); (b)MVE: O(n · #CSP + n2 · #SP); (c) WAE: O(n2 · #SP);(d) MAE: O(n · #CSP + n2 · #SP); and (e) SAE:O(n · #CSP + n2 · #SP) + O((d + n) · #CSP), where the first and second terms of the last summationare the complexities associated to Equations (22) and (23), respectively. Notice that using Floyd–Warshall’s algorithm to compute all-pair shortest paths, the n2O(#SP) terms in these complexities



can all be replaced by O(n3). Also, observe that in the second term corresponding to the complexityof the SAE test, the use of the transpose graph avoids the computation of constrained shortest pathsbetween all pairs of vertices in V \ {s}. This is so because we can constrain the set of source verticesto the terminals.

We are now in position to present the three procedures we developed to eliminate variables fromthe MIP model based on the five tests just described.

� Procedure SP-SP: uses tests WVE and WAE in that order (i.e., vertices are removed from the graphprior to eliminating arcs from paths to some terminals).

� Procedure CSP-SP: uses tests MVE and MAE in that order (same rationale as before).� Procedure CSP-CSP: uses test SAE alone, as it is the most expensive one, requiring the computation

of more constrained shortest paths.

As said earlier, the constrained shortest path problem is NP-hard. Exact algorithms exist to solvethe problem that are efficient in practice (see Irnich and Desaulniers, 2005). Our implementationsuse the label-setting algorithm available in the BOOST library (see Boost, 2018).

The results obtained by the MIP models reduced by the application of the three above proceduresare discussed next. In the discussion that follows, the model originated from each procedure X isnamed MIP-X. Besides, it is worth mentioning that all running times reported include the time con-sumed by the respective procedure to perform shortest and constrained shortest path calculations.They are negligible compared to the solver’s running time.

4.2. Computational experiments with reduced MIP models

Tables 2–4 exhibit the data relative to the execution of the MIP models from procedures SP-SP, CSP-SP, and CSP-CSP, respectively. As for Table 1, rows highlighted in dark gray ( ) refer to instanceswhose optimality was proven within the time limit of 2400 seconds, while rows in light gray ( )correspond to instances for which a feasible solution was found but the duality gap remained open.The first column in each of these tables identifies the instance, the next column shows the numberof constraints after the execution of procedure SP-SP. The third column shows the reduction in thenumber of constraints in percentage when compared to the original model. The fourth and fifthcolumns give the same values relative to the number of model variables. The four last columns areanalogous to their counterparts in Table 1.

We first analyze the number of instances for which the optimum was computed and those for whicha feasible solution was found that could not be proved to be optimal. These data are summarizedin Table 5.

It is clear that preprocessing is crucial to improve performance. The number of instances solved bythe reduced models compared to the original MIP model increased more than four times in all cases.In addition, the total number of instances for which a feasible solution was found was multiplied byfactor of 3. As expected, MIP-CSP-CSP performed better than the other models, achieving optimalityin 18 cases (46.15% of the test problems) and feasibility in 31 cases (79.49% of the test problems).It is interesting that, despite MIP-SP-SP being based on the weakest tests, this model was the onlyone where CPLEX encountered a feasible solution for instance Berlin-100-25. This behavior is likely



Table 2Computational results for the reduced MIP model using procedure SP-SP

Instance ConstraintsReduction(%) Variables

Reduction(%)

Time(seconds) Nodes

Upperbound

Lowerbound

Washington-9-3 375 0.00 246 14.58 0.07 0 2886.87 2886.87Washington-20-3 3585 41.68 5026 13.04 0.09 0 1417.89 1417.89Washington-20-5 6017 0.00 5503 2.77 4.74 291 1712.36 1712.36Washington-20-10 5935 0.00 4832 13.09 522.12 38,857 2918.79 2918.79Washington-40-7 29,869 0.52 25,952 9.00 728.55 18,639 3785.69 3785.69Washington-40-10 29,989 0.78 26,229 8.67 TLE 22,498 3819.16 3568.65Washington-40-20 24,619 0.63 15,332 33.45 837.53 21,857 7839.08 7839.08Berlin-100-5 93,474 27.78 115,677 3.20 267.77 1010 4486.06 4486.06Berlin-100-10 99,013 19.74 105,478 6.99 350.48 1502 7223.52 7223.52Berlin-100-17 126,324 1.63 106,710 9.80 TLE 3915 − 8973.68Berlin-100-25 122,600 0.24 97,007 13.77 TLE 3299 12,010.64 10,505.51Berlin-125-5 152,854 15.58 160,960 2.74 396.05 456 5029.15 5029.15Berlin-125-10 126,276 33.48 144,262 5.48 863.64 3767 7461.74 7461.74Berlin-125-21 160,775 11.23 147,967 10.46 TLE 667 12,471.37 11,300.60Berlin-125-31 172,948 5.13 140,398 15.42 TLE 1362 − 13,587.44Berlin-150-5 146,590 50.21 264,549 2.72 222.11 593 4364.78 4364.78Berlin-150-10 159,052 43.44 247,770 4.24 942.03 3135 6111.17 6111.17Berlin-150-25 254,675 5.77 220,447 10.88 TLE 1593 13,167.43 12,986.85Berlin-150-38 261,191 7.53 207,147 19.99 TLE 2579 18,183.20 16,510.38Berlin-sparse-100-5 58,718 7.83 52,705 1.85 607.97 2203 5130.09 5130.09Berlin-sparse-100-10 55,870 9.39 49,600 3.88 TLE 8738 − 8306.57Berlin-sparse-100-17 60,042 7.53 50,067 8.47 TLE 6419 10,541.22 9701.54Berlin-sparse-100-25 57,362 7.85 46,623 9.82 TLE 4609 − 11,544.89Berlin-sparse-125-5 87,399 17.25 87,847 2.39 354.11 1608 6775.90 6775.90Berlin-sparse-125-10 73,350 30.02 84,720 4.94 2038.45 9153 8233.32 8233.32Berlin-sparse-125-21 93,008 9.47 78,556 9.45 TLE 3879 13,245.12 12,884.78Berlin-sparse-125-31 100,757 2.41 71,779 17.26 TLE 6341 18,403.51 18,235.45Berlin-sparse-150-5 93,067 34.90 117,924 2.10 325.14 2799 6975.62 6975.62Berlin-sparse-150-10 105,866 25.59 114,822 4.08 1146.19 4196 9027.46 9027.46Berlin-sparse-150-25 125,542 8.91 102,539 10.64 TLE 5723 18,602.94 18,481.10Berlin-sparse-150-38 136,350 5.99 100,489 17.09 TLE 3459 22,884.24 20,912.59Geneva-sparse-400-5 593,788 48.76 989,702 0.91 TLE 1371 − 12,646.11Geneva-sparse-400-10 676,230 42.85 1,003,302 1.94 TLE 975 23,672.61 20,136.55Geneva-400-67 1,116,235 0.25 877,607 8.08 TLE 0 − −Geneva-sparse-400-100 1,155,233 0.00 877,943 10.92 TLE 0 − 1687.61Geneva-sparse-500-5 797,300 53.33 1,447,443 0.76 TLE 142 − 14,350.42Geneva-sparse-500-10 886,328 48.08 1,434,710 1.53 TLE 26 21,499.70 17,249.45Geneva-sparse-500-83 1,662,611 0.09 1,271,560 9.66 TLE 0 − 7722.70Geneva-sparse-500-125 1,661,763 0.00 1,205,895 13.65 TLE 0 − 17,362.47

due to the strategic decisions made to traverse the enumeration tree that are implemented in thesolver.

In regard to the decrease in the model size, the overall reductions in percentage in the number ofconstraints and variables with respect to the original MIP reveal that the three procedures reachedquite similar results. The statistics for each model are tabulated in Table 6.



Table 3Computational results for the reduced MIP model using procedure CSP-SP


Reduction(%)

Time(seconds) Nodes

Upperbound

Lowerbound

Washington-9-3 375 0.00 246 14.58 0.03 0 2886.87 2886.87Washington-20-3 3584 41.70 5026 13.04 0.09 0 1417.89 1417.89Washington-20-5 6017 0.00 5498 2.86 5.13 265 1712.36 1712.36Washington-20-10 5935 0.00 4832 13.09 525.78 38,857 2918.79 2918.79Washington-40-7 29,868 0.53 25,204 11.63 128.32 8023 3785.69 3785.69Washington-40-10 29,988 0.79 25,421 11.49 1312.62 36,985 3819.16 3819.16Washington-40-20 24,618 0.64 14,628 36.51 396.22 15,290 7839.08 7839.08Berlin-100-5 93,473 27.78 115,556 3.30 260.16 892 4486.06 4486.06Berlin-100-10 99,012 19.74 105,319 7.13 251.94 1983 7223.52 7223.52Berlin-100-17 126,323 1.63 106,263 10.17 TLE 3927 − 9019.25Berlin-100-25 122,599 0.24 95,501 15.11 TLE 3575 − 10,556.68Berlin-125-5 152,853 15.58 160,903 2.78 456.28 626 5029.15 5029.15Berlin-125-10 126,275 33.48 144,161 5.55 1854.86 8971 7461.74 7461.74Berlin-125-21 160,774 11.23 147,918 10.49 TLE 1589 − 11,387.82Berlin-125-31 172,947 5.13 140,272 15.50 TLE 1711 14,506.84 13,580.59Berlin-150-5 146,589 50.21 264,511 2.74 185.08 574 4364.78 4364.78Berlin-150-10 159,051 43.44 247,355 4.40 535.56 1324 6111.17 6111.17Berlin-150-25 254,674 5.77 220,376 10.91 TLE 1263 13,167.43 12,986.85Berlin-150-38 261,190 7.53 207,109 20.00 TLE 2216 18,357.56 16,459.07Berlin-sparse-100-5 58,717 7.83 52,598 2.05 917.07 4240 5130.09 5130.09Berlin-sparse-100-10 55,869 9.39 49,487 4.09 TLE 10,577 − 8368.67Berlin-sparse-100-17 60,041 7.53 49,923 8.73 TLE 6277 10,216.15 9775.91Berlin-sparse-100-25 57,361 7.85 46,395 10.26 TLE 5080 − 11,598.80Berlin-sparse-125-5 87,398 17.25 87,836 2.40 517.51 2079 6775.90 6775.90Berlin-sparse-125-10 73,349 30.02 84,709 4.95 1714.86 10,565 8233.32 8233.32Berlin-sparse-125-21 93,007 9.48 78,532 9.47 TLE 4082 13,245.12 12,658.57Berlin-sparse-125-31 100,756 2.41 71,760 17.28 TLE 5252 18,403.51 18,117.20Berlin-sparse-150-5 93,066 34.90 117,916 2.10 434.32 4917 6975.62 6975.62Berlin-sparse-150-10 105,865 25.59 114,797 4.10 1356.18 6777 9027.46 9027.46Berlin-sparse-150-25 125,541 8.91 102,221 10.92 TLE 7411 18,602.94 18,320.83Berlin-sparse-150-38 136,349 5.99 100,420 17.15 TLE 3887 − 20,958.16Geneva-sparse-400-5 593,787 48.76 989,327 0.95 TLE 5426 18,256.32 13,507.01Geneva-sparse-400-10 676,229 42.85 1,003,006 1.97 TLE 3484 24,451.03 20,842.71Geneva-400-67 1,116,234 0.25 870,515 8.83 TLE 0 − −Geneva-sparse-400-100 1,155,233 0.00 877,580 10.96 TLE 0 − −Geneva-sparse-500-5 797,299 53.33 1,447,299 0.77 TLE 3640 18,683.50 16,012.28Geneva-sparse-500-10 886,327 48.08 1,434,522 1.54 TLE 0 − 17,181.96Geneva-sparse-500-83 1,662,610 0.09 1,271,107 9.69 TLE 0 − 7722.70Geneva-sparse-500-125 1,661,763 0.00 1,205,502 13.68 TLE 0 − 17,362.47

These very close results are not enough to explain the differences in performances over the threereduced MIPs. As the elimination of some constraints (such as constraints (5)) is directly implied byvariable eliminations, we focus on the latter case. We compute the average rank of each model for all39 instances, in such a way that the smaller the rank of the model, the higher is its variable reduction.



Table 4Computational results for the reduced MIP model using procedure CSP-CSP


Reduction(%)

Time(seconds) Nodes

Upperbound

Lowerbound

Washington-9-3 375 0.00 246 14.58 0.02 0 2886.87 2886.87Washington-20-3 3584 41.70 5026 13.04 0.08 0 1417.89 1417.89Washington-20-5 6017 0.00 5491 2.99 4.71 967 1712.36 1712.36Washington-20-10 5935 0.00 4832 13.09 612.11 53,979 2918.79 2918.79Washington-40-7 28,470 5.18 24,640 13.60 17.49 5169 3785.69 3785.69Washington-40-10 29,528 2.31 25,094 12.63 2223.51 89,257 3819.16 3819.16Washington-40-20 24,178 2.41 14,408 37.47 202.57 13,728 7839.08 7839.08Berlin-100-5 89,308 31.00 115,441 3.40 165.24 653 4486.06 4486.06Berlin-100-10 94,292 23.57 105,131 7.29 189.58 1304 7223.52 7223.52Berlin-100-17 125,157 2.54 106,185 10.24 TLE 4496 − 9059.20Berlin-100-25 120,874 1.65 95,479 15.13 TLE 3303 − 10,617.35Berlin-125-5 149,873 17.23 160,811 2.83 378.48 451 5029.15 5029.15Berlin-125-10 117,395 38.16 143,950 5.68 344.71 3226 7461.74 7461.74Berlin-125-21 160,045 11.64 147,898 10.50 TLE 1766 12,187.03 11,376.17Berlin-125-31 172,228 5.53 140,215 15.53 TLE 1617 14,514.44 13,578.22Berlin-150-5 146,589 50.21 264,511 2.74 125.39 881 4364.78 4364.78Berlin-150-10 156,379 44.39 247,283 4.43 416.53 1353 6111.17 6111.17Berlin-150-25 252,049 6.74 220,367 10.91 TLE 694 13,593.25 12,857.60Berlin-150-38 261,190 7.53 207,107 20.01 TLE 3354 17,985.01 16,480.89Berlin-sparse-100-5 57,527 9.70 52,564 2.12 409.06 2184 5130.09 5130.09Berlin-sparse-100-10 54,689 11.30 49,468 4.13 TLE 11,707 − 8446.12Berlin-sparse-100-17 59,458 8.43 49,910 8.76 TLE 6854 10,216.15 9785.29Berlin-sparse-100-25 57,361 7.85 46,392 10.27 TLE 5569 − 11,707.45Berlin-sparse-125-5 87,398 17.25 87,832 2.41 155.21 714 6775.90 6775.90Berlin-sparse-125-10 71,869 31.43 84,693 4.97 1473.27 7858 8233.32 8233.32Berlin-sparse-125-21 93,007 9.48 78,531 9.47 TLE 3395 14,552.92 12,638.85Berlin-sparse-125-31 100,037 3.11 71,737 17.31 TLE 4687 18,403.51 18,105.72Berlin-sparse-150-5 89,486 37.40 117,877 2.14 411.18 2604 6975.62 6975.62Berlin-sparse-150-10 104,975 26.21 114,785 4.11 1454.45 5583 9027.46 9027.46Berlin-sparse-150-25 121,166 12.08 102,147 10.98 TLE 7369 18,602.94 18,569.49Berlin-sparse-150-38 135,487 6.58 100,440 17.13 TLE 4083 22,090.23 20,935.68Geneva-sparse-400-5 490,802 57.65 988,877 0.99 TLE 6523 18,135.72 14,521.78Geneva-sparse-400-10 604,529 48.91 1,002,680 2.01 TLE 737 28,315.98 19,799.40Geneva-400-67 1,111,568 0.67 869,521 8.93 TLE 0 − −Geneva-sparse-400-100 1,155,233 0.00 877,567 10.96 TLE 0 − 1687.61Geneva-sparse-500-5 737,399 56.84 1,447,070 0.78 TLE 9033 18,001.09 16,252.26Geneva-sparse-500-10 847,457 50.36 1,434,406 1.55 TLE 950 26,238.92 17,536.11Geneva-sparse-500-83 1,662,610 0.09 1,270,911 9.70 TLE 0 − 7722.70Geneva-sparse-500-125 1,661,763 0.00 1,205,306 13.69 TLE 0 − 17,362.47

Recall that average ranks are calculated so that when k methods have the ith best rank, they allget an average rank of k(2i + k − 1)/(2k). The reason for the superior performance of MIP-CSP-CSP

becomes more apparent now, as it was ranked first in 35 cases, with four ties, and ranked secondin the remaining four cases. Of course, as it relies on stronger tests, MIP-CSP-CSP was expected to



Table 5Instances solved to optimality and to feasibility after the different reduction procedures

MIP MIP-SP-SP MIP-CSP-SP MIP-CSP-CSP

#Optimal 4 17 18 18#Feasible 6 12 10 13Total 10 29 28 31

Table 6Reduction in percentage in the numbers of constraints and variables obtained with each reduction procedure

Constraints (%) Variables (%)

MIP-SP-SP MIP-CSP-SP MIP-CSP-CSP MIP-SP-SP MIP-CSP-SP MIP-CSP-CSP

Average 16.05 16.05 17.62 8.71 9.06 9.19Median 7.85 7.85 9.48 8.67 9.47 9.47Standard deviation 17.62 17.62 18.64 6.59 6.95 7.07

Table 7Number of instances for which each reduction procedure obtained a given average rank

Average rank MIP-SP-SP MIP-CSP-SP MIP-CSP-CSP

1 3 0 142 5 10 12.5 1 1 03 8 6 2

exhibit larger reductions. Notice, however, that the extra cost incurred by these tests strictly paidoff for 31 of the 39 instances. The question that remains open is how this affects the computingtimes.

To analyze the computing times, we concentrate on the 17 instances solved to optimality by thethree reduced MIPs, discarding instance Washington-40-10 that could not be solved by MIP-SP-SP.Again, we make use of average ranks. A cell in the Table 7 shows the number of instances that amethod got a given average rank.

One can check from the data that the greater reductions obtained by MIP-CSP-CSP translate directlyinto gains in running times. The average computing times (in seconds) on the 17 instances were565.12 for MIP-SP-SP, 561.14 for MIP-CSP-SP, and 374.12 for MIP-CSP-CSP. Therefore, on average, MIP-CSP-CSP saved about one-third of the computing time needed by the two other methods. A somewhatunexpected result is that MIP-SP-SP was faster than any other method for three instances. Althoughwe were not able to find a clear explanation for this, again, such behavior may have been caused bythe solver’s internal heuristics.



5. A relax-and-fix heuristic for MRPQOS

We propose a heuristic for the MRPQOS based on the relax-and-fix approach (cf. Wolsey, 1998). Thismethod starts with a relaxation of an integer formulation of the problem being investigated. Then,a series of iterations are executed in which three basic steps are made: (a) a subset of the continuousvariables in the model are forced to take integer values, (b) the optimum of the resulting model iscomputed, and (c) the variables in the chosen subset are fixed to their optimal values in subsequentiterations. If the current model is found to be infeasible in step (b), then the heuristic fails and nosolution is returned.

5.1. Algorithm and pseudocode

In the case of MRPQOS, the algorithm of the relax-and-fix heuristic relies on the integer program-ming formulation proposed in Section 3.1. The linear relaxation of MIP is obtained replacingconstraints (14), (15), and (16) by their respective relaxations, that is, by constraints

0 ≤ xki j ≤ 1, ∀ (i, j) ∈ A′, k ∈ D,

0 ≤ xqi j ≤ 1, ∀ (i, j) ∈ A′, q ∈ S,

0 ≤ yi j ≤ 1, ∀(i, j) ∈ A′, k ∈ D ∪ S.

The pseudocode of the relax-and-fix RNF heuristic is presented in Algorithm 1. In line 2, MODEL isinitialized with the linear relaxation of the MIP-CSP-CSP formulation in Section 3. Line 5 computes ashortest path P from s to k constrained to a maximum delay of �d using the same costs μi j associatedto the arcs (i, j) of A as in the MIP formulation. The delay consumed in path P is used to determinethe order in which the terminals are processed. Line 10 sorts the terminals in nondecreasing orderof the delay consumption of the respective paths computed in line 5. The intuition behind this stepis that the higher the position of a terminal k in the sorted list, the more difficult it should be tofind a “cheap” path from s to k that respects the delay limit. So, after this line, the terminals can bethought as being sorted from the easiest to the hardest ones to connect.

The loop in lines 11–21 attempts to obtain a feasible arborescence by constructing a path joinings to one terminal at each iteration. The order in which the terminals are treated is such that themost difficult to connect, according to the discussion above, are considered first. At the beginningof an iteration, an arborescence rooted at s is available that connects all terminals visited in theprevious iterations (i.e., those whose labels are stored in index[�], for � > count). This arborescencecorresponds to the binary variables whose values were fixed in earlier iterations (in line 18). Thenext terminal to be connected is determined in line 12. To find the best way to connect the currentterminal k to this partial solution, the arc variables associated to k, which were relaxed before, arenow forced to be binary in line 13. Then, the resulting mixed integer model is optimized in line 14.If the status of the model is INFEASIBLE, the heuristic fails to produce a solution for MRPQOS andstops in line 16. Otherwise, a feasible path was found that connects k to the current arborescence.To force this path to take part of the solution being built, in line 18 the arc variables correspondingto k are set to their optimal values computed in line 14. Line 20 updates the counter for the nextiteration. Finally, if the loop has not been prematurely interrupted in line 16, the algorithm returnsa feasible solution in line 22.



Algorithm 1. Relax-and-fix heuristic

5.2. Computational experiments with the RNF heuristic

The solutions produced by the RNF heuristic are now compared with those generated by the MIP

and the MIP-CSP-CSP models and reported in Tables 1 and 4, respectively. The results from MIP wereconsidered in this analysis because, among the options offered in this work, this is the easiest one toimplement as it only requires loading the MIP model in a commercial solver. On the other hand, itmakes sense to compare the RNF heuristic with the MIP-CSP-CSP as the latter yielded the best resultsso far, at least in what concerns optimal solutions. We now assess these methods relatively to theirability to compute, in an efficient way, solutions that are not necessarily optimal but still have goodquality. The computational setup is the same as in the previous experiments, including the runningtime limit of 2400 seconds.

Table 8 summarizes the results obtained by the three methods for the instances in our bench-mark. The first column identifies the instance. The second and third columns show the cost ofthe best solution obtained and the computation time for the MIP model, i.e., without preprocess-ing. The fourth and fifth columns give the same information, but relative to model MIP-CSP-CSP.



Table 8Comparison between the RNF heuristic, the MIP model and the MIP-CSP-CSP model

MIP MIP–CSP–CSP RNF heuristic

InstanceBestinteger

Time(seconds)

Bestinteger

Time(seconds)

Bestinteger

Time(seconds)

Washington-9-3 2886.87 0.13 2886.87 0.02 2886.87 0.01Washington-20-3 1417.89 1.93 1417.89 0.08 1663.55 0.09Washington-20-5 1712.36 7.36 1712.36 4.71 1712.36 0.76Washington-20-10 2918.79 TLE 2918.79 612.11 3675.51 1.41Washington-40-7 3785.69 TLE 3785.69 17.49 3930.41 4.21Washington-40-10 3821.84 TLE 3819.16 2223.51 4459.22 9.72Washington-40-20 8142.29 TLE 7839.08 202.57 7868.21 10.03Berlin-100-5 − TLE 4486.06 165.24 4693.36 99.24Berlin-100-10 8523.66 TLE 7223.52 189.58 7486.42 101.93Berlin-100-17 − TLE − TLE 10,634.18 193.16Berlin-100-25 − TLE − TLE 12,007.01 125.81Berlin-125-5 − TLE 5029.15 378.48 5029.15 193.63Berlin-125-10 7665.14 TLE 7461.74 344.71 7476.91 53.29Berlin-125-21 (+) − TLE 12,187.03 TLE 13,134.23 299.98Berlin-125-31 (+) − TLE 14,514.44 TLE 15,772.31 373.67Berlin-150-5 − TLE 4364.78 125.39 4364.78 38.53Berlin-150-10 − TLE 6111.17 416.53 6611.79 109.61Berlin-150-25 (−) − TLE 13,593.25 TLE 13,474.47 409.73Berlin-150-38 (+) − TLE 17,985.01 TLE 18,016.60 364.52Berlin-sparse-100-5 5130.09 2116.12 5130.09 409.06 5271.67 11.14Berlin-sparse-100-10 − TLE − TLE 9807.49 29.07Berlin-sparse-100-17 (+) − TLE 10,216.15 TLE 14,010.70 45.83Berlin-sparse-100-25 − TLE − TLE 14,322.67 47.36Berlin-sparse-125-5 − TLE 6775.90 155.21 7149.93 24.32Berlin-sparse-125-10 − TLE 8233.32 1473.27 9331.71 25.58Berlin-sparse-125-21 (+) − TLE 14,552.92 TLE 14,808.38 56.38Berlin-sparse-125-31 (+) − TLE 18,403.51 TLE 19,508.88 71.05Berlin-sparse-150-5 − TLE 6975.62 411.18 7107.21 33.53Berlin-sparse-150-10 − TLE 9027.46 1454.45 9169.39 21.69Berlin-sparse-150-25 (+) − TLE 18,602.94 TLE 19,694.28 85.85Berlin-sparse-150-38 (+) − TLE 22,090.23 TLE 25,493.95 141.88Geneva-sparse-400-5 (−) − TLE 18,135.72 TLE 18,013.16 87.98Geneva-sparse-400-10 − TLE 28,315.98 TLE − FAIL

Geneva-400-67 − TLE − TLE − FAIL

Geneva-sparse-400-100 − TLE − TLE − FAIL

Geneva-sparse-500-5 (+) − TLE 18,001.09 TLE 18,087.17 209.46Geneva-sparse-500-10 (−) − TLE 26,238.92 TLE 21,912.81 690.63Geneva-sparse-500-83 − TLE − TLE − FAIL

Geneva-sparse-500-125 − TLE − TLE − FAIL

Finally, the two last columns refer to the RNF heuristic. As before, TLE stands for “time limitexceeded.”

The data reveal that while MIP produced feasible solutions in only ten cases of the 39 tested,MIP-CSP-CSP and RNF did it for 31 and 34 instances, respectively. The poor performance of MIP even



when we give up on optimality confirms that, without preprocessing, the MIP model is not veryuseful. So, in the analysis that follows, we discarded this model for further considerations and turnour attention to the two other methods.

Among the 18 instances with known optima, i.e., those whose rows are highlighted in dark gray( ), RNF succeeded to obtain them four times. In the remaining 14 cases, the average deviation fromthe optimum was 5.89%, with the median being only 3.20% and the maximum attaining 25.93%(observed for instance Washington-20-10).

There are 13 instances for which a feasible solution was found by MIP-CSP-CSP but whose op-timality could not be proved, for which their rows are highlighted in light gray ( ). In thisgroup, instance Geneva-sparse-400-10 is the only one for which MIP-CSP-CSP found a feasible so-lution and the RNF heuristic failed to do so. The remaining 12 instances have their names inTable 8 followed by a “+” sign when the cost of the solution produced by RNF has a largercost than that coming from MIP-CSP-CSP, otherwise a “−” sign is shown. One can see that inthree of these instances, RNF found a better solution than MIP-CSP-CSP, while the opposite oc-curred in the nine other cases. We compute the gap in percentage between the solutions of thetwo methods through the formula ((zRNF − zMIP-CSP-CSP)/zMIP-CSP-CSP) × 100, where zX is thecost of the solution obtained by method X . The average gap was 5.44% with a median valueof 3.81% and, although the extreme cases are far from the average gap (the maximum was37.14% for Berlin-sparse-100-17 and the minimum −16.49% for Geneva-sparse-500-10), in gen-eral, one can say that RNF performs only slightly worse than MIP-CSP-CSP with respect to solutionquality.

It should be noticed that RNF returned feasible solutions for Berlin-100-17, Berlin-100-25, Berlin-sparse-100-10, and Berlin-sparse-100-25. In these four cases, MIP-CSP-CSP was unable to find feasiblesolutions. However, the true benefits of using RNF are better illustrated by the analysis of computingtimes.

The average and median running times for RNF over the 34 instances for which it has suc-ceeded were 116.80 and 54.84 seconds, respectively. These values drop, respectively, to 41.04and 23.01 seconds (with a maximum of 193.63 seconds), if we consider only the instanceswith known optima. For this latter group of instances, the same statistics for MIP-CSP-CSP are467.87 and 273.64 seconds (with a maximum of 2223.51 seconds), i.e., one order of magnitudehigher.

From the previous analysis, the RNF heuristic appears as a viable alternative to tackle prob-lem MRPQOS, as it most often produces solutions of high quality in short computing times. Inspite of that, improvements in Algorithm 1 are possible. For example, the constrained short-est path computed in line 5 could also consider the jitter as the bounding resource. Our deci-sion to use the delay to play this role was based on the fact that, in our benchmark, delay wasmore restrictive than jitter in general. Of course, we could simply rerun the algorithm using jit-ter rather than delay, and pick the best solution between the two runs. This would likely dou-ble the computing times but, as we privileged the efficiency of the heuristic, we opted for notdoing so.

Additional enhancements to the RNF heuristics can be conceived. For instance, to reduce thefailures of the heuristic and to encounter better solutions, one could think of embedding the mainloop of lines 11–21 into an external loop that is repeated for a limited number of iterations andwhere the greedy choice driven by the sorting in line 10 is randomized in the same way as in the



Fig. 3. Input graph for instance Washington-40-7.

Fig. 4. Solution found by RNF heuristic for instance Washington-40-7.

construction phase of the GRASP metaheuristic (cf. Resende and Ribeiro, 2016). Nevertheless, itis worth noting that the feasibility version of the MRPQOS is most likely NP-hard, since it is closelyrelated to the problem of finding a minimum cost Steiner tree with bounded diameter and otherdifficult problems on optimal trees (cf. Ho et al., 1991; Ding and Xue, 2014). As a consequence,even proving that an instance has no feasible solution could be a quite challenging task. In ourbenchmark, no solution was found by any method for four instances. Therefore, in case they wereactually infeasible, even the randomization discussed above would not be helpful.

To complete the presentation of the experimental results, Figs. 3 and 4 illustrate, respectively,the input graph and the solution produced by the RNF heuristic for instance Washington-40-7. Theinput graph for the largest instance solved by the heuristic, Berlin-100-10, is displayed in Fig. 5. Inthese drawings, the green vertex is the source, while the red ones are the terminals.



Fig. 5. Input graph for instance Berlin-100-10.

6. Conclusions

This paper investigated a multicast routing problem with QoS constraints on VANETs. An integerprogramming formulation for the problem was proposed that forms the basis of a relax-and-fixheuristic designed with the goal of producing feasible solutions of good quality. In addition, pre-processing procedures relying on simple and constrained shortest paths were developed that reducethe model size to the point of making it viable to compute. Realistic test instances were generatedand computational experiments were reported that confirm the importance of the preprocessingand the effectiveness of the relax-and-fix heuristic.

As the methods proposed here immediately benefit from any improvement in the preprocessingprocedures, the development of stronger and more efficient tests to reduce the model size is apromising research direction. Also, considering the practical application, it would be desirable toconceive heuristics to find feasible solutions very fast, even at the expense of some loss in quality.The latter issue is studied in a forthcoming paper.



Acknowledgments

The work of Celso C. Ribeiro was partially supported by CNPq research grants 303958/2015-4 and425778/2016-9 and by FAPERJ research grant E-26/202.854/2017. The work of Cid C. de Souzawas supported by CNPq research grant 304727/2014-8.

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