European Journal of Operational Research 233 (2014) 220–233

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

Optimal relay node placement in delay constrained wireless sensornetwork design

0377-2217/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.ejor.2013.08.031

⇑ Corresponding author. Address: FPM-28, IIM, Lucknow 226 013, India. Tel.: +919936009555.

E-mail address: nigam.ashutosh@iiml.ac.in (A. Nigam).

Ashutosh Nigam ⇑, Yogesh K. AgarwalDecision Sciences Group, Indian Institute of Management, Lucknow, India

a r t i c l e i n f o

Article history:Received 16 June 2012Accepted 20 August 2013Available online 30 August 2013

Keywords:Relay node placementCutting plane/facetPolyhedral theoryProjectionBranch and cutLagrangian-relaxation

a b s t r a c t

The Delay Constrained Relay Node Placement Problem (DCRNPP) frequently arises in the Wireless SensorNetwork (WSN) design. In WSN, Sensor Nodes are placed across a target geographical region to detectrelevant signals. These signals are communicated to a central location, known as the Base Station, for fur-ther processing. The DCRNPP aims to place the minimum number of additional Relay Nodes at a subset ofCandidate Relay Node locations in such a manner that signals from various Sensor Nodes can be commu-nicated to the Base Station within a pre-specified delay bound. In this paper, we study the structure of theprojection polyhedron of the problem and develop valid inequalities in form of the node-cut inequalities.We also derive conditions under which these inequalities are facet defining for the projection polyhe-dron. We formulate a branch-and-cut algorithm, based upon the projection formulation, to solve DCRNPPoptimally. A Lagrangian relaxation based heuristic is used to generate a good initial solution for the prob-lem that is used as an initial incumbent solution in the branch-and-cut approach. Computational resultsare reported on several randomly generated instances to demonstrate the efficacy of the proposedalgorithm.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

A Wireless Sensor Network (WSN) consists of spatially distrib-uted Sensor Nodes (SNs) to monitor physical or environmental con-ditions, such as temperature, pressure, motion or pollutants. TheseSNs transmit the sensed data through wireless communication to aBase Station (BS) (Clare, Pottie, & Agre, 1999). SNs may be placed in-side the event to be monitored or in the proximity of the same.These features ensure a wide range of applications for WSN in variedareas, e.g. health-care, military operations, and environmental mon-itoring. WSN may be deployed in a vast geographical area (e.g.oceans, forests) in order to detect critical events such as forest-fire,tsunami, and floods. Using WSN, doctors can remotely monitorphysiological condition of their patients. WSNs can be an essentialpart of military operations with their ability to perform key strategictasks, e.g. battlefield surveillance, reconnaissance of rival armiesetc. (cf. Akyildiz, Su, & Sankarasubramaniam, 2002).

Transmission radius of SNs (the range beyond which they can-not transmit the signals) is typically several tens of meters. Due tothe limited transmission radius and the vastness of target geo-graphical region, usually a multi-hop wireless communication,using some additional Relay Nodes (RNs), is required to facilitate

end-to-end communication between SNs and the BS. As the costof RNs ranges from tens to hundreds of dollars, minimizing thenumber of additional RNs without compromising the quality of sig-nals is an important aspect of the WSN design.

Objective of the Delay-Constrained Relay Node Placement Prob-lem (DCRNPP) is to design a multi-hop wireless mesh networkwith minimum number of additional RNs in order to facilitatewireless-communication between each of the SN and the BS. Theplacement of RNs should ensure that the delay on the paths be-tween BS and the SNs is restricted within a pre-specified delaybound. DCRNPP studied in this paper is motivated by an importantclass of WSN, where locations of the Candidate Relay Nodes (CRNs)are known a priori. For example, in forest fire detection, a set ofsites where the CRNs can be placed may be known beforehand.In brief, the key features of DCRNPP, studied in this paper, arethe following.

� The locations of SNs and CRNs are known beforehand.� Transmission radius of the SN/RN allows only certain links to be

permitted in the graph.� The objective is to obtain a sub-graph with minimum number of

CRNs selected that connects all SNs to BS.� Placement of RNs must ensure that there exists at least one path

from each SN to the BS, for which the cumulative delay does notexceed a pre-specified delay bound D.

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 221

The version of DCRNPP, with relaxation of the delay constraint,known as the Relay Node Placement Problem (RNPP) is broadly re-lated to the Steiner tree problem (STP) in graphs (Gondran & Min-oux, 1984). Classical STP is a NP-Hard problem and is extensivelystudied by various researchers (Robins & Zelikovsky, 2005; Kar-pensiki & Zelikovsky, 1993; Arora, 1998; Chopra & Rao, 1994a;Chopra & Rao, 1994b).

The Prize collecting Steiner tree Problem (PCSP) (or the Nodeweighted Steiner tree Problem, NSP), where node weights alongwith edge weights are specified, can be considered as a general-ization of both STP and RNPP (Duin & Volgenant, 1987; Segev,1987). Approximation algorithms for PCSP are proposed by vari-ous researchers (Klein & Ravi, 1995; Demaine & Hajiaghayi,2009; Remy & Steger, 2009; Canuto, Resende, & Ribeiro, 2001;Klau et al., 2004). Fischetti (1991) studied the facial structure ofa generalization of PCSP, known as the Steiner arborescence (ordirected Steiner tree) problem, and pointed out that the PCSPcan be transformed into it. Engevall, Lundgren, and Värbrand(1998) proposed another ILP formulation for the PCSP, based onthe shortest spanning tree problem formulation, which was intro-duced originally by Beasley (1989) for the Steiner tree problem. Acutting plane algorithm for the PCSP based on generalized sub-tour elimination constraints was proposed by Lucena and Re-sende (2004).

The RNPP was studied as The Steiner Tree Problem with Min-imum Number of Steiner Points and Bounded Edge Length (STP-MSPBEL) by Lin and Xue (1999). They showed the problem tobe NP-complete and proposed a polynomial time 5-approxima-tion algorithm for the problem. Cheng, Du, Wang, and Xu(2008) studied the same problem and proposed a 3-approxima-tion and a 2.5-approximation algorithm. Voss (1999) studiedthe STP with hop constraints. The problem was shown to beNP-hard and a minimal spanning tree based heuristic was pro-posed to obtain a good feasible solution. Kim, Bang, and Choo(2006) studied the delay and delay variation constrained multi-casting STP. The problem is similar to the one studied by Voss,with a delay constraint instead of the hop constraint, and a con-straint on delay variation between two sources. They proposed apolynomial time heuristic algorithm for the problem. Costa, Cor-deau, and Laporte (2008) studied the STP with revenue, budget,and hop constraints. They proposed a greedy heuristic for gener-ating initial solution. The initial solution was improved by the de-stroy and repair or the tabu search algorithm. Gouveia, Paias, andSharma (2008) studied rooted distance-constrained minimumspanning tree problem, and proposed a path based formulation.They presented a column generation scheme and a Lagrangianrelaxation based approach combined with sub-gradient optimiza-tion procedure to solve the problem. Misra, Hong, Xue, and Tang(2008) studied the constrained relay placement problem for con-nectivity and survivability, and proposed an approximation algo-rithm for the same. Their model took into account thetransmission radius as the edge length bound. Bhattacharya andKumar (2010) studied DCRNPP and showed the problem to beNP-Hard. They presented a local search based greedy heuristicto provide an approximate solution for the problem.

Another class of problem closely related to the RNPP is the Net-work Design Problem with Relays (NDPR). The NDPR is defined onan undirected graph G = (V,E,K), where V and E are the vertex andedge sets, respectively. Set K = {o(k) 2 V,d(k) 2 V} is a set of commu-nication pairs or commodities. Here, o(k) and d(k) denote the originand destination of kth commodity, respectively. A cost cij is associ-ated with each edge (i, j) 2 E and a fixed cost fi, of installing a relay atvertex i, is associated with each vertex i 2 V. The objective of NDPRis to select a subset E0 # E and a subset V0 # V in such a way thatthe total cost of network (edge cost and the relay installation cost)

is minimized, and there exists a path linking the origin o(k) and des-tination d(k) for each commodity k 2 K in which the length betweenany two consecutive nodes does not exceed a preset upper bound.Cabral, Erkut, Laporte, and Patterson (2007) developed a lowerbound procedure and several heuristics for NDPR. They comparedthese algorithms on several randomly generated test instances. Li,Aneja, and Huo (2011) developed a Branch-and-Price algorithmfor directed version of NDPR, using an arc-path formulation. Konak(2012) presented a new formulation for NDPR based on set cover-ing constraints. Using the new formulation, he proposed a GeneticAlgorithm based heuristic to solve NDPR.

To the best of our knowledge, there is no algorithm in the liter-ature that solves DCRNPP optimally. Apart from developing abranch and cut based exact algorithm to solve DCRNPP, this paperalso examines the polyhedron structure of the problem and pro-poses a projection formulation for DCRNPP. With the help of com-putational experiments on several randomly generated testinstances, we demonstrate that the proposed algorithm based uponprojection formulation is able to optimally solve problem instancesof size up to 50 SNs and 200 CRNs, within reasonable CPU time.

The paper is organized as follows. In Section 2, we describe amathematical formulation for DCRNPP that involves an exponen-tially large number of path variables. Since not all the columns inthis formulation are known explicitly, a column generation ap-proach is presented in Section 3. The column generation approachsolves the LP relaxation of the path-based formulation to computea valid lower bound for the optimal solution. In Section 4, we de-scribe a projection formulation for the problem, which involvesvariables corresponding to CRNs only. We identify a set of validinequalities for the projection formulation, known as the node cutinequalities. These inequalities are facet defining for the projectionpolyhedron under certain mild conditions. In Section 5, we presenttwo separation algorithms to generate violated node cut inequali-ties for the projection formulation. A heuristic to generate a goodfeasible solution, which is used as an initial incumbent in thebranch and cut algorithm, is discussed in Section 6. In Section 7,we discuss the implementation details for the branch and cut algo-rithm that is used to solve DCRNPP optimally. Computational re-sults on various randomly generated test instances are reportedin Section 8. In Section 9, we summarize this work and identifypossible areas for future research.

2. Problem formulation

In this section, we describe a path-based formulation for theDCRNPP. The problem is defined on an undirected graphG = (V,E). The SN set T = {1,2, . . . ,m}, CRN set R = {1,2, . . . ,n} andthe BS (node 0) constitute the node set of the graph i.e.V = T [ R [ {0}. The edge set in the graph is defined asE = {(i 2 V, j 2 V): lij 6 r}, where lij is the Euclidian distance, betweennode i and node j, bounded by the transmission radius r. Anon-negative delay dij is associated with each edge (i, j) 2 E andmay be defined as some function of the Euclidean distances lij.We define the set of neighbors for each node i 2 V, asN i ¼ fj 2 V : ði; jÞ 2 Eg.

The set Pk is defined as set of all paths between BS and the SNk 2 T. Set P(=

Sk2TPk) contains all paths between BS and all SNs.

The set Pk,D( # Pk) is defined as set of all paths between the BSand SN k 2 T within the delay bound D. Set PD consists of all de-lay-constrained paths between the BS and all SNs, i.e.PD =

Sk2TPk,D.

Following variables are used in the problem-formulation:

� For each CRN j 2 R, binary variable yj indicates whether a RN isplaced at CRN location j or not.

222 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

� For each path p 2 Pk,D, binary variable xp indicates whether pathp is available between the BS and the SN k or not.

For any path p 2 Pk,D, the set of CRNs and the set of edges on thepath are defined as Rp and Ep, respectively. We define coefficient ap

j

as 1 if j 2 Rp and 0 otherwise. The DCRNPP can be formulated as thefollowing integer program:

F1 :

MinimizeXj2R

yj ð1Þ

subject toX

p2Pk;D

apj xp6 yj; 8k 2 T; 8j 2 R ð2Þ

Xp2Pk;D

xp P 1; 8k 2 T ð3Þ

xp 2 f0;1g; 8p 2 Pk;D ð4Þyj 2 f0;1g; 8j 2 R ð5Þ

The objective function (1) aims to place RNs at minimum num-ber of CRN locations. Constraints (2) state that a path is availableonly if all the CRNs on that path are selected. Constraints (3) arethe connectivity constraints for each SN, and ensure that there isat least one delay-constrained path available between each SNand the BS.

3. Column generation

The formulation F1 has an exponentially large number of pathvariables. This naturally suggests the use of column generation ap-proach, well known as being effective for problems with exces-sively many columns (Gilmore & Gomory, 1961; Savelsbergh,1997; Desaulniers, Dumas, Solomon, & Soumis, 2005). Since col-umn generation is well documented in the literature, we brieflyintroduce the master problem and corresponding pricing subprob-lem in this section.

3.1. Master problem

As the number of columns in formulation (F1) can be very large,the columns are progressively introduced into master problem. LetLF1 be the linear relaxation of F1, i.e. the problem without integral-ity constraints for variables xp and yj. The restricted LP relaxationRLF1 is initiated with small arbitrary set of paths P0(�PD), and thenfurther columns are iteratively added to the restricted formulation.The dual variables associated with the optimal solution of RLF1 areused to define a pricing subproblem that computes paths betweenthe BS and the SNs with a negative reduced cost, if any. These pathsare then added to the current RLF1 to obtain next RLF1, which issolved again to obtain new dual variables. This iterative procedureis repeated until no more paths with a negative reduced cost can befound. At this point, the optimal solution for LF1 has been obtained.This solution provides a lower bound on the optimal solution ofDCRNPP.

Initialization of RLF1: Since the initial set of columns in RLF1 isrequired to ensure feasibility, RLF1 can be initialized with columnsthat correspond to the set of minimum delay paths between eachof the SN and the BS. The minimum delay path between a SNand the BS can be computed by finding the shortest path betweenthem, while the edge length is defined as delay on the correspond-ing edge.

3.2. Pricing subproblem

Let z⁄ be the optimal solution for RLF1, and kkj and lk be the

values of dual variables corresponding to constraints (2) and

(3), respectively. Then z⁄ is the optimal solution of LF1 if theredoes not exist any path with negative reduced cost between theBS and any of the SNs. The pricing subproblem aims to find a de-lay-constrained shortest path with most negative reduced cost.The subproblem can be formulated as the following integerprogram:

SP :

Minimize �cp ¼mink2T

Xj2Rp

kkj � lk

0@

1A

subject to p 2 PD

ð6Þ

In order to speed up the process, at each iteration we identifymore than one columns with negative reduced cost instead of add-ing only one column with most negative reduced cost. We identifya delay-constrained shortest path between each SN and the BSwith the most negative reduced cost. The pricing subproblem SPcan be decomposed into m subproblems (SP(k)) (one for eachk 2 T). The subproblem is defined as:

SPðkÞ :

Minimize �cp;k ¼Xj2Rp

kkj � lk

subject to p 2 Pk;D

ð7Þ

SP(k) may be considered as the Delay Constrained Shortest PathProblem (DCSPP), where edge length for each edge (i, j) 2 E is de-fined as:

cij ¼kk

i þ kkj

� �2

with kk0 ¼ kk

k ¼ 0� �

The DCSPP is known to be a NP-Hard problem (Dror, 1994).There are no known polynomial-time or pseudo-polynomial-timealgorithms to solve this problem. Dynamic programming is agood candidate approach to solve the DCSPP. However, thedimensionality of the associated state-space is a major issue thataffects the run-time performance of dynamic programmingapproach.

3.3. Solving the pricing subproblem

In our implementation, we have used a well-known dynamicprogramming based algorithm to solve DCSPP. The dynamic pro-gramming based algorithm assumes that the delay associated witheach edge is an integer value. The assumption helps in establishingstates for dynamic programming in form of (i,s) for each i 2 V andfor s = 0, 1, . . . , D. We define f(i,s) as the length of the shortest pathbetween BS and node i 2 S [ T with delay at most s. The valuesf(i,s) can be computed using the following recursion:

f ði; sÞ ¼ minj2N i ;dji6s

ðcji þ f ðj; s� djiÞÞ ð8Þ

f ð0; sÞ ¼ 0; 8s ¼ 0;1; . . . ;D ð9Þ

A node labeling method by (Desrochers, 1988) along with animprovement suggested by Gouveia et al. (2008), to reduce thenumber of state variables is employed to solve DCSPP using dy-namic programming based algorithm.

3.4. The column generation algorithm

The column generation procedure is initialized with the set ofminimum delay paths between BS and all the SNs. The restrictedmaster problem RLF1 is solved to optimality. The dual variables,obtained from RLF1, are used to construct the pricing subproblem.The subproblem SP (k) for each k 2 T is solved using the dynamic

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 223

programming based algorithm. Columns with negative reducedcost are added to RLF1. If no columns with a negative reduced costcan be found, then an optimal solution for LF1 is obtained and nofurther iterations are required. The flowchart of the column gener-ation algorithm is presented in Fig. 1.

4. Projection formulation

In the previous section, we presented a column generation ap-proach to find a lower bound for DCRNPP. As we shall demonstratelater with the help of computational experiments, the columngeneration approach is computationally inefficient due to variousreasons, some of which are as follows:

� The pricing subproblem (DCSPP) is solved using a dynamic pro-gramming algorithm. The dynamic programming based algo-rithm has a burdensome dimensionality aspect and can beunacceptably slow in practice [Car08].� Often, many iterations are required to prove optimality, while

the optimal value of RLF1 does not change significantly fromone iteration to another [Nab10]. This leads to a significantincrease in the computation time.� The path-based formulation (F1) has an exponentially large

number of variables and solving an integer program (in orderto obtain the optimal solution for DCRNPP) with these manyvariables is practically unviable.

To overcome these issues pertaining to the column generationtechnique, in this section we present an alternate formulation forDCRNPP. The new formulation eliminates path variables from F1.Before presenting this formulation, we discuss the concept of nodecuts in a graph. These node cuts correspond to the node cut inequal-ities for DCRNPP that provide the structure of the projectionformulation.

4.1. Node cuts and node cut inequalities

A node cut (c # R) is a set of CRNs, deletion of which discon-nects all feasible paths between the BS and a SN k 2 T (Moss &Rabani, 2007). In other words, at least one CRN from a node cut

Fig. 1. Overview of the column-generation procedure.

must be selected to ensure connectivity between BS and SN k. Aminimal node cut is the smallest possible node cut, i.e. a node cutis minimal if none of its subset is a node cut. Let C be the set ofall node cuts in a graph, a minimal node cut c 2 C satisfies the fol-lowing property:

c n fjg R C; 8j 2 c ð10Þ

A singleton node cut is a node cut with single element in it, i.e.jcj = 1. A node cut inequality is defined as:Xj2c

yj P 1; ð11Þ

where c 2 C is a minimal node cut and C is the set of all minimalnode cuts. Node cut inequalities ensure that at least one CRN fromeach minimal node cut is selected in a feasible solution.

Theorem 1. Node cut inequalities (11) are valid for DCRNPP.

Proof. From the definition of minimal node cuts, the proof of thetheorem is straightforward. If all CRNs from a minimal node cutare deleted, the induced graph is disconnected. Hence, every feasi-ble solution of DCRNPP must have at least one CRN selected fromeach node cut c 2Ck. h

4.2. Projection in the space of CRN variables

Let H be the set of all feasible solutions (x,y) to the path basedformulation F1, and letHc ¼ ConvðHÞ, where Conv (S) is the convexhull of a set S. Let T ¼ fy 2 Bnj9x 3 ðx; yÞ 2 Hg and T c ¼ ConvðT Þ.In other words, T c is the projection of Hc in the subspace of binaryvariables yj. We show that the node cut inequalities correctly de-scribe the projection polyhedron T c , i.e.T c ¼ y 2 Bnj

Pj2cyj P 18c 2 C

n o.

Theorem 2. Polyhedron T c described by node cut inequalities (11)is a valid projection for the polyhedron Hc in the space of y -variables.

Proof. In order to prove this theorem, we need to prove the fol-lowing statements:

I. All integer solutions ðx; yÞ 2 H satisfy constraints (11).II. If for an integral vector y 2 Bn; 9= x 3 ðx; yÞ 2 Hc then y does

not satisfy constraints (11).

Theorem 1 establishes (11) as valid inequalities for H, hence allfeasible solutions ðx; yÞ 2 H naturally satisfy constraints (11) andthus statement I is proved.

In order to prove statement II, consider a binary solutionvector �y0 2 Bn such that 9= �x0 3 ð�x0; �y0Þ 2 Hc , while �y0 satisfies thenode cut inequalities (11). However, since there are no feasiblesolutions using the selected CRNs (CRNs for which �y0

j ¼ 1) itimplies that there are no delay-constrained paths between BS andat least one of the SNs with respect to the solution vector �y0. Letk⁄ 2 T be one such SN for which there are no delay-constrainedpaths. Surely, 8p 2 Pk� ;D9jp 2 Rp such that �y0

jp¼ 0 (if 9p0 2 Pk� ;D such

that 8j 2 Rp; �y0j ¼ 1 then path p0 is a feasible path between the BS

and the SN k⁄ using the selected CRNs). The set �c ¼S

p2Pk� ;Dfjpg is avalid node cut with

Pj2�c�y0

j ¼ 0. A minimal node cut c# �c, can befound with

Pj2c�y0

j ¼ 0 violating one of the node cut inequalities(11). This contradicts the assumption that binary vector �y0

satisfies all node cut inequalities. Hence, for an integral vectory 2 Bn, if 9= x 3 ðx; yÞ 2 Hc then y does not satisfy constraints(11). h

224 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

Theorem 3. Polytope T c is full dimensional if and only if there are nosingleton node cuts for any of the SN k 2 T.

Proof. Let’s assume that there exists a singleton node cut for anyk 2 T, in the form of j�k

� �. Consequently, any y 2 T c cannot have

yj�k¼ 0, because then there are no paths between the BS and the

SN k. Hence, if the stated condition is not satisfied, any y 2 T c musthave yj�k

¼ 1, and therefore T c cannot be full dimensional.To show that the stated condition is sufficient, let’s create

following n vectors for each CRN j0 2 R:

yj0j ¼

0; if j ¼ j0

1; Otherwise

�

If the stated condition is satisfied, clearly these vectors, and they-vector with all 1’s (that corresponds to all CRNs selected) providen + 1 affinely independent integer solutions in T c . h

4.3. Facet conditions for node cut inequalities

Theorem 3 established the necessary and sufficient conditionsfor full dimensionality of polyhedron T c . We assume from nowon that the polyhedron T c is full dimensional. In this sub-section,we will discuss the conditions under which the node cut inequali-ties (11) are facet defining for the projection polyhedron T c.

Theorem 4. A node cut inequality (11), with respect to a minimalnode cut c, is facet defining for T c, if and only if "j 2 Rnc,$j⁄ 2 c suchthat c [ {j}n{j⁄} is not a node cut.

Proof. Let F be the face of polytope T c defined by the node cut

inequality (11), i.e. F ¼ yjy 2 T c;P

j2cyj ¼ 1n o

. For inequality (11)

to be facet defining, we need to demonstrate thatdimðFÞ ¼ dimðT cÞ � 1. Since T c is full dimensional, clearlydimðT cÞ ¼ n. In order to prove the sufficiency of the mentionedcondition, we need to construct n affinely independent feasiblesolution vectors that satisfy inequality (11) as equality.

We define the following q vector for each CRN t 2 c, whereq = jcj:

ytj ¼

0 if j 2 c n ftg1 Otherwise

�

Since c is a minimal node cut, none of its subset can be a node cut.Set cn{t} is clearly not a node cut as it is a subset of minimal nodecut c. As all CRNs are selected in yt

j except the CRNs in cn {t} (whichis not a node cut), yt

j ; 8t 2 c is a feasible solution vector for DCRNPP.Vectors yt

j also satisfy node cut inequality (11) as equality.Now let’s define the following n � q vectors for each t0 2 Rnc:

yt0j ¼

0 if j 2 ft0g [ c n ft�g1 Otherwise

�

where t⁄ 2 c is such that c [ {t0}n{t⁄} R C. Note that if the conditionof the theorem is satisfied, there exists such a t⁄ for every t0 2 Rnc.Since the set c [ {t0}n{t⁄} is not a node cut and all CRNs except thenodes in c [ {t0}n {t⁄} are selected as the part of given solution vec-tor, clearly each yt0

j 8t0 2 R n c is a feasible solution for DCRNPP andalso satisfies inequality (11) as equality.

Now consider the following n � n matrixN ¼ ðy1; . . . ; yqjyqþ1; . . . ; ynÞ. Matrix N can be partitioned andviewed as:

We want to show that n vectors of N are affinely independent. Thatis any multiplier vector k such that Nk = 0, eTk = 0 implies that k = 0,where eT is the row vector of all 1’s of appropriate size. Let’s con-sider Ni, as row i > q of N. It has a zero in position i and all other ele-ments are one. Thus, Mik = 0 and eTk = 0 imply ki = 0. Thus ki = 0"i > q. This coupled with the fact that q � q sub-matrix is an identitymatrix results in ki = 0"i 6 q. Hence, n columns of N are affinelyindependent. In this way, we can construct n affinely independentfeasible solution vectors that satisfy inequality (11) as equality. Thisproves the sufficiency condition dimðFÞ ¼ dimðT cÞ � 1.

To show the necessity of these conditions, let N be a squarematrix of some n affinely independent feasible vectors y1, . . . , yn

that satisfy (11) as equality. Let the top q rows correspond to CRNsin node cut c. Clearly, y1, . . . , yn does not contain the 0 -vector, andhence these n vectors are also linearly independent. Now let’sconsider the sub-matrix N0 consisting of only top q rows of N. Thereexist some q columns of N0 that are linearly independent, andcondition of the theorem is satisfied. h

4.4. Projection formulation

As node cut inequalities correctly describe the polyhedron T c

and are also facet defining for the projection polyhedron under cer-tain mild conditions, these inequalities can be used to formulateDCRNPP. Using node cut inequalities, we propose the following for-mulation for DCRNPP:

F2 :

MinimizeXj2R

yj

subject toXj2c

yj P 1 8c 2 C

yj 2 f0;1g; 8j 2 R

ð12Þ

Comparing formulations F1 and F2, we observe that the pathvariables are excluded from the projection formulation (F2).However, there are an exponentially large number of constraintsin the form of node cut inequalities in the projection formula-tion. In the next section, we discuss separation algorithms togenerate the violated node cuts in a graph for a given DCRNPPsolution.

5. Separation algorithms

In this section, we present separation algorithms to gener-ate violated node cuts. We define the master problem (LF2)as LP-relaxation of the projection formulation F2. Given a frac-tional solution a of LF2, a node cut �c is said to be violated by aifP

j2�caj < 1. The separation algorithms discussed in this sec-tion find such violated node cut for any suboptimal solutionof LF2.

We discuss two different separation algorithms in this section.The first separation algorithm is used to generate node cuts forthe unconstrained problem. These unconstrained node cuts ensurebackbone connectivity in the graph, i.e. it ensures that each of theSNs is connected to the BS even if there is no delay-constraint. Thesecond separation algorithm finds node cuts in delay-constrainedgraph. These node cuts make sure that BS and all SNs are connectedwithin the pre-specified delay bound D.

For the unconstrained version, the violated node cuts can befound by transforming the node-weighted graph into an edgeweighted graph and then finding the minimum weight edge cutusing the standard max flow algorithms. Interested readers mayfind more details on polynomial time algorithms for max flow

Fig. 3. Algorithm to find a minimal node cut within a violated node cut.

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 225

problem in (Ahuja, Magnanti, & Orlin, 1993). Violated node cuts, ina delay-constrained graph, are found using the algorithm discussedin detail in Section 5.2.

5.1. Node cuts for unconstrained problem

Finding a node cut violated by the current solution of LF2 isequivalent to finding a minimum weight (s � t) cut in a graphwith node capacities. An approach to convert a node-weightedgraph into an edge-weighted graph is discussed in (Lawler,1976). Let a be the LF2 solution. We expand the graph byreplacing each CRN j by two nodes an in-node j0 and an out-node j00. Capacity assigned to arc (j0, j00) is aj. Each arc (i, j) 2 A,is replaced with two arcs (i00, j0) and (j00, i0) of infinite capacityin the expanded graph. We note that k0 = k00 = k; " k 2 T [ {0}.The minimum weight s � t cut between the BS and SN k canbe found using a polynomial time maximum flow algorithme.g. shortest augmenting path algorithm (Ahuja & Orlin,1991) or preflow push algorithm (Goldberg & Tarjan, 1986).In our implementation, we have used the Highest Label Pre-flow Push algorithm, which has a complexity of O jV j2

ffiffiffiffiffiffijEj

p� �(Cheriyan & Maheswari, 1989).

5.2. Node cuts for constrained problem

The violated node cuts for a SN k 2 T in a delay-constrainedgraph are found with the greedy heuristic ConstNodeCut (k,D) de-scribed in Fig. 2. The heuristic iteratively finds minimum delaypaths between BS and SN k, and then deletes a CRN with mini-mum aj value on this path. Let’s define h(0,k) as the cumulativedelay on the minimum delay path between the BS and SN k. LetR�pk and E�pk be the set of CRNs and the set of edges, respectively,on the minimum delay path. The algorithm selects a CRNj� 2 R�pk such that aj� 6 aj8j 2 R�pk . CRN j⁄ is then deleted fromgraph and inserted into an empty set S. Algorithm again findsthe minimum delay path in the reduced graph induced by vertexset V0 = VnS. This iterative procedure is continued until set S cor-respond to a node cut for the graph, i.e. there are no paths be-tween the SN k and the BS within delay bound D in the graphinduced by VnS. Once a node cut S is found, a minimal node cutcan be found within set S by removing unnecessary CRNs fromS (procedure MinimalNodeCut (S,k,D) to find a minimal node cutin a node cut is detailed in Fig. 3).

Algorithm ConstNodeCut (k,D) is a heuristic approach for anyfractional solution of the master problem. However, it behaves asan exact algorithm if the DCRNPP solution is integral.

Fig. 2. Algorithm to generate a violated node cut for DCRNPP.

Theorem 5. The heuristic ConstNodeCut (k,D) always finds a violatednode cut for an infeasible integral solution.

Proof. For an integral solution a0, a violated node cut �c satisfiesthe condition that

Pj2�ca0

j ¼ 0. As a0 is an infeasible solution, $k⁄ -2 T such that there is no feasible path between the BS and SN k⁄.Therefore, 9jp 2 Rp8p 2 Pk� ;D such that a0

jp¼ 0. Since heuristic Const-

NodeCut (k,D) seeks a CRN with least a0j value on delay-constrained

paths, it will be able to construct a violated node cut S by identify-ing these CRNs with a0

j ¼ 0. Clearly, the heuristic will always find aviolated node cut (with weight 0) for an infeasible integralsolution. h

Fig. 4. Lagrangian relaxation based heuristic to find a delay constrained shortestpath.

Fig. 5. Heuristic to generate a good feasible solution for DCRNPP.

226 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

6. Lagrangian relaxation based heuristic

In this section, we describe a Lagrangian relaxation based heu-ristic to generate a good feasible solution for DCRNPP. This solutionis used as the initial incumbent for the branch and bound treesearch.

First, we discuss a Lagrangian relaxation based approach to findthe constrained shortest path. The DCSPP can be formulated as:

Fig. 6. Flow chart representation o

DCSPPðkÞ :

MinimizeXði;jÞ2Ep

lijzij ð13Þ

subject toXði;jÞ2Ep

dijzij 6 D ð14Þ

zij ¼ xp; 8ði; jÞ 2 Ep ð15ÞXp2Pk

xp ¼ 1 ð16Þ

Here lij is the edge length, and dij is the delay associated withedge (i,j) 2 A. Binary variable zij indicates whether or not edge(i, j) 2 E is selected in the delay constrained shortest path, and bin-ary decision variable xp indicates whether p 2 Pk is selected as thedelay constrained shortest path. Now consider the Lagrangianproblem derived by associating non-negative multipliers k to con-straint (14) and dualizing it in the usual Lagrangian way:

SPPðk; kÞ :

Minimize zk ¼Xði;jÞ2Ep

ðlij � kdijÞzij ð17Þ

zij ¼ xp; 8ði; jÞ 2 Ep ð18ÞXp2Pk

xp ¼ 1 ð19Þ

The formulation SPP (k,k) is a simple shortest path problem.Based upon this Lagrangian relaxation, we describe the heuristicConstSPTLag(k,D) to solve DCSPP between BS and SN k in Fig. 4.The algorithm is based upon the binary section search (Burden &Faires, 1985), and attempts to find the smallest Lagrangian multi-plier (k⁄), for which the shortest path with respect to the dualized

f the branch-and-cut routine.

Table 1Comparison of Lower bounds for column generation and node cut based approach for instances with jTj = 10; jRj = 60.

Instance jEj D Z⁄ Column generation Projection approach

ZLPCG% Col_CG tCG ZLP

PROJ% jCLP j tPROJ

TC#A 639 1000 22 93.8 743 0.9 98.9 60 0.11200 16 90.6 1143 2.1 96.3 72 0.21400 12 93.8 1782 5.3 95.8 83 0.41600 9 90.6 2245 8.9 94.4 96 0.61800 7 87.6 2753 11.4 89.3 107 0.81 7 92.9 3448 0.7 100.0 65 0.1

Ave. 91.5 4.9 95.8 0.3TC#B 855 2500 17 97.1 642 0.7 99.0 34 0.3

3000 15 93.5 838 1.8 96.3 46 0.33200 12 91.7 947 5.3 89.4 85 0.63500 12 87.5 912 7.8 94.8 96 0.74000 10 92.5 1103 20.3 96.8 105 0.81 10 95.0 746 0.6 95.7 48 0.5

Ave. 92.9 6.1 95.3 0.5TC#C 753 3000 14 96.4 743 7.9 95.2 46 0.4

3500 10 92.5 895 14.3 96.1 67 0.74000 7 92.4 962 18.4 95.3 58 0.94500 6 91.7 1228 40.3 91.7 46 0.55000 5 98.2 2561 61.4 100.0 28 0.41 5 100.0 1013 0.8 100.0 10 0.1

Ave. 95.2 23.8 96.3 0.5TC#D 988 2200 19 92.1 899 4.6 94.7 40 0.3

2500 14 91.1 946 9.7 92.0 103 0.73000 11 90.9 1143 15.3 90.1 110 0.83500 9 96.3 1508 39.8 100.0 81 0.84000 8 93.7 2179 65.7 100.0 61 0.61 8 100.0 509 0.7 100.0 33 0.1

Ave. 94.0 22.6 96.1 0.6TE#A 912 1000 14 92.6 350 1.2 94.6 203 0.7

1200 10 88.3 503 2.5 96.7 312 0.91400 8 89.6 687 6.9 90.6 451 1.11800 6 81.7 1042 13.5 89.5 789 3.02000 5 87.2 1247 19.3 95.0 1012 7.31 5 85.0 1351 1.1 100.0 240 0.6

Ave. 87.4 7.4 94.4 2.3TE#B 1123 1400 27 78.0 432 0.9 87.1 111 0.6

1600 19 84.2 586 1.7 83.7 208 2.02000 12 87.5 938 5.3 85.9 136 1.22500 9 86.6 1432 13.5 88.9 90 2.13000 8 90.6 2046 21.9 93.8 108 2.31 8 100.0 846 0.4 100.0 35 0.2

Ave. 87.8 7.3 89.9 1.4TE#C 1336 1500 18 91.7 1032 5.9 94.0 149 1.1

1800 13 80.8 2432 13.6 83.5 216 2.32200 9 94.4 3412 19.3 92.1 244 2.52400 8 86.9 4746 36.9 87.5 225 2.73000 7 92.9 5089 89.2 84.9 143 2.91 6 91.7 564 0.7 95.8 22 0.2

Ave. 89.7 27.6 89.6 2.0TE#D 1100 22 87.7 889 1.7 93.6 118 0.8

1400 13 76.8 1132 3.6 81.9 193 2.31600 11 86.4 1346 7.8 85.9 266 3.22000 8 84.4 1899 25.7 87.5 210 2.52200 7 85.7 2364 63.8 85.7 244 3.61 7 92.9 646 0.3 92.9 22 0.1

Ave. 85.6 17.1 87.9 2.1

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 227

cost vector {(lij � k⁄dij): (i, j) 2 A} satisfies the delay bound. Proce-dure MaxLagBoundfinds the upper bound (kmax) for the Binary sec-tion search and then the search is performed in the interval[0,kmax].

For generating a heuristic solution for DCRNPP, we use the algo-rithm LagHeuristic(D) described in Fig. 5. An overview of the algo-rithm is as follows. The algorithm starts with empty lists Q and L.To begin with, algorithm finds the delay constrained shortest pathw(0,k) for each SN k 2 T with edge length defined as lij = 1 "(i, j) 2 A(DCSPP is solved using the Lagrangian relaxation approach dis-cussed above). Further, we define R�Dpk and E�Dpk as the set of CRNs

and the set of edges, respectively, on the delay constrained shortestpath. CRNs on the delay constrained path to the node k⁄ 2 T (wherew(0,k⁄) 6 w(0,k), "k 2 T) are added to the empty list Q and are de-clared as free nodes. SN k⁄ is added to list L. All edges between a pairof free nodes are now assigned edge length 0 and edges betweenfree nodes and other nodes have edge length 0.5. In the followingiterations, the shortest path is computed between the BS and allSNs in TnL, and more free nodes are added to list Q from the pathw(0,k⁄) as described above. Iterations are continued until all theSNs are added to the list L i.e. jLj = m. The set of free nodes Q con-stitutes a heuristic solution for DCRNPP.

Table 2Comparison of lower bounds for column generation and node cut based approach for instances with jTj = 20; jRj = 100.

Instance jEj D Z⁄ Column generation Projection approach

ZLPCG% Col_CG tCG ZLP

PROJ% jCLP j tPROJ

TC#A 1850 1000 24 97.4 1738 150.1 96.9 104 0.81200 18 94.1 2612 345.3 93.7 145 1.71400 13 88.5 5304 687.6 88.2 195 2.31600 12 87.9 7998 912.4 87.3 294 4.21800 11 – – > 1000 83.2 337 5.21 11 95.5 10,193 127.5 94.3 126 7.1

Ave. 92.8 444.6 90.6 3.6TC#B 2517 2100 17 88.8 4538 532.6 90.0 181 2.2

2300 14 87.5 8936 789.8 90.7 213 2.72500 10 – – > 1000 89.0 246 4.32700 6 – – > 1000 83.3 378 9.83000 3 – – > 1000 100.0 532 11.31 3 100.0 82 0.9 100.0 8 0.6

Ave. 92.1 441.1 92.2 5.2TC#C 2082 2100 29 84.5 778 58.3 86.4 156 1.9

2500 18 80.6 973 89.9 81.0 312 4.23000 15 83.3 1291 142.3 84.2 448 8.43500 13 80.0 5638 289.6 82.2 415 8.74000 10 79.5 13,482 623.2 81.4 404 11.71 10 95.0 443 5.3 90.8 102 0.8

Ave. 83.8 201.4 84.3 6.0TC#D 2278 2800 22 84.1 1387 156.3 90.4 162 2.7

3000 19 94.7 1523 189.2 93.3 280 4.13200 17 89.7 3419 213.8 90.7 283 5.63500 15 80.3 12,886 746.9 85.4 319 7.64000 11 – – > 1000 92.7 432 12.31 11 95.0 589 2.8 95.4 134 0.4

Ave. 88.8 261.8 91.3 5.5TE#A 5844 600 14 92.7 2415 213.7 92.9 370 8.9

800 12 92.3 3887 403.5 91.6 451 6.31000 11 86.4 7418 898.6 93.2 499 111200 9 – – > 1000 91.7 577 11.61600 8 – – > 1000 92.8 789 14.41 8 98.4 12,145 145.8 95.9 338 5.4

Ave. 92.4 610.3 93.0 9.6TE#B 4035 1000 29 90.5 3418 201.0 91.5 589 11.3

1200 20 90.0 5286 528.3 89.0 672 18.21500 14 82.1 14,538 813.6 83.9 788 20.31800 12 – – > 1000 83.3 714 19.22000 10 – – > 1000 89.5 532 18.61 10 95.0 4583 32.9 100.0 54 0.6

Ave. 89.4 394.0 89.5 14.7TE#C 3186 1100 24 83.3 2148 176.3 85.4 337 5.8

1200 21 78.6 3663 218.3 82.2 528 11.21300 19 88.2 5418 245.7 87.3 336 8.91400 18 84.7 8942 346.9 87.5 524 14.31600 17 85.3 12,369 438.7 85.5 202 5.41 17 97.1 3532 18.7 94.1 67 0.4

Ave. 86.2 240.8 87.0 7.7TE#D 4247 1400 24 97.9 1012 121.3 99.3 61 1.4

1600 20 89.0 1837 148.9 94.5 128 2.91800 17 84.2 2626 198.7 91.2 199 4.52000 14 86.4 4312 338.6 88.6 236 6.92200 11 – – – 81.4 123 5.21 11 93.2 548 3.8 95.5 96 0.6

Ave. 90.1 162.3 91.7 3.6

228 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

7. Implementation details

In this section, we present the details of implementation ofBranch-and-Cut algorithm using the projection formulation in or-der to obtain an optimal solution of DCRNPP.

7.1. Problem reduction

Theorem 3 states that polyhedron T c is full dimensional if andonly if there are no singleton node cuts in the graph. In reductionprocedures, we eliminate singleton node cuts and redundant nodes

from the graph. A singleton node cut is a node cut with size 1. Inorder to identify such node cuts, we check whether the set{j};"j 2 R is a node cut or not. We find the minimum delay path be-tween BS and all SNs in a graph induced by vertex set Vn{j}. Ifcumulative delay on all such paths is within the delay-bound D,then {j} is not a singleton node cut. Otherwise, the set {j} corre-sponds to a singleton node cut. All the CRNs which constitute a sin-gleton node cut are assigned value 1 i.e. a�j ¼ 1.

Some of the CRNs may be irrelevant with respect to the delaybound, i.e. there may be no paths via these CRNs for any of theSNs within the specified delay bounds. These CRNs can be deletedfrom the graph without affecting the optimality of the final solu-

Table 3Results for branch-and-cut and heuristic approach for DCRNPP instances with jTj = 20; jRj = 100.

Instance jEj D Z⁄ Branch & Bound Heuristic

%LB jCLP j jCBBj BBnd tLP tBB tttl Zheur ZHeur% tHeur

TC#A 1850 1000 24 96.9 104 15 5 0.8 1.0 1.8 25 104.2 0.41200 18 93.7 145 54 457 1.7 2.4 4.1 19 105.6 0.71400 13 88.2 195 0 0 2.3 2.4 4.7 13 100.0 0.11600 12 87.3 294 4 43 4.2 4.3 8.5 12 100.0 0.31800 11 83.2 337 0 17 5.2 4.4 9.6 12 100.0 0.21 11 94.3 126 27 58 7.1 6.7 13.8 12 109.1 0.2

Ave. 90.6 3.6 3.5 7.1 103.2 0.3TC#B 2517 2100 17 90.0 181 88 156 2.2 7.2 9.4 20 117.6 0.3

2300 14 90.7 213 135 536 2.7 13.6 16.3 15 107.1 0.72500 10 89.0 246 246 1198 4.3 38.9 43.2 12 120.0 0.62700 6 83.3 378 1287 5432 9.8 59.7 69.5 7 16.7 0.63000 3 100.0 532 11.3 0 11.3 0.0 11.3 4 33.3 0.91 3 100.0 8 0 0 0.1 0.0 0.1 4 33.3 0.1

Ave. 92.2 5.2 19.8 25.0 121.4 0.5TC#C 2082 2100 29 86.4 156 393 2651 1.9 5.2 7.1 31 106.9 0.4

2500 18 81.0 312 1003 5799 4.2 27.5 31.7 22 122.2 0.73000 15 84.2 448 1069 15,106 8.7 68.1 76.8 18 120.0 0.83500 13 82.2 415 778 2733 8.7 25.7 34.4 16 123.1 0.94000 10 81.4 404 473 1230 11.7 13.9 25.6 12 120.0 1.31 10 90.8 102 127 110 0.8 0.5 1.3 13 130.0 0.3

Ave. 84.3 6.0 23.5 29.5 120.4 0.7TC#D 2278 2800 22 90.4 162 350 856 2.7 5.5 8.2 23 104.5 0.2

3000 19 93.3 280 433 2117 4.1 9.2 13.3 20 105.3 0.53200 17 90.7 283 375 2943 5.6 9.9 15.5 18 105.9 0.73500 15 85.4 319 419 3866 7.6 11.5 19.1 18 120.0 0.64000 11 92.7 432 88 127 12.3 1.7 14.0 14 127.3 0.81 11 95.4 134 69 40 0.4 0.4 0.8 13 118.2 0.1

Ave. 91.3 5.4 6.4 11.8 113.5 0.5TE#A 5844 600 14 92.9 370 0 0 8.9 21.4 30.3 15 107.1 0.4

800 12 91.6 451 73 1466 6.3 32.8 39.1 12 100 0.41000 11 93.2 499 0 0 11.0 17.9 28.9 12 109.1 0.31200 9 91.7 577 12 444 11.6 20.1 31.7 10 111.1 0.21600 8 92.8 789 121 1042 14.4 40.2 54.6 10 112.5 0.31 8 95.9 338 0 0 5.4 15.6 21.0 8 100 0.2

Ave. 93.0 9.6 24.7 34.3 106.6 0.3TE#B 4035 1000 29 91.5 589 148 2538 11.3 20.6 31.9 34 117.2 1.1

1200 20 89.0 672 289 4082 18.2 36.9 55.1 23 115.0 0.81500 14 83.9 788 582 12,314 20.3 88.9 109.2 17 121.4 0.91800 12 83.3 714 1014 15,132 19.2 103.7 122.9 13 108.3 0.92000 10 89.5 532 838 10,086 18.6 76.2 94.8 11 110.0 11 10 100.0 54 0 0 0.6 0.0 0.6 10 100.0 0.2

Ave. 89.5 14.7 54.4 69.1 112.0 0.8TE#C 3186 1100 24 85.4 337 2204 2088 5.8 29.5 35.3 25 104.2 0.9

1200 21 82.2 528 4843 18,604 11.2 119.8 131.0 25 119.0 0.71300 19 87.3 336 3242 3701 8.9 69.7 78.6 21 110.5 0.81400 18 87.5 524 226 2905 14.3 46.6 60.9 21 116.7 0.91600 17 85.5 202 119 82 5.4 1.7 7.1 18 105.9 0.41 17 94.1 67 135 46 0.3 0.4 0.7 18 105.9 0.2

Ave. 87.0 7.7 44.6 52.3 110.4 0.7TE#D 4247 1400 24 99.3 61 0 0 1.4 0.0 1.4 26 108.3 0.4

1600 20 94.5 128 112 446 2.9 22.6 25.5 22 110.0 0.71800 17 91.2 199 1046 896 4.5 79.8 84.3 20 117.6 0.92000 14 88.6 236 1132 5432 6.9 117.2 124.1 18 128.6 0.92200 11 81.4 123 2489 7889 5.2 138.7 143.9 13 118.2 1.21 11 95.5 96 0 0 0.6 0.0 0.6 13 118.2 0.1

Ave. 91.7 3.6 59.7 63.3 116.8 0.7

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 229

tion. Let d(i, j) be the delay on minimum delay path between a pairof nodes i and j. Clearly, we can replace R with the setR0 = Rn{j 2 Rjd(0, j) + d(j,k) > D; "k 2 T}.

7.2. Branch and cut algorithm

If the optimal solution of LF2 is fractional, the branch-and-cutroutine is adopted to find an optimal solution of DCRNPP. Theheuristic solution obtained with the help of Lagrangian heuristic

discussed earlier provides a good upper bound (incumbent) forthe branch-and-cut routine. In branch-and-cut tree, the violatednode cuts are added in order to improve the LP-bound for the opti-mal solution. These violated node cuts are identified using the sep-aration heuristic ConstNodeCut (k,D), which is an exact approachfor any integer solution of DCRNPP. Solving the separation problemat each node of the tree would be exorbitantly expensive. There-fore, rather than finding the violated node cuts at each and everynode of branch and cut tree, the separation problem is solved only

Table 4Results for branch-and-cut and heuristic approach for DCRNPP instances with jTj = 50; jRj = 200.

Instance jEj D Z⁄ Branch and bound Heuristic

%LB jCLP j jCBBj BBnd tLP tBB tttl Zheur ZHeur% tHeur

TC#A 6977 1000 58 97.1 463 1700 341 31.6 26.6 58.2 62 106.9 1.51200 49 93.9 612 1089 444 48.8 58.6 107.4 52 106.1 1.11400 35 87.6 701 2013 3922 87.5 138.5 226.0 39 111.4 1.21600 28 96.0 1028 285 268 116.3 99.0 215.3 32 114.3 1.22200 9 81.0 1485 2013 8576 118.4 140.6 259.0 12 111.1 0.81 9 94.4 501 181 34 5.8 5.9 11.7 11 122.2 0.9

Ave. 91.7 68.1 78.2 146.3 112.0 1.1TC#B 5566 1200 73 89.2 313 1203 13,351 14.2 46.2 60.4 81 111.0 1.2

1400 65 92.7 648 1546 5668 31.6 148.2 179.8 73 112.3 1.41600 59 88.5 738 3488 15,818 38.9 195.3 234.2 62 105.1 1.31800 54 84.5 898 6913 22,437 45.2 246.3 291.5 59 109.3 2.12000 38 84.7 746 5032 17,983 78.3 227.1 305.4 45 118.4 2.31 38 90.8 135 2130 4835 1.6 15.5 17.1 41 107.9 0.8

Ave. 88.4 35.0 146.4 181.4 110.7 1.5TC#C 5473 1200 77 94.2 289 695 3162 7.3 28.4 35.7 82 106.5 1.8

1400 65 93.3 596 718 2468 14.7 68.2 82.9 69 106.2 2.11600 58 93.1 689 1123 5638 32.7 98.3 131.0 59 101.7 2.81800 46 89.7 817 1568 8973 69.1 158.2 227.3 52 113.0 2.72000 42 88.4 1146 2991 11,389 89.6 213.7 303.3 49 116.7 3.21 42 90.5 140 2803 3262 2.3 21.6 23.9 47 111.9 1.3

Ave. 91.5 36.0 98.1 134.1 109.3 2.3TC#D 4295 1500 52 84.9 189 4837 18,538 18.7 129.7 148.4 61 117.3 2.1

1700 47 85.8 428 3648 23,618 38.3 218.6 256.9 56 119.1 3.11900 42 83.8 778 2129 16,547 69.2 179.8 249.0 50 119.0 2.82100 35 80.7 959 5887 18,726 97.6 240.6 338.2 38 108.6 3.22300 30 80.4 1413 8648 21,929 118.7 438.9 557.6 34 113.3 2.91 30 86.7 140 1894 13,566 3.9 30.8 34.7 33 110.0 0.7

Ave. 83.7 57.7 206.4 264.1 114.6 2.5TE#A 11,546 600 46 92.3 1548 1219 1810 63.2 116.1 179.3 53 115.2 1.8

800 41 96.9 1838 1532 989 125.9 80.2 206.0 46 112.2 1.61000 35 89.3 2128 3412 12,350 115.9 210.9 326.8 38 108.6 1.21200 27 94.1 1034 1245 3472 99.6 188.6 288.2 31 114.8 0.91600 10 88.1 2752 5032 12,589 145.7 418.8 564.5 11 110 2.91 10 91.5 1018 2556 1810 20.8 44.6 65.4 11 110 1.1

Ave. 92.0 101.6 188.6 290.2 111.1 1.5TE#B 12,090 800 45 90.6 1879 2016 12,789 79.8 189.7 269.5 52 115.6 1.9

1000 30 94.2 2463 789 5639 98.3 167.3 265.6 38 126.7 2.31200 21 94.6 2898 328 1238 112.7 155.8 268.5 25 119.0 3.11400 12 85.4 1527 5619 18,948 109.3 248.7 358.0 15 125.0 4.81600 7 85.7 3047 7848 22,738 189.8 618.3 808.1 9 128.6 4.71 7 92.9 73 158 10 5.2 4.9 10.1 9 128.6 0.2

Ave. 90.6 99.2 230.8 330.0 123.9 2.8TE#C 14,167 800 62 90.7 1784 1638 588 44.3 188.7 233.0 71 114.5 3.2

1000 48 87.7 2632 2548 7648 78.7 213.2 291.9 58 120.8 4.71200 39 87.8 4513 5339 22,137 118.3 735.2 853.5 45 115.4 4.91600 31 81.1 4868 4389 17,889 128.6 646.3 774.9 36 116.1 4.51800 21 78.6 577 14,538 42,617 20.7 1178.2 1198.9 27 128.6 5.11 21 88.1 141 1954 439 3.7 29.9 33.6 24 114.3 0.1

Ave. 85.7 65.7 498.6 564.3 118.3 3.8TE#D 11,312 800 44 87.4 798 2489 4480 154.8 218.9 373.7 50 113.6 2.1

1000 40 88.8 1136 1083 778 178.6 138.7 317.3 47 117.5 2.71200 35 89.3 2483 978 1127 193.2 159.6 352.8 40 114.3 5.31600 31 86.7 1823 1212 1506 156.2 188.7 344.9 39 125.8 5.11800 27 83.3 418 5876 17,838 127.9 823.7 951.6 30 111.1 4.91 27 88.9 141 875 925 2.3 9.9 12.2 27 100.0 0.2

Ave. 87.4 135.5 256.6 392.1 113.7 3.4

230 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

when an integer solution that is better than current incumbentsolution is obtained at any node (Agarwal, 2013).

Normally, if an integer solution is found that is better than thecurrent incumbent solution, the incumbent should be updated.However, before saving the solution as an incumbent, we mustcheck for the feasibility of this solution. If the solution is feasible,it is stored as an incumbent. Otherwise, the violated node cutinequalities obtained using the heuristic ConstNodeCut (k,D) areadded to the LP at the current node, and B&B process in continued.In all likelihood, addition of new inequalities will make the

solution non-integral and further branching will resume, until an-other integer solution is found, which will again be tested for fea-sibility. A flow-chart of the branch-and-cut algorithm is presentedin Fig. 6.

8. Computational experiments

In order to test the effectiveness of our approach, we con-ducted several computational experiments on a series of ran-domly generated test problems. Section 8.1 describes the

A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233 231

procedure that was adopted to generate these random instances.In Section 8.2, we compare the lower bounds and CPU times forthe two approaches we discussed in this paper, i.e. the columngeneration approach and the node cut based approach. This com-parison clearly establishes the superiority of the node cut basedapproach. Section 8.3 reports the optimal solutions and heuristicsolutions to demonstrate the strength of the heuristic approachdiscussed in Section 6.

All computations were conducted on an Intel Core I5 processorwith 2.53 gigahertz clock speed and 2 gigabyte RAM runningMicrosoft Windows XP (SP2) operating system. CPLEX version12.1 was used to solve LPs and IPs (CPLEX, 2010).

8.1. Test instances

Waxman random network model was used to generate the ran-dom network topologies (Waxman, 1988). The model first decideslocation of the nodes (CRNs and SNs) randomly and then decideswhether there exists an edge between a pair of node (i, j) withthe following probability equation:

Peði; jÞ ¼ ge�lði;jÞqr

Typical range for the parameters g, q is between 0 and 1. Largervalues of g result in graphs with higher link densities, while smallvalue of q increases the density of short links relative to longerones. In our experiments, we take the values of g and q as 0.7 each.

Since the locations of the BS may have an impact on the diffi-culty in solving the problem, we generated two set of WSN in-stances. In one set, the BS is placed at the center (TC), and in theother BS is placed at the corner (TE).

� TC instances: BS is placed at coordinate (0,0). SNs and CRNs arerandomly distributed in a square area within ±50 unit of BS.� TE instances: BS is placed at coordinate (0,0). SNs and CRNs are

randomly distributed in a square of side 100 unit in the positivequadrant.

The graphs are generated and tested to ensure the connectivitybetween the BS and all SNs. For each test instance, the edge delaydij is an integer computed as dij ¼ l2

ij

j k, where lij is the Euclidean

distance between node pair i and j.For all instances, we present results for five delay bound val-

ues along with the results for the unconstrained version of theproblem. The minimum value of delay bound is chosen in sucha manner that for any delay bound value less than this value, cor-responding DCRNPP instance is infeasible. The maximum delaybound value is such that the optimal solution for this value issame as the optimal solution for unconstrained version of theproblem.

8.2. Comparison of lower bounds

In Tables 1 and 2, we compare the lower bounds obtained withthe column generation approach and the node cut based approach.These tests were performed on two classes of DCRNPP instances.Table 1 reports the comparative analysis for instances withm = 10 and n = 50. Table 2 reports the same for instances withm = 20 and n = 100. For larger problems, CPU time required to solvethe column generation problem is too high. Even for smaller sizeinstances, column generation approach is unviable with large delaybound values. The strength of bounds is expressed as a percentageof the optimal solution obtained with the branch-and-cut routinediscussed earlier. While comparing the two approaches, a timelimit of 1000 s was imposed on the solution time.

Legends for Tables 1 and 2:

jEj

Number of edges in the graph D Delay bound Z⁄ Optimal solution for DCRNPP ZLP

CG%

Optimal value of LF1 as a % of Z⁄

ZLPPROJ%

Optimal value of LF2 as a % of Z⁄

jCLPj

Number of node cuts generated to solve LF2 ColCG Number of columns generated to solve LF1 tCG CPU time (in sec) to solve LF1 tPROJ CPU time (in sec) to solve LF2

The LP bound provided by the projection approach is signifi-cantly better than the bound obtained with the column generationapproach in most instances. Even those instances, where, LP boundobtained with node cut approach is not better, the bound is reason-ably closer to the one obtained with the column generation tech-nique. One major advantage of the node cut based approach,however, is in terms of computational efficiency. CPU time forthe projection approach is significantly better in comparison tothe same for the column generation technique. Reduction in CPUtime may be attributed to, (1) use of dynamic programming in col-umn generation technique, which in most of the practical cases isan inefficient approach and (2) number of column required to ob-tain optimality in column generation approach are too many, whilethe number of node cuts to obtain LP lower bound is quite small incomparison. From our discussion on dynamic programming ap-proach in Section 3.3, it is evident that the state space requirementin dynamic programming approach is proportional to the numberof nodes and the delay bound value. Due to the same reason,CPU Time for the column generation approach grows exponentiallywith an increase in problem size and delay bound value, and weare unable to report comparative analysis of the two approachesfor a few larger size instances.

8.3. Optimal and heuristic solutions

In Tables 3 and 4, we report the optimal as well as heuristicsolutions for various DCRNPP instances. Table 3 presents the com-putational results for instances with m = 20 and n = 100. While, Ta-ble 4 presents the results for DCRNPP instance of size m = 50 andn = 200.

Legends for Tables 3 and 4 are as following:

%LB

Optimal value of LF1 as a % of Z⁄

jCLP j

Number of node cuts generated to solve LF1

jCBBj

Number of node cuts generated in Branch and boundtree other than the root node of the tree

BBnd

Number of branch-and-bound tree nodes ZHeur Heuristic solution for DCRNPP ZHeur% Heuristic solution as a % of Z⁄

tLP

CPU time (in s) required to find optimal solution of themaster problem LF1

tBB

CPU time (in s) spent in the branch-and-cut tree tttl CPU time (in s) to solve DCRNPP optimally (tLP + tBB) tHeur CPU time (in s) to solve heuristic

Comparison of computational efforts required to solve variousinstances with different delay bound reveals that as the delaybound value increases, CPU time required to solve DCRNPP in-creases significantly. However, the CPU time required to solveDCRNPP instances without any delay constraint is quite low. Itmay be attributed to the fact that the separation algorithm, whichis used to find node cuts in constrained graph, is a heuristic ap-

232 A. Nigam, Y.K. Agarwal / European Journal of Operational Research 233 (2014) 220–233

proach. The separation heuristic explores all feasible paths be-tween the BS and a SN and deletes a CRN on the path until it findsa node cut (which may or may not be violated by the master prob-lem solution). As the delay bound value increases, number of fea-sible paths between the BS and SN increases significantly leadingto a higher run-time of separation heuristic. CPU time requiredto solve the unconstrained is significantly less because the separa-tion algorithm for unconstrained node cuts is an exact algorithm.The run-time efficiency of this algorithm does not depend uponthe number of feasible paths in the graph, as it is based upon themax-flow algorithm in transformed graph (see Section 5.1).

It is noteworthy that for all instance, the optimal solution valuereduces as the delay bound value is increased. The reason is thatthe delay values do not satisfy triangular inequality; hence, for atighter (smaller) delay bound, more hops are required to connecta SN and the BS.

The experiments also demonstrate that the TE instances areharder to solve in comparison to the TC instances. This can beattributed to the fact that the number of edges (jEj) in the TE in-stances is more in comparison to the TC instances of the similarsize. Due to the higher edge density, TE instances are more difficultto solve than TC instances (see the survey on hop-constrained min-imum spanning tree by Dahl, Gouveia, & Requejo (2006)).

9. Conclusion

In this paper, we discussed two different approaches to solveDCRNPP. We presented a path-based formulation with an expo-nentially large number of variables. We developed a column gener-ation method to obtain a lower bound for the optimal solution ofDCRNPP. Since, the method uses dynamic programming basedalgorithm to solve the subproblem, and due to convergence issueswith the column generation approach, the path-based formulationwas found unsuitable to solve the problem optimally.

Next, we introduced the projection formulation for the problemin the space of variables corresponding to CRNs only. This formula-tion involves an exponentially large number of constraints in theform of node cut inequalities. We proposed a branch-and-cut basedexact algorithm using this projection formulation. The proposedalgorithm is the first exact algorithm for DCRNPP. Experiments showthat the algorithm was able to find the optimal solution for instanceswith 50 SNs and 200 CRNs. For still larger problem instances, a heu-ristic approach remains a viable alternative. We have also proposed aLagrangian relaxation based heuristic that provides close-to-opti-mal solution for all of the test instances.

We are currently studying some higher order inequalities, i.e.those with right-hand-side 2 or more, and conditions under whichthese inequalities are facet defining. These inequalities may be de-rived by combining several node cut inequalities. We believe thatthere is scope for extending these results to the survivable WSNdesign and to the design of WSN with heterogeneous CRNs (i.e.where a range of CRNs with varying cost and communication rangeare available). There is also a need to develop more effective heu-ristics, which can be used to solve larger size problem instancesmore efficiently.

Acknowledgements

Authors would like to thank two anonymous referees for theirvaluable comments, which have helped in substantially improvingthe content, structuring and presentation of this paper.

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