Approximation algorithms and heuristics for a 2-depot, heterogeneous Hamiltonian path problem

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust. Nonlinear Control 2011; 21:1434–1451Published online 25 February 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1701

Approximation algorithms and heuristics for a 2-depot,heterogeneous Hamiltonian path problem

Riddhi Doshi, Sai Yadlapalli, Sivakumar Rathinam and Swaroop Darbha∗,†

Texas A&M University, College Station, TX 77843, U.S.A.

SUMMARY

This article addresses an important routing problem that arises in surveillance applications involving twoheterogeneous vehicles. As the addressed routing problem is NP-Hard, we develop an approximationalgorithm and heuristics to solve the problem. Our approach involves solving the routing problem in twomain steps: Partitioning and Sequencing. Partitioning involves finding a distinct set of targets to be visitedby each vehicle. Sequencing provides the order in which each vehicle must visit the subset of targetsassigned to it. The problem of partitioning is tackled by solving a linear program (LP) obtained by relaxingsome of the constraints of an integer programming model for the problem. We consider two LP models forpartitioning. The first LP model is obtained by mainly relaxing both the integrality and degree constraints,whereas the second model relaxes mainly the integrality constraints. Once the targets are partitioned, thesequencing problem can be solved either by Hoogeveen’s algorithm or by the Lin–Kernighan heuristicto yield an approximately optimal solution. Computational results show that the algorithms based on thesecond LP model, on an average, provided better (closer to the optimum) solutions as compared withthose based on the first LP model. We also observed that for both the LP models, the average quality ofsolutions given by the heuristics were found to be within 4% of the optimum, whereas the average qualityof solutions obtained from the approximation algorithms were within 8–20% of the optimum dependingon the problem size. Copyright � 2011 John Wiley & Sons, Ltd.

Received 25 September 2010; Revised 8 December 2010; Accepted 21 December 2010

KEY WORDS: approximation algorithms; heuristics; Hamiltonian path; heterogeneous vehicles; vehiclerouting

1. INTRODUCTION

Unmanned aerial vehicles (UAVs), unmanned underwater vehicles (UUVs), and unmanned groundrobots are routinely involved in military applications for border patrol, mine clearance, reconnais-sance, maritime surveillance expeditions, etc. Their civilian applications usually include remotesensing, traffic monitoring, search and rescue, scientific research, etc. The missions employingthese vehicles usually operate with constraints on time and resource. It thus becomes very crucialto find an optimal path of travel for the vehicles involved. In most cases, a heterogeneous collec-tion of vehicles differing in either structure or function or both is employed for the completionof a mission. In this article, we mainly concern ourselves with some commonly encounteredrouting problems for such missions. This problem is more realistic and more challenging than itshomogeneous counterpart due to the inherent differences among the vehicles.

We begin by classifying the heterogeneity of these vehicles into two basic categories: structuralheterogeneity and functional heterogeneity. Vehicles that are structurally heterogeneous mainly

∗Correspondence to: Swaroop Darbha, Texas A&M University, College Station, TX 77843, U.S.A.†E-mail: [email protected]

Copyright � 2011 John Wiley & Sons, Ltd.

APPROXIMATION ALGORITHMS AND HEURISTICS FOR A 2DHHPP 1435

differ in the design and dynamics. Thus, they may differ based on their fuel consumption, themaximum speed at which they can travel, the maximum payload capacity, etc. Realistically, somestructural differences are always present between any pair of vehicles. A collection of vehicles isfunctionally heterogeneous if not all vehicles may be able to visit a target. This type of functionalheterogeneity is present when the vehicles may occasionally be equipped with disparate sensorsdue to the respective payload restrictions on each vehicle. In this case, the targets may then bepartitioned into disjoint subsets: targets to be visited by specific vehicles and targets that any ofthe vehicles can visit. This article addresses routing problems for a collection of structurally andfunctionally heterogeneous vehicles.

In this article, we primarily concentrate on the following 2-depot, heterogeneous Hamiltonianpath problem (2DHHPP):

Given a set of targets and two heterogeneous vehicles located at distinct depots, find aHamiltonian path‡ for each vehicle such that each target is visited at least by one vehicle,the vehicle–target constraints are satisfied and the sum of the costs of the paths traveled byboth the vehicles is minimized.

The 2DHHPP is a generalization of the Hamiltonian path problem (HPP) which is known to beNP-Hard [1]. Hence, there are no constant factor approximation algorithms possible for a generalcase of this problem unless P =NP. However, if all the costs satisfy the triangle inequality,§ an�-approximation algorithm [2] could possibly be used for finding solutions to such problems.

Definition 1An �-approximation algorithm is characterized by the following properties: the algorithm runs inpolynomial time to find a feasible solution to a given problem; for any instance of the problem,the cost of the feasible solution found by the algorithm is guaranteed to be at most equal to �times the optimal cost.

In addition to approximation algorithms, one can also develop heuristics to find good feasiblesolutions for these types of problems. Heuristics can find a feasible solution to the problem inpolynomial time, however, there may be no a priori guarantees on the quality of the solutionobtained. This article presents some approximation algorithms and heuristics for the 2DHHPP. Italso presents a computational study comparing the performance of these algorithms.

The remainder of the article is organized as follows. In Section 2, we review and discuss theprior work related to our problem. In Section 3, we formulate the 2DHHPP. In Sections 4 and 5, wepresent the approximation algorithms and the heuristics. In Section 6, we perform a computationalstudy to numerically evaluate the performance of the algorithms. In Section 7, we conclude thearticle by summarizing the work and discussing possible improvements.

2. RELATED WORK

In the graph theory [3], a Hamiltonian path is defined as a path in a directed or an undirectedgraph that visits each vertex of the graph exactly once. The HPP is thus a problem of finding aHamiltonian path in a given graph. In this article, however, we use HPP to denote a problem offinding a minimum-cost Hamiltonian path in the graph. The 2DHHPP is then a generalization ofthe HPP for two heterogeneous vehicles, each stationed at a distinct depot, subject to additionalconstraints on vehicle–target assignment.

‡A Hamiltonian path for a vehicle is a path that starts at the depot and visits each of the targets assigned to thevehicle exactly once before halting at one of its assigned target.§For any pair of distinct nodes i and k in a graph, the cost of traveling from node i to node k directly is not greater

than the cost of traveling from node i to node k through a sequence of intermediate nodes.

Copyright � 2011 John Wiley & Sons, Ltd. Int. J. Robust. Nonlinear Control 2011; 21:1434–1451DOI: 10.1002/rnc

1436 R. DOSHI ET AL.

2.1. Hamiltonian path problem and traveling salesman problem

A Hamiltonian cycle problem (HCP) is then a problem of finding a tour (in a graph) that visitseach vertex exactly once. The traveling salesman problem (TSP) is equivalent to the problem offinding the minimum-cost Hamiltonian cycle in a graph and thus is a generalization of the HCP.

We observe that the TSP, HCP, and HPP are closely related and belong to a class of routingproblems. Several methods including exact algorithms, heuristics, approximation algorithms, andtransformation methods have been developed to address this class of routing problems [4]. Ifthe costs do not satisfy the triangle inequality, it is currently known that there cannot exist aconstant factor approximation algorithm for the TSP unless P =NP [2]. The following sectionswill primarily discuss some noted work in the field of approximation algorithms for single andmulti depot routing problems.

2.2. Single vehicle problems

For a single vehicle, symmetric TSP (STSP), there are two approximation algorithms thatare commonly used: the 2-approximation algorithm that doubles the minimum spanning tree(MST) to find a feasible tour and the algorithm given by Christofides [5], known to give asolution no worse than 1.5 times the optimal solution. The Christofides algorithm produces asolution by combining the MST with a weighted non-bipartite minimum cost perfect matchingon its odd-degree nodes. The performance guarantee of the Christofides algorithm has not beenreduced for over three decades now and finding a smaller approximation factor remains anopen problem. However, for a specific case of TSP where the distances between the nodes areeither one or two, a 7/6-approximation algorithm is presented in [6]. Also, for a EuclideanTSP, Arora presented a polynomial time approximation scheme (PTAS) which for a fixedc>1 is known to give a (1+1/c)-approximation to the optimum TSP tour in O(n(logn))O(c)

time [7].For a single vehicle HPP (SHPP), Hoogeveen in [8] employs an adaptation of Christofides’

algorithm to prove a 3/2-approximation algorithm for cases where the starting node/depot for apath is specified and a 5/3-approximation algorithm for cases where both the starting point and theterminal point is specified. Chekuri and Pal [9] address the asymmetric HPP (AHPP) and presenta O(logn)-approximation algorithm for the AHPP.

Several linear/integer programming models like those described in [10–12] have also beensuccessfully employed to solve the TSP, HPP and their variants. These combinatorial approachesare known to give close to optimal or optimal solutions, however, the time taken to obtain asolution can be very large depending upon the number of targets and the resource constraints thatthe vehicle is subjected to. There are also heuristics like the Lin–Kernighan heuristic (LKH) [13]which are relatively fast and find high-quality solutions.

2.3. Multi-vehicle problems

For a homogeneous, multiple TSP (MTSP) and its variations, Kara and Bektas present some integerlinear programming formulations in [14]. A branch-and-bound-based method for large-scale MTSPmay be found in [15]. A cutting plane algorithm for MTSP can be found in [16]. An overview ofthe commonly used formulations for the MTSP can also be found in [17].

For multi depot TSP (MDTSP), HPP (MDHPP) and their variants, 2-approximation algorithmswere presented in [18, 19]. For a Euclidean multi depot vehicle routing problem, Cardon et al. haveproposed a PTAS that ensures a (1+�)-approximation for �>0 with running time (qd/�)O(q3d2/�2) +O(n logn), where n denotes the total number of customers, and q represents the maximum numberof customers that can be visited by any of the vehicles located at the d depots [20]. Rathinam andSengupta also presented a 3/2-approximation algorithm in [21] for two variants of a 2-depot HPP.The main focus of this article is the development of approximation algorithms and heuristics for a2 depot, heterogeneous routing problem. We also present a computational study for the algorithmsin order to obtain a better insight on their actual performance.



3. PROBLEM FORMULATION

Let T represent all the targets to be visited and D ={d1,d2} denote the two depots (initial locations)corresponding to the first and the second vehicles, respectively. Let E be the set of all the edgesjoining any two vertices in V =T ∪ D. Now, consider the undirected graph G = (V, E). Let thecost of traversing edge e∈ E for the first (second) vehicle be C1

e (C2e ). Let e :={i, j} denote the

edge joining any two vertices i, j ∈V . We will assume that the costs satisfy the triangle inequality,i.e. for every i, j,k ∈V , e1 :={i, j},e2 :={ j,k},e3 :={i,k}, C1

e1+C1

e2�C1

e3and C2

e1+C2

e2�C2

e3.

Furthermore, we also assume that there are vehicle–target constraints owing to which the firstvehicle is required to visit a set of targets R1 ⊆T and the second vehicle is required to visit a setof targets R2 ⊆T with R1 ∩ R2 =∅. Note that the sets R1, R2 are specified a priori and only acommon target present in T \{R1 ∪ R2} can be visited either by the first or the second vehicle.

Let the number of targets visited by the i th vehicle be ki . In the trivial case when vehicle idoes not visit any target, i.e. ki =0, the corresponding travel cost for the i th vehicle is zero. Inthe general case when ki >0, let the sequence of vertices visited by the i th vehicle be denoted asSi = (di , pi

1, pi2, . . . , pi

ki), where pi

1, . . . , piki

∈T . The total travel cost corresponding to this sequence

for the i th vehicle is defined as Costi =Ci{di ,pi

1}+∑ki

j=2 Ci{pi

j−1,pij }

. The objective of the 2DHHPP

is to find a sequence of vertices to be visited for each vehicle, i.e. S1, S2, such that each target isvisited exactly once, the vehicle–target constraints are satisfied and the total travel cost of both thevehicles, Cost1 +Cost2, is minimized.

4. APPROXIMATION ALGORITHM VIA A 2-COMPONENTHETEROGENEOUS MINIMUM SPANNING FOREST

We extend a basic approach available for few variants of the TSP [2, 4, 22] to devise an approxi-mation algorithm for the 2DHHPP. The main steps in this approach are as follows:

(1) Remove the degree constraints in the given variant of the TSP to find a suitable relaxationthat can be solved in polynomial time. Solving this relaxation yields a network that spansall the vertices with minimum cost. For example, for a TSP, the network is an MST. Fora multiple TSP, the network is a minimum spanning forest such that no two depots areconnected.

(2) Double the edges in the minimum spanning network to obtain a connected, Eulerian subgraphfor each vehicle.

(3) Find an Eulerian tour in each connected subgraph. The Eulerian tour traverses each edge inits connected subgraph exactly once.

(4) By short cutting each Eulerian tour, a path or a tour can be obtained for each vehicle suchthat each target is visited exactly once.

The important step in the above approach is to find an appropriate relaxation of the givenrouting problem that can be solved in polynomial time. By removing the degree constraints ofthe 2DHHPP, we obtain a relaxation that requires calculating an optimal heterogeneous minimumspanning forest (HMSF). A heterogeneous spanning forest consists of two disjoint trees rooted atd1 and d2, so that all the targets in T are spanned, the depots are not connected and each target inRi is connected to di (i =1,2). The cost of the tree rooted at d1 is computed with the edge costsassociated with the first vehicle, whereas the cost of tree rooted at d2 is computed with the edgecosts associated with the second vehicle. An optimal HMSF is a heterogeneous spanning forestwhere the sum of the cost of the two trees is a minimum. The problem of finding an optimalHMSF is referred to as the HMSF problem in this article. Though it is not clear whether thereis a polynomial time algorithm that can find an HMSF, in this article, we prove that there is a4-approximation algorithm for the HMSF problem.

The difficulty in developing an approximation algorithm for the HMSF lies in finding a suitablepartition of the target vertices that must be visited by each of the vehicles. We do this by posing



the HMSF problem as a multi-commodity flow problem as follows: Suppose there are n distinctcommodities corresponding to each of the n targets, and at least one unit of each commodity isrequired to be delivered to its corresponding target by either of the vehicles. If the commodity isdelivered by the i th vehicle to a target, then that commodity is routed through those edges that onlycarry commodities from the i th vehicle. The HMSF problem may be posed as the construction oftrees for the vehicles such that their combined cost is minimum and at least one unit of commodityspecific to each target may be delivered by either one of the vehicles.

Before we present the approximation algorithm for the HMSF problem, we formulate thismulti-commodity flow problem as an integer program.

4.1. Integer programming formulation of the HMSF problem

Consider a digraph G ′ = (V, E ′) formed by replacing each edge {i, j} in E with two directed arcs(i, j) and ( j, i) in E ′. Let fi j be the binary variable that denotes whether arc (i, j) is used by thefirst vehicle and similarly let gi j be the binary variable that denotes whether arc (i, j) is used by thesecond vehicle. Let xe and ye represent the binary variables that decide whether edge e={i, j}∈ Eis present in the route of the first vehicle and second vehicle, respectively. Edge e is present inthe tour (xe =1) of the first vehicle if and only if any one of the directed arcs ((i, j) or (i, j))corresponding to e is chosen, i.e. either fi j =1 or f j i =1. Similar requirements also apply to ye

also. Let pki j denote the flow of kth commodity originating from the first depot and flowing from

node i to node j . Similarly, let qki j denote the flow of kth commodity originating from the second

depot and flowing from node i to node j. Let �k be the total quantity of the kth commodity shippedto the kth target from the first depot and let �k be the corresponding total quantity shipped fromthe second depot. The integer programming formulation of the HMSF problem is as follows:

CHMSF∗ =min∑e∈E

(C1e xe +C2

e ye) (1)

Capacity constraints:

0 � pki j� fi j ∀i, j ∈T ∪d1 ∀k ∈T, (2)

0 � qki j�gi j ∀i, j ∈T ∪d2 ∀k ∈T . (3)

Directed edge constraints:

fi j + f j i = xe ∀e={i, j}, i, j ∈V, (4)

gi j +g ji = ye ∀e={i, j}, i, j ∈V . (5)

Flow balance constraints for first vehicle:

∑j∈T

(pki j − pk

ji ) = �k ∀k ∈T and i =d1,

∑j∈T ∪{d1}

(pki j − pk

ji ) = 0 ∀i,k ∈T and i =k,

∑j∈T ∪{d1}

(pki j − pk

ji ) = −�k ∀i,k ∈T and i =k.

(6)



Flow balance constraints for second vehicle:

∑j∈T

(qki j −qk

ji ) = �k ∀k ∈T and i =d2,

∑j∈T ∪{d2}

(qki j −qk

ji ) = 0 ∀i,k ∈T and i =k,

∑j∈T ∪{d2}

(qki j −qk

ji ) = −�k ∀i,k ∈T and i =k.

(7)

Vehicle–target constraints:

�k = 1 ∀k ∈ R1, (8)

�k = 1 ∀k ∈ R2. (9)

Vehicle–target constraints for common targets:

�k +�k =1 ∀k ∈T \ R1 ∪ R2. (10)

For all i, j ∈V,k ∈T,e :={i, j}

xe, fi j ∈ {0,1} pki j ,�k ∈�+, (11)

ye,gi j ∈ {0,1} qki j ,�k ∈�+. (12)

In the above formulation, the capacity constraints (2) and (3) state that the flows, pki j ,qk

i j , are con-strained in amount by the capacity of the arc (i, j). The directed edge constraints (4) specify thatedge e is present in the tour (xe =1) of the first vehicle if any one of the directed arcs ((i, j) or( j, i)) corresponding to e is chosen, i.e. either fi j =1 or f j i =1. Similar constraints are specifiedfor the second vehicle in (5). The shipment of the kth commodity can originate only at one of thedepots and can only be delivered to the kth target. These flow balance constraints are expressedin (6) and (7). As the vehicles are functionally heterogeneous, there are vehicle–target constraintswhich state that any target in R1 must be visited by the vehicle at depot 1 and any target in R2must be visited by the vehicle at depot 2. These constraints are formulated in (8) and (9). Anycommon target not in R1 or R2 must be visited by one of the two vehicles. This constraint isexpressed in (10).

The complexity of determining an optimal solution for an HMSF is not clear. However, in thisarticle, we provide a 4-approx algorithm for the HMSF problem in the following section.

4.2. Approximation algorithm for the HMSF problem

The pseudocode of the approximation algorithm is first given below. The details of this approxi-mation algorithm are presented next.

Algorithm 1 Pseudocode of the approximation algorithm for the HMSF problem1: Solve a linear programming relaxation of the integer program corresponding to the H M SF

problem. This step is used to find the fractional quantities of each commodity shipped fromboth the depots.

2: Assign each target to the vehicle at the depot that ships the maximum amount of commodity tothe target. If both the depots ship equal amounts of commodity to a target, break ties arbitrarily.

3: For each vehicle, find a minimum spanning tree covering its assigned targets and its depotusing the respective traveling costs associated with the vehicle.



(1) Replace the vehicle–target constraints in (8), (9), the commodity constraints (10), and theintegrality constraints in (11), (12) with the following set, and solve the resulting linearprogramming problem:

�k � 1 ∀k ∈ R1, (13)

�k � 1 ∀k ∈ R2, (14)

�k +�k � 1 ∀k ∈T \ R1 ∪ R2, (15)

xe, ye, fi j ,gi j � 0 ∀e :={i, j}, i, j ∈V,k ∈T . (16)

Note that the relaxed problem (call it LP∗) can be solved in polynomial time as the numberof variables and constraints only scale polynomially with the number of targets.

(2) Find the optimal fractional quantities of each commodity shipped from both the depots.Partition the targets into two disjoint groups according to which depot ships the maximumamount of commodity to the targets. If both depots ship equal amounts of commodity toa target, break ties arbitrarily. In essence, let X= R1 ∪{k :k ∈T \{R1 ∪ R2},�k��k} be theset of targets assigned to the first depot d1 and let Y :=T \X be the targets assigned to thesecond depot d2.

(3) Find a tree spanning the targets X and the depot d1 of minimum cost. This minimum costspanning tree (MST) can be computed using the cost of edges associated with the vehiclestarting at depot d1. Similarly, we can find a minimum cost tree spanning the targets Y andthe depot d2 using the cost of the edges associated with the vehicle at depot d2. Clearly, thisis a feasible solution to the HMSF problem.

Let the optimal cost of the MST spanning all the nodes in X=X∪{d1} corresponding to thefirst vehicle be denoted by C1

MST1. Similarly, let the optimal cost of the second MST spanning all

the nodes in Y=Y∪{d2} corresponding to the second vehicle be C2MST2

. We now state one of themain results of this article:

Theorem 4.1The cost of the feasible solution produced by the HMSF algorithm is within four times the costof the relaxed linear program and hence, is less than 4CHMSF∗ . That is, the approximation factorof the HMSF Algorithm is 4.

We first outline the gist of the proof before presenting the details of the proof. Let the optimalcost of the relaxed linear program LP∗ be denoted as CLP∗ . Corresponding to this cost, let �∗

k , �∗k

be the optimal quantities of kth commodity shipped from d1 and d2, respectively. We formulate anew linear program LP1 by replacing the constraints (15) in LP∗ with the following constraints:

�k � 1 ∀k ∈X\ R1, (17)

�k � 1 ∀k ∈Y\ R2. (18)

Let the optimal cost of this new LP be denoted by CLP1 . The main steps in the proof ofTheorem 4.1 are as follows:

(1) We first prove that the optimal cost of the LP∗ is at least equal to half the optimal cost ofthe new linear program, LP1. That is, CLP∗�(1/2)CLP1 .

(2) In LP∗, note that the only equations that couple the first set of variables, {xe, fi j , pki j ,�k∀e :=

{i, j}, i, j ∈V,k ∈T } with the second set of variables {ye,gi j ,qki j ,�k∀e :={i, j}, i, j ∈V,k ∈T }

is through (15). In essence, if one were to replace these coupling equations using (17), (18),there would be no constraint that relates both these sets of variables. Therefore, the newlinear program, LP1, decomposes into two subproblems. In the first subproblem, LP1(X),the objective is to minimize

∑e∈E C1

e xe subject to the constraints in (2), (4), (6), (13), (17)



and xe, fi j , pki j ,�k ∈�+. Similarly, in the second subproblem, LP1(Y), the objective is to

minimize∑

e∈E C2e ye subject to the constraints in (3), (5), (7), (14), (18) and ye,gi j ,qk

i j ,�k ∈�+. Let the optimal cost of these two subproblems be defined as CLP1(X) and CLP1(Y),

respectively. As the linear program, LP1 decouples into two subproblems LP1(X) and LP1(Y),it is relatively easy to note that

CLP1 =CLP1(X) +CLP1(Y). (19)

3. In the final step, we prove the following results:

CLP1(X) � 12 C1

MST1,

CLP1(Y) � 12 C2

MST2.

(20)

Summarizing the results in each of the above steps, we obtain

CLP∗ � 12 CLP1

= 12 CLP1(X) + 1

2 CLP1(Y)

� 14 C1

MST1+ 1

4 C2MST2

. (21)

In other words, if the results in the outline of the proof are correct, then the sum of the costof the two MSTs obtained from the HMSF algorithm is at most equal to four times the optimalLP relaxation cost of the HMSF problem. Since this optimal LP relaxation cost is a lower boundon the optimal cost of the HMSF problem, it follows that the approximation factor of the HMSFalgorithm is 4. Therefore, to prove Theorem 4.1, it is sufficient to prove the following two lemmas:

Lemma 1

CLP∗� 12 CLP1 .

Lemma 2

CLP1(X) � 12 CMST1,

CLP1(Y) � 12 CMST2 .

4.3. Proof of Lemma 1

Let an optimal solution of LP∗ be denoted by Sol∗ ={x∗e , y∗

e , f ∗i j ,g∗

i j , pk∗i j ,qk∗

i j ,�∗k ,�

∗k∀e=

{i, j}, i, j ∈V,k ∈T }. We now prove that 2Sol∗ is also a feasible solution for the linear program,LP1. This would prove Lemma 1. Note that for any k ∈X, �∗

k�(1/2) or 2�∗k�1. Similarly, any

k ∈Y, 2�∗k�1. It is easy to check that scaling the optimal solution, Sol∗, by a factor of 2 will

satisfy all the constraints corresponding to LP1. Therefore, 2Sol∗ is a feasible solution for LP1.Hence, 2CLP∗�CLP1 .

4.4. Proof of Lemma 2

To prove this lemma, we first show that CLP1(X)�(1/2)C1MST1

. The proof for CLP1(Y)�(1/2)C2MST2

follows exactly the same steps.To prove CLP1(X)�(1/2)C1

MST1, let us first summarize all the constraints and the objective of

LP1(X) as follows:

CLP1(X) =min∑e∈E

C1e xe,



subject to

0 � pui j� fi j ∀i, j ∈V, u ∈T (22)

fi j + f j i = xe ∀e :={i, j}, i, j ∈V, (23)∑j∈T

(pki j − pk

ji ) = �k ∀k ∈T and i =d1, (24)

∑j∈T ∪{d1}

(pki j − pk

ji ) = 0 ∀i,k ∈T and i =k, (25)

∑j∈T ∪{d1}

(pki j − pk

ji ) = −�k ∀i,k ∈T and i =k, (26)

�k � 1 ∀k ∈X, (27)

�k � 0, fi j�0, pki j�0 and xe�0 for all e:={i, j}, i, j,∈V,k ∈T . (28)

Now, consider a target vertex t ∈X and a set S ⊂V such that the depot d1 ∈ S and t /∈ S.Constraints (27) state that vertex t must receive at least one unit of commodity from the depotd1. From the max-flow min-cut theorem [2, 23], there is a flow of at least one unit from the depot(d1 ∈ S) to a terminal node, t , in V \S if and only if

∑i∈S, j∈V \S fi j�1. This implies that

∑e:={i, j},i∈S, j∈V \S

xe�1 for all S ⊂V, d1 ∈ S, t /∈ S.

Therefore, the constraints in (22)–(27) can be further relaxed and replaced with the followingset of constraints: ∑

e∈�(S)xe�1 for S ⊂V, S∩X =∅, X\S =∅,

where �(S)={e :={i, j} : i ∈ S, j ∈V \S}. Therefore, the new linear program obtained by relaxingthe constraints in (22)–(27) is written as follows:


C1e xe

subject to∑

e∈�(S)xe � 1 for S ⊂V, S∩X =∅, X\S =∅, (29)

xe � 0 for e∈ E . (30)

If we define f (S) is equal to one whenever S ⊂V, S∩X =∅,X\S =∅ and is equal to zero otherwise,the linear program LP(X) can be rewritten as follows:


C1e xe,

subject to∑

e∈�(S)xe � f (S) for S ⊂V, (31)

xe � 0 for e∈ E . (32)

The above formulation of LP2(X) is actually a linear programming relaxation of a well-knownproblem in the literature called the Steiner Tree problem. Given an undirected graph G = (V, E)



with edge costs and a subset of nodes, X⊂V , the objective of the Steiner Tree problem is tofind a tree of minimum cost spanning all the nodes in X. The resulting tree may or may not havethe optional nodes (i.e. nodes in V \X). The optional nodes are often referred to as the Steinernodes. Now, we use a result due to Goemans and Williamson [24] that is available for problemsof this type to deduce that CLP2(X)�(1/2)C1

MST1. If the edge costs satisfy the triangle inequality,

Goemans and Williamson showed that the cost of the MST over the vertices in X is at most twicethe optimal cost of LP2(X). This result is stated in the following theorem:

Theorem 4.2 (Goemans and Williamson [24])If the costs satisfy the triangle inequality

C1MST1

�(

2− 2

|X|

)CLP2(X).

For more details on the above theorem, the readers are referred to [2, 24]. Using the abovetheorem it is clear that CLP1(X)�CLP2(X)�(1/2)C1

MST1. Hence, Lemma 2 is proved.

4.5. 8-Approximation algorithm for the 2DHHPP

The approximation algorithm for the 2DHHPP combines the algorithm in the previous section withHoogeveen’s algorithm [8] available for constructing a sequence for each vehicle. The pseudocodeof this algorithm is first presented below with the details of the algorithm explained in the ensuingdiscussion.

Algorithm 2 Pseudocode of the approximation algorithm for the 2DHHPP problem.1: Apply the approximation algorithm for the 2HMSF problem to find a minimum spanning tree

corresponding to each vehicle.2: For each vehicle, use Hoogeveen’s algorithm [8] on each minimum spanning tree to find a

path that starts from its respective depot and visits each of its assigned targets exactly once.

(1) Construct a feasible solution, for the HMSF problem using the 4-approximation algorithmpresented in the previous section. Let the two MSTs in this feasible solution correspondingto the two vehicles be denoted as MST1 and MST2, respectively.

(2) For i =1,2, do the following:

• Find all the wrong degree nodes, Wi , in MSTi . Any vertex u is defined as a wrong degreenode if the following holds true:

◦ u is a target and its degree in MSTi is odd.◦ u is a depot and its degree in MSTi is even.

It is easy to check that Wi is always odd [8].• Find a partial matching, i.e. a set of edges E(M∗

i )⊂ E , of cardinality (Wi −1)/2 thatmatches Wi −1 nodes in MSTi such that the cost of partial matching,

∑e∈E(M∗

i ) Cie, is

minimized.• Add the set of edges in E(M∗

i ) to MSTi to obtain a graph Ei . It follows from the result in[8] that Ei can either be an Eulerian graph or a graph with exactly two odd vertices, oneof which must be a depot. Using the graph Ei , one can find an Eulerian path that startsat the depot and visits each of the edges in Ei exactly once [8]. This path can then beshort cut to obtain a path, PATHi , that starts at depot i and visits each of the targets in Eiexactly once.

The above algorithm for 2DHHPP has an approximation factor of 8. To show this, let CiPATH =∑

e∈PATHiCi

e be the total cost of all the edges present in PATHi for i =1,2. Similarly, let CiMATCH =∑

e∈E(M∗i ) Ci

e, CiMSTi

=∑e∈MSTi

Cie be the total cost of edges in E(M∗

i ) and MSTi , respectively.



If the costs satisfy the triangle inequality, it is known that CiMATCH�Ci

MSTi[25]. Therefore, we

have the following result:∑

i=1,2Ci (PATH) �

∑i=1,2

CiMATCH +Ci

MSTi

� 2∑

i=1,2

CiMSTi

.

Therefore, combining the above result and the approximation factor in Theorem 4.1, it followsthat

∑i=1,2

Ci (PATH) � 8CHSMF∗

� 8C2DHHPP∗,

where C2DHHPP∗ is the optimal cost of the 2DHHPP.

4.6. Heuristic for the 2DHHPP

The 2DHHPP can also be solved by applying the LKH instead of Hoogeveen’s algorithm to thepartition obtained by solving the linear programming relaxation corresponding to the HMSF. Thepseudocode of this heuristic is as follows:

Algorithm 3 Pseudo code of the heuristic for the 2DHHPP1: Solve the linear programming relaxation (LP∗) of the integer program corresponding to the

HMSF problem. This step is used to find the fractional quantities of each commodity shippedfrom both the depots.

2: Assign each target to the vehicle at the depot that ships the maximum amount of commodity tothe target. If both the depots ship equal amounts of commodity to a target, break ties arbitrarily.

3: For each vehicle, use the LKH [26] to find a path that starts from its depot and visits each ofits assigned targets exactly once.

The LKH employs a variable k-opt algorithm to find the shortest tour/path in a given graph. Itbegins by randomly selecting a feasible tour on the graph. At each iteration, for increasing valuesof k, the heuristic checks whether an interchange of k-edges between the current tour and the restof the graph will result in a shorter tour. These edges are selected such that a feasible tour maybe formed at any stage of the algorithm. This process is continued until no further improvementis possible or until all possible exchanges are exhausted.

The search strategy and the termination criterion for the LKH play a very crucial role in itsperformance. Also, the selection of a good starting tour may reduce the computation time forfinding the final solution. Helsgaun in [26] devised a better implementation which restricted thesearch for the replacement edges within the set of edges that are ranked according to their closenessto the dual solution of the TSP obtained by the Held–Karp relaxation. This reduces the search setdrastically. Also, the starting tour is calculated such that most of the edges belong to the optimaldual solution or are close to it. These improvements along with many others described in [26, 27]have drastically improved the performance of the LKH and enabled it to provide optimal solutionsfor many large instances of TSP.

5. ALTERNATIVE APPROXIMATION ALGORITHM AND HEURISTIC

For both the approximation algorithm and the heuristic presented in the previous section, theprocedure used for partitioning the targets into two disjoint sets utilized a linear programmingrelaxation (LP∗) which did not include any degree constraints on the nodes. In an alternative



approach, one can partition the targets into two disjoint sets by solving the LP∗ with the followingadditional set of degree constraints on each of the nodes:

∑j∈V

fi j � �i ∀i ∈T, (33)

∑i∈V

fi j = � j ∀ j ∈T, (34)

∑j∈V

gi j � �i ∀i ∈T, (35)

∑i∈V

gi j = � j ∀ j ∈T, (36)

∑j∈T

fd1 j � �i ∀i ∈T, (37)

∑j∈T

gd2 j � �i ∀i ∈T, (38)

∑j∈T

f jd1 � 1, (39)

∑j∈T

g jd2 � 1. (40)

Using the same procedure as developed in the previous section, it can be verified that theapproximation algorithm that uses the new LP∗ with the degree constraints also has an approxi-mation factor of 8. However, solving a tighter linear programming relaxation can lead to tighterbounds and better feasible solutions. Computational results presented in the following section alsocorroborate this intuition.

6. COMPUTATIONAL STUDY

6.1. Details of implementation

All the algorithms were implemented in C++. The CPLEX callable libraries available from IBM’s[28] ILOG Concert Technology have been embedded in the implementation for solving the relaxedlinear and integer programs. The Boost Graph Library (BGL) [29] was used to find an MST.The minimum cost matching required for this algorithm is obtained through Blossom V [30], animplementation of Edmonds’ blossom algorithm [31].

For applying the LKH to the partitions, we used the implementation developed by Helsgaun[26, 27], which is available online [32]. This implementation of the LKH is known to give veryhigh-quality solutions relatively fast. In order to be able to use this implementation, the costsof traveling between any two nodes were rounded up to a positive integer value. Also, sincethis implementation does not solve a HPP, we used the implementation for the asymmetric TSPassuming the costs of all the edges incoming to the depot are zero.

Fifty random instances were generated for each problem size ranging from 15 to 50 targets(in increments of 5). For each instance, the position of each of the targets and the depots wasuniformly chosen from a test area of size 5000×5000sq. units. The distance to travel betweenany two nodes was calculated assuming that each vehicle is modeled as a Reed–Shepp car [33]. AReed–Shepp car is a car that can travel both forwards and backwards and has a lower bound on theturning radius at each point on its path. The optimal distance of traveling between any two nodesfor a Reed–Shepp vehicle given the initial and the final heading required at the nodes was solvedby Reed and Shepp in [33]. For each instance of the problem, an angle was uniformly chosenfrom the interval [0,2�] for each node. The cost (i.e. optimal distance) of traveling between anytwo nodes was then calculated using the results by Reed and Shepp [33] for each vehicle. The



two vehicles were assumed to be structurally heterogeneous with the first vehicle having a turningradius of 150 units and the second vehicle having a turning radius of 180 units. We simulatedfunctional heterogeneity by assigning (approximately) one-fifth of all the targets to the first vehicleand another one-fifth of the targets to the second vehicle. Thus, both structural and functionalheterogeneity were introduced.

All the linear programming relaxations and the integer program were solved using CPLEX. Asdiscussed in the previous sections, two different linear programming relaxations were consideredfor both the approximation algorithms and the heuristics; the first linear programming relaxationdefined by LP∗ and another relaxation obtained by adding the degree constraints (Section 5) toLP∗. The optimal solution for the 2DHHPP was obtained by solving an integer program formedby adding the degree constraints and the following restriction on the domain of the variables toLP∗: For all i, j ∈V,e :={i, j}, xe, fi j , ye,gi j ∈{0,1}.

6.2. Results

All the tests were implemented on an Intel� Xeon� X5450 3.00 GHz/16 GB machine. The deviationof the lower bound obtained by solving a linear programming relaxation compared with the optimalcost for instance I is defined as

DeviationI = Costlb(I )−Costopt(I )

Costopt(I )100%, (41)

where

Costlb(I )=Optimal cost corresponding to a linear programming relaxation of 2DHHPP forinstance I .Costopt(I )=Optimal cost of the 2DHHPP obtained by solving the Integer Program of 2DHHPPfor instance I .

The average deviation of the lower bound for both the linear programming relaxations of the2DHHPP is shown in Figure 1. As expected the lower bounds found using LP∗ with degreeconstraints were found to give smaller deviations on average as compared with just using LP∗.Specifically, the average deviation found using LP∗ with degree constraints was within 0.5% forall the problem sizes. One of the advantages of having a tight linear programming relaxation isthat the quality of the feasible solutions can be compared directly with the relaxation as opposedto comparing it with an optimal solution which might be more difficult to find. Given an algorithm

15 20 25 30 35 40 45 50

0

1

Number of targets

Ave

rage

dev

iatio

n of

Low

er B

ound

(%

)

LP*LP*+degree constraints

Figure 1. Comparison of the lower bounds obtained by the two LP relaxations.



15

(a)

20 25 30 35 40 45 506

8

10

12

14

16

18

20

22

24

26

Number of targets

Using LP*Using LP*+degree constraints

15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

Number of targets

Using LP*Using LP*+degree constraints

(b)

Figure 2. (a) Average solution quality (in %) obtained by the approximation algorithms using the twolinear programming relaxations and (b) average solution quality (in %) obtained by the heuristics using

the two linear programming relaxations.

alg and an instance I , the following equation was used to calculate the quality of the solutionsproduced by the algorithm on I :

QualityI = Costalg(I )−Costopt(I )

Costopt(I )100%, (42)

where

Costalg(I )=Cost of the solution obtained by applying algorithm alg on the instance I .

The average quality of the solutions produced by the approximation algorithms using the twolinear programming relaxations is presented in Figure 2(a). In addition, the standard deviations inthe quality of the solutions found by the approximation algorithms are shown in Figure 3(a). Theseresults show that for the tested instances, although the approximation algorithms use two differentlinear relaxations (with one relaxation finding very tight bounds), there is not much difference inthe overall quality of the solutions produced by both the approximation algorithms. In particular,



15 20 25 30 35 40 45 502

(a)

4

6

8

10

12

14

16

18

20

Number of targets

Using LP*

Using LP*+degree constraints

15 20 25 30 35 40 45 500

1

2

3

4

5

6

Number of targets

Using LP*

Using LP*+degree constraints

(b)

Figure 3. (a) Standard deviation (in %) of the quality of the solutions obtained by the approximationalgorithms using the two linear programming relaxations and (b) standard deviation (in %) of the quality

of the solutions obtained by the heuristics using the two linear programming relaxations.

the average quality produced by both the approximation algorithms is within 20%. Although thea priori upper bound proved for the quality of any solution found by the approximation algorithmswas 700% (corresponds to an approximation ratio of 8), the worst solution quality obtained forthe tested instances was approximately 56%.

The average quality of the solutions produced by the heuristics using the two linear programmingrelaxations is shown in Figure 2(b). The standard deviations in the quality of the solutions obtainedby the heuristics are shown in Figure 3(b). For all problem sizes, the average quality of thesolutions obtained by the heuristic using the linear programming relaxation (LP∗) with degreeconstraints was better than the average quality of the solutions obtained by the heuristics usingLP∗. These results show that for each of the problem sizes, the heuristics proposed here outperformthe approximation algorithms with regards to finding solutions of better quality.

The computation time of all the algorithms was dominated by the time required to solve theirrespective linear programming relaxations. Figure 4 shows the average computation times required



15 20 25 30 35 40 45 500

100

200

300

400

500

600

700

800

Number of targets

OptimalLP*+degree constraintsLP*

Figure 4. Average computation time for the IP and the LP relaxations in seconds.

for solving the linear programming relaxations and the integer program as the number of targetswere varied from 15 to 50. For a problem size of 50 targets, the average time required to solve theinteger program was approximately 780 s, whereas the heuristic (using LP∗ with degree constraints)took approximately 195 s to find solutions with a quality of 1.09%.

7. CONCLUSIONS

We developed approximation algorithms and heuristics for a two depot, heterogeneous vehiclerouting problem. Our basic approach for the approximation algorithms and the heuristics followstwo main steps; in the first step, we solve an LP relaxation of the multi-commodity formulationof the routing problem in order to assign each target to one of the two vehicles; in the secondstep, we use a sequencing algorithm such as Hoogeveen’s algorithm or the LKH to find a suitablesequence of targets to visit for each vehicle. We considered two LP models for partitioning thetargets in the first step. The first LP model is obtained by mainly relaxing both the integrality anddegree constraints, whereas the second model relaxes mainly the integrality constraints. All thealgorithms were implemented in a C++ environment with the help of CPLEX and BGLs. Thecomputational results show that the algorithms based on the second LP model, on an average,provided better solutions as compared with those based on the first LP model. Also, we observedthat for both the LP models, the average quality of solutions produced by the heuristics werefound to be within 4% of the optimum, whereas the average quality of solutions obtained fromthe approximation algorithms were within 8–20% of the optimum depending on the problemsize.

The basic ideas discussed in this article can be extended to routing problems involving multipleheterogeneous vehicles also. In particular, a straightforward extension of the approximation algo-rithms presented for the 2DHHPP to the multiple vehicle case yields an algorithm with an approx-imation ratio of 4m, where m is the number of depots/vehicles. Apart from minimizing the totaldistance traveled by all the vehicles, future work can also focus on other objectives such as mini-mizing the maximum distance traveled by the vehicles. In addition, the angle of arrival at each ofthe targets is chosen a priori in this article. Although there are few approximation algorithms forrouting problems with motion constraints [22], the problem of finding an optimal solution or tightlower bounds to these routing problems is still open.



ACKNOWLEDGEMENTS

The authors thank Dr Waqar Malik for providing us the code for calculating the optimal Reed–Sheppdistances. This research has been supported by AFOSR award No. FA9550-10-1-0392 and NSF awardNo. ECCS-1015066.

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Approximation algorithms and heuristics for a 2-depot, heterogeneous Hamiltonian path problem