Applied Probability Trust (21 October 2006)

PROBABILISTIC ANALYSIS OF UNIT DEMAND VEHICLE ROUTING PROBLEMS AGUST´ıN BOMPADRE,∗ Operations Research Center, MIT MOSHE DROR,∗∗ Operations Research Center, MIT JAMES B. ORLIN,∗∗∗ Sloan School of Management, MIT

Abstract We analyze the Unit Demand Euclidean Vehicle Routing Problem (VRP) where n customers are modeled as uniformly i.i.d. points and have unit demand. We show new lower bounds on the optimal cost for the metric VRP and we analyze them in this setting. We prove that there exists a constant cˆ > 0 such that the Iterated Tour Partitioning (ITP) heuristic given by Haimovich and Rinnooy Kan [9] is a 2 − cˆ approximation algorithm with probability arbitrarily close to one as the number of customers goes to infinity. It has been a long standing open problem whether one can improve upon the factor of 2 given in [9]. We also generalize this result and previous ones to the multi-depot case. Keywords: Vehicle Routing Problems; Heuristic; Probabilistic Analysis of Algorithms AMS 2000 Subject Classification: Primary 68W40 Secondary 90B06

1. Introduction 1.1. Unit demand vehicle routing problem We study the Unit Demand Euclidean VRP where customers x1 , . . . , xn and depot y1 are given as points in the plane, and the distance between points is the Euclidean distance. Each customer has unit demand. There are an unlimited number of identical vehicles, and each one has capacity Q ∈ IN. The route of each vehicle starts and

ends at the depot y1 . Each vehicle cannot deliver more than its capacity Q. The

cost of a solution is the sum of the traversing cost of each vehicle. In the problems ∗

Postal address: Optiant Inc., 4 Van de Graaff Drive, Burlington, MA 01803, United States.

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A. BOMPADRE, M. DROR AND J. B. ORLIN

we consider the objective is to route vehicles to deliver the demand of every customer while minimizing the overall cost. Except for some special cases, the VRP is an NP-hard problem. In their seminal paper [9], Haimovich and Rinnooy Kan provided a worst case and a probabilistic analysis of the VRP (see also [10]). In [9], a lower bound on the cost of VRP with metric distance is proved which is the maximum between the cost of a Traveling Salesman Problem (TSP) tour and the so-called radial cost. When Q is fixed, the Unit Demand Euclidean VRP admits a Polynomial Time Approximation Scheme (PTAS). This means that for any ! > 0 there exists a 1 + !-approximation algorithm. The first PTAS for this case appeared in [9]. Subsequently, Asano et al. [3] improved its running time using the PTAS for the Euclidean TSP (see Arora [2] or Mitchell [12]). In [9], the authors also analyzed the problem from a probabilistic point of view when the locations of customers are modeled as i.i.d. points in the plane, and Q is a function of the number of customers. The analysis showed that the lower bound they √ propose is asymptotically optimal when Q = o( n) or n = o(Q2 ). As a result, the Iterated Tour Partitioning heuristic (ITP) by Haimovich and Rinnooy Kan [9] becomes asymptotically optimal in both cases. For the rest of the cases, the ITP heuristic is within a factor of 2 of the optimal cost. That is, under no assumptions on Q as the number of customers goes to infinite, the probabilistic analysis of the ITP heuristic is no better than its worst case analysis. In this paper, we show that the ITP heuristic is indeed better on average that what its worst case analysis suggests, by proving that its approximation ratio is strictly smaller that 2. The primary contributions of this paper are as follows:

1. We improve the approximation ratio of the ITP heuristic of Haimovich and Rinnooy Kan [9] for the VRP. This improvement, though slight, does resolve a long standing open question of whether any improvement was possible. 2. We provide nonlinear valid inequalities (Lemma 3) that are useful for improving bounds for VRP problems. 3. We show that for n points uniformly distributed in the unit square and for every

Probabilistic Analysis of VRPs

3

p with 0 < p ≤ 1, there is a constant c(p) such that c(p) lim P(at least pn points have a neighbor at distance ≤ √ ) = 0. n

n→∞

This result implies non-trivial lower bounds on the cost of combinatorial optimization problems such as the minimum latency problem. 4. We extend the probabilistic analysis of VRP to the multi-depot case. To be more precise, we first improve the radial cost lower bound on the cost of the √ metric VRP given by [9]. The improvement we present is Ω( n) with probability 1 when customers are given as uniformly distributed points in the square [0, 1]2 , and the distance is the Euclidean distance. As a result, the approximation ratio of the ITP heuristic is strictly better than 2 with probability 1 as the number of customers goes to infinite. As a side effect, our analysis also shows a further improvement on the √ approximation ratio when Q = Θ( n). This parametric case is the borderline between the cases analyzed by Haimovich and Rinnooy Kan [9]. The results proven for the Unit Demand Euclidean VRP are generalized in the second part of the paper to the multi-depot case, where the number and location of the depots is fixed in advance. Related papers are Li and Simchi-Levi [11] and Stougie [14]. Li and Simchi-Levi [11] performed a worst-case analysis of the multi-depot vehicle routing problem when the distance satisfy the triangle inequality, and showed how to reduce this problem to a single-depot case with triangle inequality. Their analysis is valid for metric spaces, and thus it cannot take advantage of the properties of the Euclidean metric in the plane. Stougie [14] studied a two-stage multi-depot problem, where on the first stage the central planner decides how many depots to build (and where), and on the second stage he/she deals with a multi-depot VRP. His analysis is probabilistic, since the customers of the second stage are i.i.d. points in the unit square. The objective in this problem is to minimize the sum of the costs of both stages. Unlike our approach, the number and location of depots are variables. His approach does not translate in our context, since the optimal solution of the first stage simplifies the analysis of the second stage problem. We introduce a natural generalization of the Iterated Tour Partitioning heuristic (ITP) to the multi-depot case and we show that the results proved for the single-depot case carry through. The rest of the paper is organized as follows: in Section 2, we present new and known

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A. BOMPADRE, M. DROR AND J. B. ORLIN

lower bounds on the optimal cost of the VRP. In Section 3, we analyze the value of the lower bound in a probabilistic setting. We present lower and upper bounds on the cost of combinatorial optimization problems such as the minimum latency problem. We also prove our main results about the approximation ratio of ITP. We discuss the Multi-Depot Vehicle Routing Problem (MDVRP) in Section 4. In Section 5 we present an algorithm for the MDVRP that generalizes the ITP heuristic. In Section 6 we present lower bounds for the MDVRP. In Section 7 we analyze the lower bounds and the algorithm for MDVRP in a probabilistic setting. We summarize the results presented in the paper in Section 8. 1.2. Notation Unless otherwise stated, customers and depot are modeled as points in the plane. The location of the ith customer is denoted by xi for any 1 ≤ i ≤ n. The depot is located at y1 . The set of customers is denoted by X (n) . The distance between

customers i, j is denoted by cij or by cxi xj , the distance between a customer i and depot y1 is cy1 ,i . A solution of a VRP is denoted by a pair (K, V ) where K is the number of vehicles used, and V = {vk : k ∈ {1, . . . , K}} is the set of routings. Given a solution (K, V ), we let dki be 1 when vehicle vk ∈ V visits customer i and 0 otherwise. !n The routing cost of vehicle k is denoted by c(vk ). R = i=1 2cy1 ,i /Q is the so-

called radial cost as per Haimovich and Rinnooy Kan [9]. The radial cost is a lower bound on the cost of the VRP; see [9], or Lemma 3 below. We denote by c(V RP ) or by c(V RP (X (n) )) the cost of an optimal VRP. We let c(T SP ) or c(T SP (X (n) ))

denote the cost of an optimal traveling salesman tour. Given a probability space and a probability measure, the probability that event A occurs is denoted by P(A). The probability of event A conditioned on event B is P(A|B). The complement of event A is A. We use upper case letters (e.g., X) to denote random variables and lower case letters (e.g., x) to denote a realization of a random variable.

2. Lower bounds on the optimal cost of the VRP We assume in this section that all distances satisfy the triangle inequality. We refer to such problems as metric. In this section we present new and known lower bounds

Probabilistic Analysis of VRPs

5

on the cost of a metric VRP. The main result of this section is summarized in Lemma 5, which improves the lower bound by Haimovich and Rinnooy Kan [9]. We state Haimovich and Rinnooy Kan’s lower bound on the cost of a metric VRP. Lemma 1. (Haimovich and Rinnooy Kan [9].) The cost of the metric VRP is at least max{R; c(T SP )}. The Iterated Tour Partitioning heuristic ITP(α) for the unit demand VRP defined in [9] (see also Altinkemer and Gavish [1]) receives an α−optimal TSP as part of the input and outputs a solution with cost at most R + α(1 −

1 Q )c(T SP ).

We briefly describe

this heuristic. It receives a tour (y1 , i1 , i2 , . . . , in , y1 ) of cost at most αc(T SP ) as part of the input and selects the best valid routing among Q solutions constructed as follows.

For each 1 ≤ t ≤ Q, the solution routet for the VRP is the union

t t of the Kt = % n−t Q & + 1 routings v1 = (y1 , i1 , . . . , it , y1 ),v2 = (y1 , it+1 , . . . , it+Q , y1 ),

v3t = (y1 , it+Q+1 , . . . , it+2Q , y1 ), . . . , v$t n−t %+1 = (y1 , i($ n−t %−1)Q+t+1 , . . . , in , y1 ). That Q

Q

is, routet transforms the original tour (y1 , i1 , . . . , jn , y1 ) into routings with Q customers

each (except possibly the first and the last routing). The average cost of these solutions is at most R + (1 −

1 Q )αc(T SP ).

At least one of these solutions has cost at most their

average. By Lemma 1, max{R; c(T SP )} is a lower bound on the optimal cost of the VRP, and therefore one of the solutions considered routet is within 1 + (1 −

1 Q )α

the

optimal cost.

In the Euclidean setting there exists a PTAS for the TSP (see [2], [12]). Therefore the ITP is a 2 −

1 Q

approximation algorithm in this case. We denote by c(V RP IT P )

the cost of the solution generated by the Iterated Tour Partitioning heuristic when it receives an optimal TSP as part of the input. When Q is not fixed, namely it is part of the input, the approximation ratio is asymptotically 2, as it was shown for a family of instances by Li and Simchi-Levi [11]. We state this result as a lemma. Lemma 2. (Li and Simchi-Levi [11].) The Iterated Tour Partitioning heuristic ITP(1) is a 2-approximation algorithm. Corollary 2 of [7] presents a lower bound on the cost of the VRP. The impact of this lower bound on the worst case analysis of the VRP is discussed in that paper. In Lemma 3 given below we state a stronger version of this bound. The new lower bound

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A. BOMPADRE, M. DROR AND J. B. ORLIN

is more suitable for the probabilistic analysis we perform in this paper. Definition 1. Given a routing vk = (y1 , ik1 , ik2 , . . . , iksk , y1 ) that starts and ends at depot y1 , we associate to it the sequence of customers lk = (ik1 , ik2 , . . . , iksk ). For any k customers i, j visited by vk , let lij be the length of the unique path determined by vk

from i to j that does not pass through the depot y1 . If i or j are not visited by vk , we k let lij = 0. Given a solution (K, V ) of the VRP, its associated sequences l1 , . . . , lK are

the associated sequences of v1 , . . . , vK ∈ V . k k = lji . For example, let vk = (y1 , 1, 2, 3, 4, 5, 6, y1 ), Since the distance matrix is symmetric, lij k k where cy1 ,1 = cy1 ,6 = cs,s+1 = 2 for all 1 ≤ s ≤ 6. In this example, l2,6 = l6,2 = 8.

Lemma 3. Let the pair (K, V ) be a solution of the metric VRP. For each 1 ≤ k ≤ K, 1 ≤ i ≤ n, let dki ∈ {0, 1} denote whether vehicle vk ∈ V visits customer i or not.

The cost of (K, V ) is at least K " n "

k=1 i=1

dk 2cy1 ,i !n i

t=1

dkt

+

K "

"

k=1 i,j∈{1,...,n}

Proof. To simplify notation, let us define zik = !n Since j=1 zjk = 1, we can rewrite (1) as K " n "

2cy1 ,i zik +

k=1 i=1 K " n "

k=1 i=1 K "

K "

j=1

t=1

dk t

"

.

k k k lij zi zj

=

"

k k k lij zi zj

=

k k k lij zi zj

=

$

≤

k=1 i,j∈{1,...,n}

2cy1 ,i zik zjk +

k=1 i,j∈{1,...,n} K " #

dk Pn i

(1)

k=1 i,j∈{1,...,n}

n K " " 2cy1 ,i zik ( zjk ) +

"

dk dk k !ni j k 2 . lij ( t=1 dt )

K "

"

k=1 i,j∈{1,...,n}

"

k (cy1 ,i + lij + cy1 ,j )zik zjk

k=1 i,j∈{1,...,n} K "

"

c(vk )zik zjk .

k=1 i,j∈{1,...,n}

The last inequality holds because of the following reasoning: if zik zjk = 0 then k (cy1 ,i + lij + cy1 ,j )zik zjk = c(vk )zik zjk = 0; otherwise vk starts and ends at depot y1 , and k visits customers i, j and thus (cy1 ,i +lij +cy1 ,j )zik zjk ≤ c(vk )zik zjk because of the triangle

Probabilistic Analysis of VRPs

7

inequality. We use the equality K "

!n

k j=1 zj

"

= 1 again to simplify the last expression:

c(vk )zik zjk

=

k=1 i,j∈{1,...,n} K "

k=1

n n " " c(vk )( zik )( zjk ) = i=1

j=1

K "

c(vk ).

k=1

! Since any vehicle vk delivers holds.

!n

t=1

dkt ≤ Q units of demand, the following observation

Observation 1. Let the pair (K, V ) be a solution of the metric VRP, let dki ∈

{0, 1} denote whether vehicle vk ∈ V visits customer i or not. Therefore, the term !K !n !n dk 1 P i k=1 i=1 2cy1 ,i n dk is at least the radial cost R = i=1 2cy1 ,i Q . t=1

t

The lower bound on the cost of a solution of the metric VRP given by Lemma 3

contains two sums. One sum is at least the radial cost, as Observation 1 states. The second sum is an improvement over the radial cost. This improvement will be analyzed in a probabilistic setting in the next section. We give some definitions first. Definition 2. Given a solution (K, V ) of a VRP, a vehicle vk is called half-full if it visits at least

Q 2

customers. Let A ⊆ X (n) be the set of customers visited by half-full

vehicles. A solution satisfies the fullness property if |A| ≥ n −

Q 2.

The following lemma says that there always exists an optimal solution that satisfies the fullness property. Lemma 4. There exists an optimal solution (K, V ) such that |A| ≥ n −

Q 2.

Proof. Let (K, V ) be an optimal solution such that the associated set A has maximal cardinality. Either there is at most one vehicle that visits at most there are at least two. In the first case, |A| ≥ n −

two vehicles that visit at most

Q 2

Q 2.

Q 2

of the customers or

In the second case, we can replace

of the customers by one vehicle without increasing

the routing cost. In this case we found an optimal solution with an associated set A' bigger than A, contradicting the maximality of A. We give a name to the quadratic term in expression (1).

!

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A. BOMPADRE, M. DROR AND J. B. ORLIN

Definition 3. Given a solution (K, V ), let QC(K, V ) =

K "

"

k=1 i,j∈{1,...,n}

dk dk k !ni j k 2 . lij ( t=1 dt )

(2)

Let QC be the maximum value of QC(K, V ) among all optimal solutions (K, V ) that satisfy the fullness property of Definition 2. It is apparent that the radial cost plus the quadratic term QC is a lower bound on the cost of a solution (K, V ) that satisfies the fullness property. Therefore, the following lemma holds. Lemma 5. The cost of VRP is at least max{R + QC; c(T SP )}. 3. Probabilistic analysis The main technical result of this section is Lemma 7, which shows that the quadratic √ term (2) is Ω( n) under a probabilistic model. This in turn implies the main result of this paper (Theorem 2), which states that the approximation ratio of the ITP heuristic is strictly better than 2 under a probabilistic model. We start by giving the classic result by Beardwood, Halton and Hammersley [5] concerning the asymptotic behavior of the cost of the TSP. Let X1 , X2 , . . . be a sequence of i.i.d. points having a distribution on [0, 1]2 . Let f be the density of the absolutely continuous part of the distribution of Xi . There exists a constant β > 0 that does not depend on the distribution of Xi such that, with probability one, the cost of an optimal subtour through the first n points satisfies that % c(T SP (X (n) )) √ lim = β f 1/2 dx. n→∞ n

(3)

Computing the exact value of β is an open problem. However, Beardwood et al. showed that 0.62 ≤ β ≤ 0.93.

From now on, we assume that the customers X1 , . . . , Xn are independent random

variables with distribution U [0, 1]2 . Although the results proven in this paper also hold for more general random variables, the restriction to uniform random variables is made √ in order to simplify proofs. In this case, the ratio between the cost of TSP and n converges a.s. to β. The following theorem is proved in [9]. Informally, the result says

Probabilistic Analysis of VRPs

9

√ that the radial cost dominates the cost of the TSP when the capacity Q = o( n), and the opposite happens when n = o(Q2 ). As a result, the ITP heuristic is asymptotically optimal in both cases. Theorem 1. ([9].) Let X1 , X2 , . . . be a sequence of i.i.d. uniform random points in [0, 1]2 with expected distance µ from the depot, and let X (n) denote the first n points of the sequence. • If limn→∞

Q √ n

= 0, then c(V RP (X (n) ))Q = 2µ a.s. n→∞ n lim

• If limn→∞

Q √ n

= ∞, then c(V RP (X (n) )) √ = β a.s. n→∞ n lim

where β > 0 is the constant of [5]. This theorem implies that the ITP heuristic is asymptotically optimal whenever limn→∞

Q √ n

= 0 or limn→∞

Corollary 1. If limn→∞

Q √ n

Q √ n

= ∞.

= 0 or if limn→∞

Q √ n

= ∞ then

c(V RP IT P ) c(V RP )

= 1 a.s.

In order to prove that the ITP heuristic is strictly better than a 2-approximation algorithm on average, we have to prove that the quadratic term (2) has a non-negligible growth as a function of n. Informally, the quadratic term captures part of the interdistance between customers that is neglected by the radial cost. This cost is related to the distance of a generic customer i to its closest neighbor j *= i. Let p be a parameter. If we define a threshold value and we consider a customer an isolated customer whenever its distance to its closest neighbor is at least the threshold, then the following lemma says that we can define the threshold as a function of p in order to guarantee that the proportion of isolated customers is at least 1 − p with probability approaching 1 as √ n → ∞. This lemma is critical to prove that the quadratic term (2) is Ω( n) with

probability 1.

Lemma 6. For any 0 < p ≤ 1 there exists a value c(p) > 0 such that c(p) lim P(at least pn customers have a neighbor at distance ≤ √ ) = 0. n→∞ n

(4)

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A. BOMPADRE, M. DROR AND J. B. ORLIN

Proof. First divide the unit square into K 2 = αpn subsquares, each with a side whose length is

√1 . αpn

We assume that αpn is the square of an integer. This assumption

is used to simplify the proof. This proof carries through even if αpn is not a square by defining K as the (unique) integer such that K 2 ≤ αpn < (K + 1)2 . We will soon

choose α as a function of p.

Suppose that n points are dropped at random, corresponding to the selection of n squares at random (with replacement). We will call a selected square isolated if no other selected square is a neighbor. (Neighbors include the square and its 8 adjacent squares.) Otherwise, we call it non-isolated. Then, P(at least pn customers have a neighbor at distance ≤

√1 ) αpn

P(at least pn selected squares have a selected neighbor) =

≤

(5)

P(at least pn selected squares are non-isolated). Suppose that we select points randomly one at a time. As the k−th point is randomly selected, the probability that it is adjacent to a point that already has been randomly selected is at most

9(k−1) αpn ,

which is bounded from above by

9n αpn

=

9 αp .

This upper

bound is independent of the locations of the first k − 1 points. Therefore, the total number of non-isolated points is bounded by 2X, where X is the number of successes in n Bernoulli experiments with success probability

9 αp .

The factor of 2 comes from

the fact that if point k is adjacent to point i < k, then i is also adjacent to k. Thus, P(at least pn selected squares are non-isolated) ≤ P(at least

pn 2

successes out of n with probability of success of

Therefore, if we choose α so that

18 αp

< p (that is, α >

18 p2 ),

9 αp ).

(6)

then by the weak

law of large numbers, limn→∞ P(at least pn selected squares are non-isolated) = 0. Therefore, any value of c(p) smaller than

√ p √ 3 2

satisfies equation (4).

!

This lemma is similar in flavor to known bounds on the minimum distance between a fixed point and n i.i.d. points. It differs in the fact that it is an upper bound on the probability of a set of non-independent events (the minimum distances among the n points) to happen. We will use it to prove a lower bound on the quadratic term (2). It also provides non-trivial lower bounds on the cost of combinatorial optimization problems such as the minimum weighted matching, the traveling salesman problem

Probabilistic Analysis of VRPs

11

and the minimum latency problem as the next corollary shows. We will invest some time analyzing the minimum latency problem since we are not aware of asymptotic results for this problem. The minimum latency problem (see e.g., Blum et al. [6]) is to find a tour of minimum latency through all customers. The latency of a customer ik , denoted by cik , is the time spent to arrive at ik . The latency of a tour is the sum of waiting times of all customers. That is, given a solution (y1 , i1 , i2 , . . . , in , y1 ) of the latency problem (we denote by i0 the depot y1 ), its cost is n "

cik =

k=1

n−1 " k=0

(n − k)cik ,ik+1 .

We observe that the lower bounds we obtain in the following corollary for a minimum weighted matching and traveling salesman problem are of the same order of magnitude as the asymptotic results proved for these problems by Papadimitriou [13] and Beardwood et al. [5] respectively. The lower bound for the minimum latency problem is also of the same order of magnitude as the optimal cost of this problem, as we show in Proposition 1. Corollary 2. The cost of a minimum weighted matching and the cost of TSP when √ points are uniformly distributed in [0, 1]2 are Ω( n) with probability 1. The cost of minimum latency problem when points are uniformly distributed in [0, 1]2 is Ω(n1.5 ) with probability 1. Proof. For a fixed 0 < p < 1 and for c(p) such that equation (4) holds, minimum √ √ (1−p)n = Ω( n) with probability 1. Selecting weighted matching has cost at least 2c(p) n √

p

p = 1/3 and c(p) smaller than 3√2 imply that minimum weighted matching has cost at √ least 0.04 n with probability 1. This bound also holds for the TSP, since the minimum weighted matching is a relaxation of the TSP and gives a lower bound for this problem. Given a solution (y1 , i1 , i2 , . . . , in , y1 ) of the minimum latency problem, its cost !n−1 is k=0 (n − k)cik ,ik+1 . Informally, Lemma 6 says that a significant percentage of

customers are isolated. That is, for a fixed 0 < p < 1 and for c(p) such that equation (4) holds, at least (1 − p)n arcs used by any solution of the latency problem have cost at least

c(p) √ . n

Since the cost function of the minimum latency problem is a weighted

sum of the arcs used, where the arcs used first have greater weight, the cost of any

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A. BOMPADRE, M. DROR AND J. B. ORLIN

solution is at least n−1 "

c(p) (1 − p)2 n2 c(p) = Ω(n1.5 ). (n − k) √ ≥ √ 2 n n

(7)

k=pn

√ p √ , 3 2 1.5

In particular, by selecting p = 1/5 and c(p) smaller than that the minimum latency problem has cost at least 0.03n

Equation (7) implies

with probability 1.

!

The next proposition complements Corollary 2 for the minimum latency problem. Namely, it shows that the minimum latency problem has cost O(n1.5 ) when the depot and the customers are points in [0, 1]2 . We observe that this result holds in the worstcase sense, namely, without assuming any probabilistic distribution of the points. Proposition 1. The cost of minimum latency problem when customers and depot are located in [0, 1]2 is O(n1.5 ). Proof. We will use an optimal solution for the TSP through the n customers to construct a solution for the minimum latency problem. Let route0 = (i1 , . . . , in ) be an optimal traveling tour through the n customers. Few [8] proved that the optimal cost √ of the TSP through any n points in [0, 1]2 is at most 2n. The cost of route0 is thus √ !n−1 2n. We consider the following n routings for the latency k=1 cik ,ik+1 + cin ,i1 ≤ problem.

For each 1 ≤ l ≤ n, let routel be (y1 , il , il+1 . . . , in , i1 , i2 , . . . , il−1 , y1 ).

That is, route1 = (y1 , i1 , . . . , in , y1 ), route2 = (y1 , i2 , . . . , in , i1 , y1 ), . . . , routen =

(y1 , in , i1 , . . . , in−1 , y1 ). For each 1 ≤ l ≤ n, the latency cost of routel is ncy1 ,il + !n−1 !l−1 k=l (n + l − k − 1)cik ,ik+1 + (l − 1)cin ,i1 + k=1 (l − k − 1)cik ,ik+1 . Since each arc

(ik , ik+1 ) of the tour route0 appears in each possible place in routes route1 , . . . , routen ,

the sum of the latency costs of the routings route1 , . . . , routen is n " l=1

n−1 "

ncy1 ,il + (

l=0

n−1 "

l)(

k=1

√ n(n − 1) √ cik ,ik+1 + cin ,i1 ) ≤ n2 2 + 2n. 2

√ The average cost of these routes is at most n 2 +

(n−1) √ 2n 2

= O(n1.5 ). Since at least

one of the routes route1 , . . . , routen has cost at most the average, we proved that there exists a solution of the minimum latency with cost O(n1.5 ).

!

√ We will use Lemma 6 to prove that the quadratic term (2) is Ω( n) with probability 1 (Lemma 7). We give some definitions first.

Probabilistic Analysis of VRPs

13

Definition 4. Given a parameter c, we say that a customer i is non-isolated with respect to c if it has a neighbor at distance at most

√c . n

Proposition 2. Given a sequence l = (i1 , i2 , . . . , is ) with s > 1 customers where b of them are non-isolated w.r.t. c, the following inequality holds. "

i,j∈{i1 ,...,is }

Proof. We can express the sum

lij c s ≥ √ ( − 2b). 2 s n 6

!

lij i,j∈{i1 ,...,is } s2

as

!s−1 t=1

2t(s−t) cit ,it+1 . s2

Let δ(t) = 0

whenever it and it+1 are both non-isolated customers w.r.t. c and 1 otherwise. Then cit ,it+1 ≥

δ(t)c √ n

and therefore "

i,j∈{i1 ,...,is }

lij s2

s−1 " 2t(s − t) δ(t)c √ s2 n t=1

=

s−1 " 2t(s − t) t=1

=

s2

cit ,it+1 ≥

s−1 " 2t(s − t) c " 2t(s − t) ( − )√ . 2 s s2 n t=1

(8)

t:δ(t)=0

In order to bound from below the right hand side of this inequality we will use 2 !s−1 !s−1 2 s(s−1) (s−1)s(2s−1) the identities and ≤ s (s−1) . The sum t=1 t = t=1 t = 2 6 3 !s−1 2t(s−t) is then t=1 s2 s−1 " 2t t=1

s

−

s−1 " 2t2 t=1

s2

1 s 2 ≥ (s − 1) − (s − 1) = (s − 1) ≥ . 3 3 6

In order to bound from below the sum

!

t:δ(t)=0

−2t(s−t) s2

(9)

we observe that among

all possible distributions of b non-isolated customers in a sequence with s customers, this sum reaches its minimum value when the non-isolated customers are located in the center of the sequence l. The reason is that the weight

−2t(s−t) s2

is a quadratic

convex function of t, that reaches its minimum at t = s/2. That is, this sum reaches its minimum when the non-isolated customers in the sequence l are ia , ia+1 , . . . , ia+b−1 s−b where a is either , s−b 2 - or , 2 - + 1. (In particular, a is at most ! the sum t:δ(t)=0 2t(s−t) is at most s2 a+b−1 " t=a

s−b 2

+ 1.) Therefore,

a+b−1 " 2ts 2t(s − t) b(s + 1) b(2a + b − 1) ≤ ≤ 2b. ≤ = 2 2 s s s s t=a

(10)

Inequalities (9) and (10) imply that the right hand side of inequality (8) is at least √c ( s n 6

− 2b).

!

14

A. BOMPADRE, M. DROR AND J. B. ORLIN

√ When Q ≥ 2, the quadratic term QC defined in (2) is Ω( n) with probability 1 as

the next lemma shows.

Lemma 7. Assuming that Q ≥ 2, there exists c¯ > 0 such that √ lim P(QC ≥ c¯ n)

n→∞

exists and is equal to 1. Proof. Let (K, V ) be any optimal solution that satisfies the fullness property. By Lemma 4 there is always an optimal solution that satisfies the fullness property. This implies that at most one vehicle vk of this solution visits only one customer. We will √ prove that there exists c¯ > 0 such that limn→∞ P(QC(K, V ) ≥ c¯ n) = 1 for any such √ (K, V ), which implies that limn→∞ P(QC ≥ c¯ n) = 1 holds. Let 0 < p < 1, c > 0 be parameters to be fixed afterwards. Each vehicle vk ∈ V has associated a sequence lk

with sk customers and bk of them are non-isolated customers w.r.t. c. Proposition 2 implies that K "

"

k=1 i,j∈{1,...,n}

dk dk k !n i j k 2 lij ( j=1 dj )

≥

c n √ ( − 2|{non isolated customers w.r.t. c}|). n 6

K " c s √ ( k − 2bk ) = n 6

k=1

(11)

If we choose p, c such that Lemma 6 holds, then lim P(|{non-isolated customers w.r.t. c}| ≤ pn) = 1

n→∞

and therefore lim P

n→∞

K #"

"

k=1 i,j∈{1,...,n}

$ dk dk c n k !ni j k 2 ≥ √ ( − 2pn) = 1. lij n 6 ( j=1 dj )

We now have all the necessary pieces to prove the main result. Theorem 2. There exists a constant cˆ > 0 such that lim P(

n→∞

exists and is equal to 1.

c(V RP IT P ) ≤ 2 − cˆ) c(V RP )

!

Probabilistic Analysis of VRPs

15

Proof. We know that c(V RP IT P ) ≤ R + c(T SP ). Lemma 5 says that c(V RP ) ≥

max{R + QC, c(T SP )}. Therefore,

c(V RP IT P ) R + c(T SP ) QC ≤ ≤2− . (12) c(V RP ) max{R + QC, c(T SP )} c(T SP ) √ We know that the ratio between the cost of TSP and n converges a.s. to a √ constant β. By Lemma 7 we know that limn→∞ P(QC ≥ c¯ n) = 1. By setting

cˆ =

c¯ β


0 is at most 0.93 (see e.g., Ausiello et al. [4]). By fixing p =

1 36 ,

the value

√

p( 16 −2p) √ 3 2β

(and therefore cˆ) is at

least 0.0046. It is possible to prove a tighter bound for the constant c(p) in Lemma 6 and to prove a tighter bound in Proposition 2. As a result, we can prove that cˆ is at least 0.015. A generalization Let µ = E(cy1 ,X ) be the expected distance of a customer X to the depot. If limn→∞

Q √ n

exists and is equal to a finite value w > 0, the strong law

of large numbers implies that limn→∞

R

2µ √ n w

= 1 a.s. The approximation ratio of the

ITP heuristic satisfies the following. Theorem 3. Let c¯ be the constant defined in Lemma 7. Assume that limn→∞

Q √ n

exists and is equal to 0 < w < ∞. The approximation ratio of the ITP heuristic

satisfies that • If

2µ w

+ c¯ ≥ β, lim P(

n→∞

• If

2µ w

c(V RP IT P ) ≤1+ c(V RP )

2µ w

β − + c¯

2µ w

c¯ ) = 1. + c¯

+ c¯ < β, lim P(

n→∞

2µ c(V RP IT P ) ≤1+ ) = 1. c(V RP ) wβ

16

A. BOMPADRE, M. DROR AND J. B. ORLIN

Proof. The results follow from the ratio R + c(T SP ) c(V RP IT P ) ≤ c(V RP ) max{R + QC, c(T SP )} and the limits lim inf n→∞ 2µ wβ .

c(T SP ) R+QC

≥

β

2µ c w +¯

, lim supn→∞

QC R+QC

≤

c¯

2µ c w +¯

, limn→∞

R c(T SP )

=

!

4. Multi-depot vehicle routing problem In this Section we generalize the VRP to a multi-depot scenario. There are n customers and m depots. Each vehicle starts and ends at the same depot. We assume that there is no restriction on the number of vehicles available at each depot. 4.1. Notation We extend the notation from the previous sections. Unless otherwise stated, customers and depots are modeled as points in the plane. The location of the ith customer is denoted by xi for any 1 ≤ i ≤ n. The location of the jth depot is denoted by yj for any 1 ≤ j ≤ m. The set of customers is denoted by X (n) . The set of depots is denoted

by Y (m) . A solution of a multi-depot vehicle routing problem (MDVRP) is denoted by a pair (K, V ) where K is the number of vehicles used, and V = {vk : k ∈ {1, . . . , K}}

is the set of routings. For each customer i, let ci = min{ci,y : y ∈ Y (m) } be its !n minimum distance to a depot. RM D = i=1 2ci /Q is the multi-depot radial cost.

We denote by c(M DV RP ) or by c(M DV RP (X (n) )) the cost of an optimal MDVRP

and by c(T SP ) or by c(T SP (X (n) ∪ Y (m) )) the cost of an optimal traveling salesman tour through all customers and depots. Let c∞ (M DV RP ) denote the cost of the

MDVRP when the capacity of vehicles is infinite. Following Definition 1, given a k routing vk = (yvk , ik1 , ik2 , . . . , iksk , yvk ) and customers i, j visited by vk , we denote by lij

the length of the unique path determined by vk from i to j that does not pass through k the depot yvk . If i or j are not visited by vk , we let lij = 0.

5. An algorithm for the multi-depot VRP We generalize the Iterated Tour Partitioning to the multi-depot case. The MultiDepot Iterated Tour Partitioning (MDITP(α)) heuristic we propose uses the ITP(α)

Probabilistic Analysis of VRPs

17

heuristic as a subroutine. First, it assigns each customer to its closest depot (see Figure 1), and then solves m independent VRP problems. More formally, the MDITP heuristic is as follows: 0) Sj = ∅ for all 1 ≤ j ≤ m;

1) For each customer xi find yj , its closest depot, and let Sj = Sj ∪ {xi };

2) For each 1 ≤ j ≤ m run the ITP(α) heuristic to approximately solve the VRP problem V RPj where yj is the only depot and the set of customers is Sj ;

The cost of the solution produced by this heuristic is the sum of the cost of the !m solutions produced by ITP(α) on each V RPj which is at most RM D +α j=1 c(T SPj ). In what follows, we denote by c(M DV RP M DIT P ) the cost of this heuristic when we

use ITP(1) as a subroutine. This analysis implies the following.

Lemma 8. The cost of M DV RP is at most RM D +

!m

j=1

c(T SPj ).

It is easy to see that the MDITP(1) heuristic is an Ω(m)-approximation algorithm under the worst case analysis, where m is the number of depots. Let us consider the following example.

Example 1. There are m depots located on the unit circle and there are m customers located close to the origin (0, 0), defined as follows. Let ! > 0, and let Q, the capacity of each vehicle be infinite. For each 1 ≤ j ≤ m, depot yj is located in the point

2πj 2πj 2πj (cos 2πj m , sin m ) and customer ij is located in the point (! cos m , ! sin m ).

In this example, the MDITP(α) heuristic will assign customer ij to depot yj and will output a solution with cost 2m(1 − !). On the other hand, the solution that uses only one vehicle with routing (y1 , i1 , i2 , . . . , im , i1 , y1 ) has cost at most 2(1 − !) + !2π. By

letting ! → 0, the ratio between the two solutions goes to m. Therefore, the MDITP(1)

heuristic is an Ω(m)-approximation algorithm. However, in Section 7 we will show that the MDITP(1) heuristic performs much better on average. Namely, its approximation ratio is the same as the one of the ITP heuristic for the single depot case on average.

18

A. BOMPADRE, M. DROR AND J. B. ORLIN

U1

U3

Depot 3

Depot 1

U2

Depot 2

Figure 1: Assigning customers to their closest depot.

6. Lower bounds on the optimal cost of multi-depot VRP In this section we generalize the results from Section 2 to the multi-depot case. The following lemma generalizes Lemma 1. Lemma 9. The cost of MDVRP is at least max{RM D ; c∞ (M DV RP )}. Proof. Let Xk be the set of customers visited by vehicle vk and let y be the depot from where vk starts and ends. Then, c(vk ) ≥ 2 max{ci,y } ≥ 2 max{ci } ≥ 2 i∈Xk

i∈Xk

!

i∈Xk

|Xk |

ci

≥

2 " ci . Q

Summing for all vehicles we obtain that c(M DV RP ) ≥ RM D .

i∈Xk

Inequality c(M DV RP ) ≥ c∞ (M DV RP ) holds since any feasible solution of the MD-

Probabilistic Analysis of VRPs

19

VRP with vehicles with capacity Q is also feasible for the infinite-capacity MDVRP. ! The next lemma relates c∞ (M DV RP ) with the cost of a TSP. Lemma 10. Let c(T SP (Y (m) )) denote the cost of an optimal subtour through the depots. Then, c∞ (M DV RP ) ≥ c(T SP (X (n) ∪ Y (m) )) − c(T SP (Y (m) )). Proof. We observe that c∞ (M DV RP ) + c(T SP (Y (m) )) is the cost of the walking tour formed by the union of the routes of an optimal solution of the infinite-capacity MDVRP plus a subtour through all the depots. This walking tour can be transformed into a TSP through all customers and depots of lesser cost by shortcutting nodes already visited. Therefore, c∞ (M DV RP ) + c(T SP (Y (m) )) ≥ c(T SP (X (n) ∪ Y (m) ))

!

holds.

The following lemma is a generalization of Lemma 3. Lemma 11. Let (K, V ) be a solution of the MDVRP. Let dki be 1 when vehicle vk visits customer i and 0 otherwise. Then, the cost of (K, V ) is at least K " n "

k=1 i=1

dk 2ci !n i

t=1

dkt

+

K "

"

k=1 i,j∈{1,...,n}

dk dk k !ni j k 2 . lij ( t=1 dt )

(14)

We generalize the fullness property given in Definition 2 to the multi-depot case. A solution of the MDVRP satisfies the fullness property if |A| ≥ n − m Q 2 . The following

lemma says that there always exists an optimal solution of the MDVRP that satisfies the fullness property. Lemma 12. There exists an optimal solution (K, V ) of the MDVRP such that |A| ≥ n − mQ 2.

Proof. Given an optimal solution (K, V ), each depot has at most one vehicle that is not half-full. Otherwise, we can merge the routings of two not half full vehicles and obtain another optimal solution that satisfies the fullness property. We give a name to the quadratic term in expression (14).

!

20

A. BOMPADRE, M. DROR AND J. B. ORLIN

Definition 5. Given a solution (K, V ) of the MDVRP, let QC(K, V ) =

K "

"

k=1 i,j∈{2,...,n}

dk dk k !ni j k 2 . lij ( t=1 dt )

(15)

Let QC be the maximum value of QC(K, V ) among all optimal solutions (K, V ) of the MDVRP that satisfy the fullness property. It is apparent that the multi-depot radial cost plus the quadratic term QC is a lower bound on the cost of any solution (K, V ) that satisfies the fullness property. Since there is at least one optimal solution that satisfies the fullness property, the following lemma holds. Lemma 13. The cost of MDVRP is at least max{RM D + QC; c(T SP (X (n) ∪ Y (m) )) − c(T SP (Y (m) ))}. Lemma 7 also holds for the multi-depot case. The constant c¯ of this lemma is the same as the one in Lemma 7. Lemma 14. Assuming that Q ≥ 2, there exists c¯ > 0 such that √ lim P(QC ≥ c¯ n)

n→∞

exists and is equal to 1. 7. Probabilistic analysis of MDVRP In this section, we extend the main results of the single depot case (e.g., Theorems 1 and 2) to the multi-depot case. 7.1. Probabilistic analysis of lower bounds We analyze the Unit Demand Euclidean MDVRP where the depots and customers are modeled as points in the plane. The customers have unit demand. The m depots y1 , . . . , ym are fixed in advance whereas the location of the n customers are i.i.d. uniform random variables on [0, 1]2 . Let U1 , . . . , Um be a disjoint partition of the square [0, 1]2 such that each Uj contains exactly one depot, namely yj . For each 1 ≤ j ≤ m, let nj ≥ 0 be the number of

Probabilistic Analysis of VRPs

21

customers that belong to Uj and let X (nj ) be the set of customers that belong to Uj . Let c(T SPj ) := c(T SP (X (nj ) ∪ {yj })) be the cost of an optimal subtour that visits all

customers of X (nj ) and depot yj . The following result holds. Lemma 15. lim

!m

j=1

n→∞

c(T SPj )

c(T SP )

=

1 (a.s.)

Proof. For each 1 ≤ j ≤ m, let fi be the restriction of the uniform density to the

set Uj . That is,

The function

f |Uj |

  1 if x ∈ U , j fj (x) =  0 otherwise.

is the density function of a customer in Uj . We will prove that % c(T SPj ) 1/2 √ = β fj dx (a.s.) (16) lim n→∞ n

for each 1 ≤ j ≤ m. Let us fix j. In order to prove equation (16) we would like to restrict the experiment to the customers that fell inside Uj and apply the result by

Beardwood et al. [5]. However, this has to be done with some care since the number of customers that fell inside Uj is random. We observe that the distribution of customers that fell outside Uj does not affect c(T SPj ). Therefore, the cost of the optimal TSP of the following experiment is probabilistically the same random variable as c(T SPj ). Let the n customers be i.i.d. points with the following distribution: with probability pj = |Uj | customer i is located on Uj according to density

fj pj ,

with probability 1 − pj

customer i is located in depot yj . The main properties of the new experiment are 1. All the points fell inside Uj .

2. The cost of an optimal tour through all points and depot yj in the new experiment has the same distribution as the cost c(T SPj ) in the original experiment. Since the absolutely continuous density part of the distribution of a customer in the new experiment is fj , the result by Beardwood et al. [5] implies that equation (16) holds for each j. Since for each j, equation (16) holds almost surely, altogether they imply that !m % " % m j=1 c(T SPj ) 1/2 √ lim =β fj dx = β f 1/2 dx (a.s.). (17) n→∞ n j=1

22

A. BOMPADRE, M. DROR AND J. B. ORLIN

The last equality holds since f =

!m

j=1

fj and the support of functions fj are mutually

disjoint. Finally, (17) and (3) imply that !m !m √ j=1 c(T SPj ) j=1 c(T SPj )/ n √ = lim lim = 1 (a.s.) n→∞ n→∞ c(T SP ) c(T SP )/ n ! The following theorem generalizes Theorem 1 to the multi-depot case. Theorem 4. Let X1 , X2 , . . . be a sequence of i.i.d. uniform random points in [0, 1]2 with expected distance µ to its closest depot, and let X (n) denote the first n points of the sequence. 1. If limn→∞

Q √ n

= 0, then c(M DV RP (X (n) ))Q = 2µ a.s. n→∞ n lim

2. If limn→∞

Q √ n

= ∞, then c(M DV RP (X (n) )) √ = β a.s. n→∞ n lim

where β > 0 is the constant in [5]. Proof. Lemmas 8, 9, 10 imply that max{RM D ; c(T SP ((X (n) ∪ Y (m) )) − c(T SP (Y (m) ))} ≤ c(M DV RP (X (n) )) ≤ RM D + !m !n RM D Q iQ = i=1 2cnQ = j=1 c(T SPj ). The ratio between RM D and n/Q is equal to n !n 2ci i=1 n . The law of large numbers implies that this ratio converges a.s. to 2µ. If

X1 , X2 , . . . are uniformly bounded and limn→∞ Pm

Q √ n

c(T SPj ) Q √ = 0 since limn→∞ j=1√n n Pm (n) (m) j=1 c(T SPj ) √ limn→∞ = limn→∞ c(T SP (X√n ∪Y )) n

= 0, then limn→∞

Pm

j=1

c(T SPj )Q n

=

is a constant. Therefore,

max{RM D ; c(T SP (X (n) ∪ Y (m) )) − c(T SP (Y (m) ))} ≤ n→∞ n/Q !m RM D + j=1 c(T SPj ) c(M DV RP (X (n) )) lim ≤ lim = 2µ n→∞ n→∞ n/Q n/Q

2µ = lim

in this case.

When limn→∞

Q √ n

= ∞, then

RM √D Q n

=

!n

2ci √nQ i=1 n nQ

=

!n

2ci √ n. i=1 n

to infinity, this ratio converges to 0 a.s. The ratio between c(T SP ((X

(n)

When n goes ∪ Y (m) ))) and

Probabilistic Analysis of VRPs

23

√ n converges to β a.s. The ratio between c(T SP (Y (m) )) and n converges to 0 since !m √ c(T SP (Y (m) )) is a constant. The ratio between j=1 c(T SPj ) and n also converges

√

to β a.s. because of Lemma 15. Therefore,

max{RM D ; c(T SP (X (n) ∪ Y (m) )) − c(T SP (Y (m) ))} √ ≤ n→∞ n !m RM D + j=1 c(T SPj ) c(M DV RP (X (n) )) √ √ lim ≤ lim =β n→∞ n→∞ n n

β = lim

in this case.

!

7.2. Probabilistic analysis of an algorithm for MDVRP The following theorem is an extension of Theorems 2 and 3. Theorem 5. There exists a constant cˆ > 0 such that lim P(

n→∞

c(M DV RP M DIT P ) ≤ 2 − cˆ). c(M DV RP )

(18)

exists and is equal to 1. Moreover, 1. If limn→∞

Q √ n

= 0, then c(M DV RP M DIT P ) = 1 (a.s.) n→∞ c(M DV RP ) lim

2. If limn→∞

Q √ n

= w > 0 and

lim P(

n→∞

3. If limn→∞

Q √ n

= w > 0 and n→∞

Q √ n

+ c¯ ≤ β,

c(M DV RP M DIT P ) ≤1+ c(M DV RP )

lim P(

4. If limn→∞

2µ w

2µ w

2µ w

β − + c¯

2µ w

c¯ ) = 1. + c¯

+ c¯ < β,

2µ c(M DV RP M DIT P ) ≤1+ ) = 1. c(M DV RP ) wβ

= ∞, then c(M DV RP M DIT P ) = 1 (a.s.) n→∞ c(M DV RP ) lim

where β is the constant from [5], µ is the expected distance of a customer to its closest depot and c¯ is the constant of Lemma 14. Proof. The proofs of these results go along the same lines as the proofs of Theorems 2 and 3. The proofs of equation 18 and 2 follow from Lemmas 8, 13, 14 and 15. The proofs of 1, 3 and 4 follow from Lemma 8 and Theorem 4.

!

24

A. BOMPADRE, M. DROR AND J. B. ORLIN

8. Summary We study the Unit Demand Euclidean Vehicle Routing Problem under a probabilistic model. We present a new lower bound for the metric VRP with unlimited number of homogeneous vehicles. This bound is at least the radial cost plus a term of the same order of magnitude as the cost of TSP when customers have unit demand, they are modelled as i.i.d. uniform random points in the plane, and the distance is the Euclidean distance. This lower bound improves the radial cost bound defined by Haimovich √ and Rinnooy Kan [9]. We show that this improvement over the radial cost is Ω( n) with probability 1 as the number of customers goes to infinity. In particular, this improvement is of the same order of magnitude as the cost of the TSP, which is also a lower bound on the cost of the Unit Demand Euclidean Vehicle Routing Problem. The improvement on the lower bound of the Unit Demand Euclidean Vehicle Routing Problem we present implies an improvement on the approximation ratio of the Iterated Tour Partitioning (ITP) heuristic for this problem. As a result, the approximation ratio of the ITP heuristic is shown to be strictly better than a 2 with probability one as the number of customers goes to infinity. In the second part of this paper we analyze the multi-depot vehicle routing problem. We give a natural generalization of the ITP heuristic for this problem. The lower bounds presented in the first part are extended to this problem. The asymptotic results for the single-depot case from the first part are generalized to the multi-depot case. References [1] Altinkemer, K. and Gavish, B. (1990). Heuristics for delivery problems with constant error guarantees. Transportation Science 24, 294–297. [2] Arora, S. (1996). Polynomial-time approximation schemes for euclidean tsp and other geometric problem. Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science 2–12. [3] Asano, T., Katoh, N., Tamaki, H. and Tokushama, T. (1997). Covering points in the plane by k-tours: towards a polynomial time approximation scheme

Probabilistic Analysis of VRPs

25

for general k. Proceedings of STOC ’97 275–283. [4] Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., MarchettiSpaccamela, A. and Protasi, M. (1999). Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. [5] Beardwood, J., Halton, J. L. and Hammersley, J. M. (1959). The shortest path through many points. Proc. Cambridge Phil. Soc. 55, 299–327. [6] Blum, A., Chalasani, P., Coppersmith, D., Pulleyblank, B., Raghavan, P. and Sudan, M. (1994). The minimum latency problem. In STOC ’94: Proceedings of the twenty-sixth annual ACM symposium on Theory of computing. ACM Press. pp. 163–171. [7] Bompadre, A., Dror, M. and Orlin, J. B. (2006). Improved bounds for vehicle routing solutions. Discrete Optimization 3, 299–316. [8] Few, L. (1955). The shortest path and the shortest path through n points. Mathematika 2, 141–144. [9] Haimovich, M. and Kan, A. H. G. R. (1985). Bounds and heuristics for capacitated routing problems. Math. Oper. Res. 10, 527–542. [10] Haimovich, M., Kan, A. H. G. R. and Stougie, L. (1988). Vehicle Routing, Methods and Studies. ch. Analysis of Heuristics for Vehicle Routing Problems, pp. 47–61. [11] Li, C. L. and Simchi-Levi, D. (1990). Worst-case analysis of heuristics for the multi-depot capacitated vehicle routing problems. ORSA J. Comput. 2, 64–73. [12] Mitchell, J. S. B. (1996).

Guillotine subdivisions approximate polygonal

subdivisions: A simple new method for the geometric k-MST problem. Proc. 7th ACM-SIAM Sympos. Discrete Algorithms 402–408. [13] Papadimitriou, C. H. (1977). The probabilistic analysis of matching heuristics. Proc. 15th Annual Allerton Conference on Communication, Control, and Computing 368–378.

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[14] Stougie, L. (1987). Design and analysis of algorithms for stochastic integer programming. CWI Tract 37, Stichting Mathematisch Centrum, Amsterdam.