Accepted Manuscript

Approximation algorithms for the arc orienteering problem

D. Gavalas, C. Konstantopoulos, K. Mastakas, G. Pantziou,N. Vathis

PII: S0020-0190(14)00218-XDOI: 10.1016/j.ipl.2014.10.003Reference: IPL 5189

To appear in: Information Processing Letters

Received date: 25 May 2014Revised date: 30 August 2014Accepted date: 3 October 2014

Please cite this article in press as: D. Gavalas et al., Approximation algorithms for the arcorienteering problem, Inf. Process. Lett. (2014), http://dx.doi.org/10.1016/j.ipl.2014.10.003

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Highlights

• We propose a polylogarithmic approximation algorithm for the AOP in directed graphs.• A constant factor approximation algorithm for the AOP in undirected graphs is presented.• An inapproximability result for the AOP is obtained.• Approximation algorithms for the Mixed Orienteering Problem are given.

Approximation Algorithms for the Arc Orienteering

Problem

D. Gavalasa,f, C. Konstantopoulosb,f, K. Mastakasc,f,∗, G. Pantzioud,f, N.Vathise,f

aDepartment of Cultural Technology and Communication, University of Aegean, GreecebDepartment of Informatics, University of Piraeus, Greece

cSchool of Applied Mathematical and Physical Sciences, National Technical University ofAthens, Greece

dDepartment of Informatics, Technological Educational Institution of Athens, GreeceeSchool of Electrical and Computer Engineering, National Technical University of

Athens, GreecefComputer Technology Institute & Press “Diophantus”, Patras, Greece

Abstract

In this article we present approximation algorithms for the Arc Orienteering

Problem (AOP). We propose a polylogarithmic approximation algorithm in

directed graphs, while in undirected graphs we give a (6 + ε + o(1)) and a

(4+ε)−approximation algorithm for arbitrary instances and instances of unit

profit, respectively. Also, an inapproximability result for the AOP is obtained

as well as approximation algorithms for the Mixed Orienteering Problem.

Keywords: Arc Orienteering Problem, Orienteering Problem, Mixed

Orienteering Problem, Approximation Algorithms, Inapproximability

∗Corresponding authorEmail addresses: dgavalas@aegean.gr (D. Gavalas), konstant@unipi.gr (C.

Konstantopoulos), kmast@math.ntua.gr (K. Mastakas), pantziou@teiath.gr (G.Pantziou), nvathis@softlab.ntua.gr (N. Vathis)

Preprint submitted to Information Processing Letters October 14, 2014

1. Introduction1

The Arc Orienteering Problem (AOP) is a single route arc routing prob-2

lem with profits defined as follows [10]: Given a quadruple (G = (V,A), t, p, B)3

where G = (V,A) is a directed graph with V = {s = u1, u2, . . . , un = l} the4

set of nodes and A the set of arcs, t : A → R+ and p : A → R

+ i.e., each5

arc a ∈ A is associated with a nonnegative travel time ta and a nonnegative6

profit pa, and a nonnegative time budget B, the goal is to find an s− l walk7

with length at most B with the maximum collected profit from the traversed8

arcs. An arc may be traversed more than once, however its profit is collected9

only once. In a similar way, we may define the AOP in undirected graphs.10

The AOP is the arc routing version of the Orienteering Problem (OP) [7],11

where the nodes (instead of the arcs) are associated with profit. The algorith-12

mic research relevant to the OP and its extensions is broad [6, 11]. The OP13

in directed and undirected graphs is NP-hard [7] and hard to approximate14

within a factor of 14811480

[3]. The best approximation algorithm for the OP in15

undirected graphs is the (2 + ε)-approximation algorithm given in [4] while16

in directed graphs is the O( log2 nlog logn

)-approximation algorithm given in [9].17

Contrary to the OP, limited body of literature deals with the AOP and18

its extensions. A metaheuristic algorithm for the AOP was proposed by19

Souffriau et al. [10] while a matheuristic approach and a branch-and-cut20

technique for the extension of the AOP to multiple tours were proposed21

in [1] and [2], respectively. In this article we study the AOP in directed22

and undirected graphs. In Section 2 we give an inapproximability result23

for the AOP and propose a polylogarithmic approximation algorithm for24

the problem using the polylogarithmic approximation algorithm for the OP25

2

in [9]. In Section 3 we present a (6 + ε + o(1))−approximation algorithm26

for the AOP in undirected graphs and a (4 + ε)−approximation algorithm27

for the unweighted version of the problem, using the (2 + ε)−approximation28

algorithm for the unweighted OP in [4]. Finally, we obtain approximation29

algorithms for the Mixed Orienteering Problem (MOP), the extension of both30

the OP and the AOP, where both nodes and arcs are associated with profit.31

2. The Arc Orienteering Problem32

In this section we first obtain an inapproximability result for the AOP.33

Then, we propose an approximation algorithm for the problem.34

The OP in directed graphs is reduced to the AOP. Given an OP instance35

an AOP instance is created as follows: Assign zero profit to each arc and for36

each node u add a node u′, the arcs (u, u′), (u′, u) of zero travel cost and pu2

37

profit and remove u’s profit. Then, an OP solution yields an equal length38

and at least the same profit AOP solution and vice versa. Using the results39

in [7] and [3], we get the following theorem.40

Theorem 1. The AOP is NP-hard and hard to approximate within 14811480

.41

Theorem 2. An f(n)−approximation algorithm for the OP in directed graphs,42

where n is the number of nodes, yields an f(m+2)−approximation algorithm43

for the AOP, where m is the number of arcs.44

Proof. Given an instance of the AOP (G = (V,A), t, p, B), |A| = m, we con-45

struct an instance of the OP in the directed network N = (V ′, A′) with V ′ =46

{s, l}∪{(u, v) : (u, v) ∈ A}, |V ′| = n′ = m+2, and A′ = {(s, (s, u)), ((v, l), l) :47

(s, u), (v, l) ∈ A} ∪ {((u, v), (v, w)) : (u, v), (v, w) ∈ A}. The travel times of48

3

the arcs in N are defined as follows: t′((u,v),(v,w)) =t(u,v)+t(v,w)

2, t′(s,(s,u)) =

t(s,u)2

49

and t′((v,l),l) =t(v,l)2

. The profit of each node (u, v) equals to p′(u,v) = p(u,v) and50

B′ = B. Then, a solution of the AOP instance yields a solution of the OP51

instance of equal total profit and length and vice versa.52

The theorem yields a O( log2 mlog logm

)−approximation algorithm for the AOP,53

applying Nagarajan and Ravi’s algorithm [9] to the metric closure of the54

constructed OP instance and transforming the solution to an AOP solution.55

3. Approximation Algorithms for the AOP in Undirected Graphs56

In this section we study the AOP in undirected graphs. Similarly to the57

previous section the problem is NP-hard and hard to approximate within58

14811480

via a reduction from the OP in undirected graphs. We obtain a con-59

stant factor approximation algorithm for the problem by reducing it to the60

Unweighted OP (UOP) in undirected graphs, the restriction of the OP with61

nodes of unit profit. First, we reduce the AOP to the special case with poly-62

nomially bounded positive integer profits using a similar technique with [4, 8].63

Lemma 3. A ρ−approximation algorithm for the AOP in undirected graphs64

with polynomially bounded positive integer profits yields a (ρ + o(1))− ap-65

proximation algorithm for the AOP in undirected graphs.66

Proof. Given an AOP instance I = (G = (V,E), t, p, B), we construct an67

instance I ′ = (G′ = (V ′, E ′), t, p′, B) with polynomially bounded positive in-68

teger profits. First, we guess the edge of highest profit (pmax) in the optimal69

walk and remove all higher profit edges. Then, we set p′e = �n3pepmax

� + 1 for70

each edge e. A feasible walk W , consisting of the distinct edges e1, e2, . . . , ek71

4

has profit profit(W ) =k∑

j=1

pej in I and profit′(W ) =k∑

j=1

p′ej > n3profit(W)pmax

72

in I ′. Hence, OPT′ > n3

pmaxOPT, where OPT(OPT′) is the optimum in73

I(I ′). On the other hand, profit′(W ) =k∑

j=1

p′ej ≤k∑

j=1

(n3pejpmax

+ 1) =⇒74

n3

pmaxprofit(W ) ≥ profit′(W ) − m ≥ profit′(W ) − mOPT

pmax, so profit(W ) ≥75

pmax

n3 profit′(W ) − mn3OPT. Then, a ρ−approximation solution W of I ′ has76

profit′(W ) ≥ OPT′ρ

, so profit(W ) ≥ 1ρpmax

n3 OPT′− mn3OPT > (1

ρ− m

n3 )OPT.77

Theorem 4. A ρ−approximation algorithm for the UOP in undirected graphs78

yields a 3ρ−approximation algorithm for the AOP in undirected graphs with79

polynomially bounded positive integer profits.80

Proof. Given an AOP instance, each edge e such that the shortest path81

from s to l passing through e exceeds the time budget is removed from the82

graph. Then, we construct an UOP instance by splitting each edge {u, v}83

into puv+1 edges as follows: Each node of the AOP instance is a node of the84

UOP instance (basic node) and for each edge {u, v} of the AOP instance,85

the UOP instance includes the auxiliary nodes {u, v}1, {u, v}2, · · · , {u, v}puv86

and the edges {u, {u, v}1}, {{u, v}1, {u, v}2}, · · · , {{u, v}puv−1, {u, v}puv},87

{{u, v}puv , v}. The travel times of {u, {u, v}1} and {{u, v}puv , v} are set to88

t{u,v}2

, while the travel time of the remaining edges is zero. The time budget89

of the UOP instance is set equal to the time budget of the AOP instance.90

An AOP solution yields an equal length UOP solution of at least the same91

profit by replacing each edge {u, v} by the segment (u, {u, v}1, · · · , {u, v}puv , v).92

On the other hand, we will show that any UOP solution yields an AOP so-93

lution of at least a third of the former’s profit (pAOP ≥ pUOP

3). A sequence of94

nodes (u, {u, v}1, {u, v}2, · · · , {u, v}puv , v) is called an appropriate segment,95

5

i.e. a segment representing the traversal of the edge {u, v} in the AOP in-96

stance, while (u, {u, v}1, · · · , {u, v}i−1, {u, v}i, {u, v}i−1 · · · , {u, v}1, u) is called97

an inappropriate segment, i.e. a segment representing the partial traversal98

of the edge {u, v} of the AOP instance. For each inappropriate segment we99

consider that i = puv, otherwise we may extend it to the equal length and100

higher profit segment with i = puv.101

In an UOP solution, if pAS(pIS) is the profit gained by the appropriate102

(resp., inappropriate) segments, then pUOP = pAS+pIS. Note that pAS equals103

to the number of visited basic nodes plus the number of auxiliary nodes104

visited in the appropriate segments while pIS equals to the number of auxiliary105

nodes visited in the inappropriate segments, since any basic node visited in106

an inappropriate segment has already been counted in pAS. If all segments107

are appropriate, an AOP solution is obtained replacing the segments by their108

representing edges. This yields an AOP solution of pAOP > pUOP

2, since pUOP109

equals to pAOP plus the number of visited basic nodes, apart from s, l. Hence,110

pUOP is less than pAOP plus the number of traversed edges, i.e. pUOP < 2pAOP.111

If however inappropriate segments exist, let IS = {s1, s2, . . . , sk} be the112

set of inappropriate segments of the UOP solution. Let also p1, p2, . . . , pk be113

the profits collected by traversing them (k∑

i=1

pi = pIS), assuming w.l.o.g that114

p1 ≥ p2 ≥ · · · ≥ pk and t1, t2, . . . , tk be the travel times associated with them.115

If p1 ≥ pUOP

3, then an AOP solution of pAOP ≥ pUOP

3is obtained returning the116

shortest path from s to l passing through the edge represented by s1.117

Otherwise, we shall replace some of the inappropriate segments by their118

representing edge traversed in both directions, i.e. the inappropriate segment119

(u, {u, v}1, · · · , {u, v}puv−1, {u, v}puv , {u, v}puv−1, · · · , {u, v}1, u) shall be re-120

6

placed by the sequence of nodes (u, v, u) in the AOP instance. A subset121

RS of IS, with∑

sj∈RS

2tj ≤k∑

i=1

ti =∑

sj∈IStj will be called a replaceable subset,122

i.e. replacing the appropriate segments by their representing edges and the123

segments in the replaceable set by their representing edges traversed in both124

directions, we obtain a feasible AOP solution. RS is a maximal replaceable125

subset, if inserting a segment (sm /∈ RS) into the set would violate the time126

constraint, i.e.∑

sj∈RS

2tj+2tm >k∑

i=1

ti. Consider a maximal replaceable subset127

MRS of segments. We distinguish between the following cases: (i) pMRS ≥ pIS3

128

or pAS + pMRS ≥ 2pUOP

3, where pMRS is the total profit of segments in MRS.129

Then an AOP solution is obtained consisting of the edges represented by130

the appropriate segments (contributing at least pAS

2profit) and the sequences131

that replace the segments in MRS (contributing pMRS profit), with profit at132

least pAS+pIS3

= pUOP

3or greater than pAS+pMRS

2≥ pUOP

3, hence pAOP ≥ pUOP

3. (ii)133

The set of inappropriate segments MRSc = IS\MRS has pMRSc >pUOP

3. Note134

that MRSc contains at least two segments, otherwise p1 ≥ pMRSc > pUOP

3, a135

contradiction. Furthermore, pMRSc > 2pIS3, hence removing the lowest profit136

segment sm ∈ MRSc, MRSc\{sm} has profit pMRSc\{sm} ≥ pMRSc

2> pIS

3. Since137

∑

sj∈MRS

2tj + 2tm >k∑

i=1

ti then∑

sj∈MRSc\{sm}2tj <

k∑

i=1

ti, hence MRSc\{sm} is138

replaceable. Then, we apply the same technique with (i) using MRSc\{sm}139

instead of MRS, obtaining an AOP solution of pAOP > pUOP

3.140

Using Lemma 3, Theorem 4 and the (2 + ε)−approximation algorithm141

for the UOP by Chekuri et al. [4] we obtain a (6 + ε+ o(1))−approximation142

algorithm for the AOP in undirected graphs with execution time nO( 1ε2

).143

The unweighted AOP (UAOP) is the special case of the AOP with edges144

7

of unit profit. Similarly to Theorem 4, a ρ−approximation algorithm for145

the UOP in undirected graphs yields a 2ρ−approximation algorithm for the146

UAOP in undirected graphs. If the optimal UAOP solution contains less than147

ρ+2 nodes, it is found by exhaustive search since ρ is a constant. Otherwise,148

OPTUOP ≥ OPTUAOP + ρ. Then, an UOP solution of pUOP ≥ OPTUOP

ρyields149

an UAOP solution of pUAOP ≥ pUOP−12

≥ OPTUAOP

2ρby replacing its appropriate150

and its half shortest inappropriate segments. Using Chekuri et al.’s algorithm151

[4] we obtain a (4 + ε)−approximation algorithm.152

The Mixed Orienteering Problem (MOP) [5, 11] extends both the OP153

and the AOP, assigning profit to both nodes and arcs. The MOP is reduced154

to the AOP in a similar way with the reduction from the OP to the AOP,155

retaining though the arcs’ profit. As a result, any approximation algorithm156

for the AOP yields an approximation algorithm for the MOP.157

Acknowledgment. We thank the anonymous reviewers for their construc-158

tive comments and for suggesting the inapproximability result for the AOP.159

This work has been supported by the EU FP7/2007-2013 (DG CONNECT.H5-160

Smart Cities and Sustainability), under Grant Agreement no. 288094 (project161

eCOMPASS) and the EU FP7 Collaborative Project MOVESMART (Grant162

Agreement No 609026).163

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