Approximation algorithms for time-dependent orienteering

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  • Information Processing Letters 83 (2002) 5762

    Approximation algorithms for time-dependent orienteeringFedor V. Fomin a,1, Andrzej Lingas b,

    a Graduiertenkolleg des PaSCo, Heinz Nixdorf Institut and University of Paderborn, Frstenallee 11, D-33102, Paderborn, Germanyb Department of Computer Science, Lund University, Box 118, S-221 00, Lund, Sweden

    Received 1 December 2000; received in revised form 28 October 2001Communicated by S. Albers

    Abstract

    The time-dependent orienteering problem is dual to the time-dependent traveling salesman problem. It consists of visiting amaximum number of sites within a given deadline. The traveling time between two sites is in general dependent on the startingtime.

    For any > 0, we provide a (2 + )-approximation algorithm for the time-dependent orienteering problem which runs inpolynomial time if the ratio between the maximum and minimum traveling time between any two sites is constant. No priorupper approximation bounds were known for this time-dependent problem. 2001 Elsevier Science B.V. All rights reserved.

    Keywords: Time-depending orienteering; Traveling salesman; Approximation algorithms; Network design

    1. Introduction

    In the orienteering problem (see, e.g., [1,9]) atraveler wishes to visit a maximum number of sites(nodes) subject to given restrictions on the length ofthe tour. This problem can be regarded as the prob-lem of traveling salesperson with restricted amount ofresources (time, gasoline, etc.) wishing to maximizethe number of visited sites. For this reason, the ori-enteering problem has been also called the general-ized traveling salesperson problem or even as the

    * Corresponding author.E-mail addresses: fomin@upb.de (F.V. Fomin),

    Andrzej.Lingas@cs.lth.se (A. Lingas).1 Part of this work was done while the author was at the Centro

    de Modelamiento Matemtico, Universidad de Chile and UMR2071-CNRS, Chile. Supported by FONDAP. Also supported by theIST Programme of the EU under contract number IST-1999-14186(ALCOM-FT).

    bank robber problem [3,9]. Even if the ratio betweenthe maximum and minimum amount of resources re-quired for traveling between two sites is constantlybounded, the orienteering problem is MAX-SNP-hardsimply because the correspondingly restricted trav-eling salesman problem is MAX-SNP-hard [5,14](cf. [4,8,13]).

    In this paper we consider a generalization of theorienteering problem which we term time-dependentorienteering (TDO, for short). In our generalization,the cost of traveling (time cost in our terminology)from any site to any other site in general depends onthe start moment.

    The orienteering problems considered in [1] areclassified as the path-orienteering, cycle-orienteering,and even tree-orienteering problems depending onwhether or not the network to be induced by theset of pairs of consecutive sites visited is supposedto have a form of a path, a cycle, or even a tree,

    0020-0190/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0020-0190(01) 00 31 3- 1

  • 58 F.V. Fomin, A. Lingas / Information Processing Letters 83 (2002) 5762

    respectively. Additionally, one can wish to have oneor more roots, which are essential sites requiredto be visited. Following this classification we willrefer to the cases of TDO problems without roots aspath (cycle, tree)-TDO and to the cases with roots asrooted-path (cycle, tree)-TDO.

    For illustration, consider the two following exam-ples of possible applications of TDO.

    Kinetic TSP [5,10]. There is given a set of targetsand one robot (intercepter) with restricted amount ofresources (e.g., fuel). The dynamic of targets is known,i.e., for each target one can specify its location atany discrete time moment. The problem is to find aprogram for the robot which allows it to hit as manytargets as possible within a given time. This problem isan example of path-TDO. If there is a target specifiedto be intercepted then we have an example of rooted-path-TDO.

    Time-Dependent Maximum Scheduling Problem(TDMS) [15]. There is given a set of tasks for a sin-gle machine. The execution of any task can start at anydiscrete moment and its execution time depends onthe starting moment. The problem is to find a sched-ule for the machine maximizing the number of taskscompleted within a given time period. This problemis equivalent to Job Interval Selection Problem stud-ied by Spieksma [15]. It can be interpreted as a spe-cial case of TDO where the time of traveling from asite a to a site b does not depend on the site b andis interpreted as the execution time of the task cor-responding to a. The Web Searching Problem stud-ied by Czumaj et al. [7] yields the following motiva-tion for TDMS. Assume that there is a central com-puter which is being used to collect all the informa-tion stored in a number of web documents, located invarious sites. The information is gathered by schedul-ing a number of consecutive client/server connectionswith the required web sites, to collect the informa-tion page by page. The loading time of any partic-ular page from any site can vary at different times,e.g. the access to the page is much slower in peakhours than in off-peak hours. We wish to downloadthe maximum number of pages within a given periodof time.

    1.1. Main results

    An algorithm is said to be a c-approximation al-gorithm for a maximization problem P if for any in-stance of P it yields a solution whose value is at least1/c times the optimum.

    Let n be the number of input sites and let k bethe ratio between the maximum and minimum timerequired for traveling between two sites.

    We present (2 + )-approximation algorithms forpath-TDOs and cycle-TDOs running in time

    O((

    2k2

    2+

    )!kn2k2( 2+ )+1

    ).

    In the corresponding rooted cases the time complex-ity increases by the multiplicative factor O(kn/).These bounds immediately carry over to the corre-sponding time-independent special cases, i.e., the un-rooted and rooted, path-orienteering problems andcycle-orienteering problems. Our algorithm is thefirst constant-factor approximation algorithm for TDOwith k = O(1) running in polynomial time. Althoughfor large k, our algorithm can be hardly claimed to bepractical because of its fairly high running time, it sug-gests that practical and efficient algorithms might bepossible.

    1.2. Related results

    The authors are not familiar with any explicitprior approximation algorithms for time-dependentorienteering (TDO). Of course, if the ratio betweenthe maximum and minimum distance is k then anyapproximation algorithm is a k-approximation one.

    For the Time-Dependent Maximum SchedulingProblem (TDMS) which can be interpreted as a specialcase of TDO, a simple greedy 2-approximation algo-rithm running in time O(mt), where m is the numberof available tasks and t is the deadline, follows fromSpieksmas algorithm [15] for Job Interval SelectionProblem. Also, it follows from the same work [15] thatTDMS is MAX-SNP-hard.

    As for the classical, i.e., time-independent, ori-enteering problem, Awerbuch et al. proved that a c-approximation algorithm to the so-called k-travelingsalesperson problem, asking for a shortest cycle vis-iting k sites (k-TSP), yields a 2c-approximation al-gorithm for the orienteering problem [2]. This result

  • F.V. Fomin, A. Lingas / Information Processing Letters 83 (2002) 5762 59

    combined with known approximation results for k-TSP yields a 6-approximation algorithm for the ori-enteering problem in metric spaces and a (2 + )-approximation algorithm in the Euclidean plane. Thelatter result has been subsumed by Arkin et al. whopresented 2-approximation algorithms for several vari-ants of the orienteering problem in the plane [1]. Morerecently, Broden has designed a 4/3-approximation al-gorithm for the very special case of the orienteeringproblem where the pairwise distances are constrainedto {1,2} [5]. Note here that the recent lower boundson the constant approximation factor for the anal-ogously restricted traveling salesperson problem [4]easily carry over to the aforementioned special caseof the orienteering problem.

    For an interesting review of results related to theorienteering problem, including several variants of thetraveling salesperson problem, the reader is referredto [1].

    The recent works by Hammar and Nilsson [10]and Broden [5] contain a number of inapproximabilityand approximability results on various restrictions ofthe problem dual to TDO, i.e., the time-dependenttraveling salesperson problem.

    2. Formal definition of TDO

    For a given set S of n sites, a time-travel functionl :S S N {0} R+ and a deadline t , thesalespersons tour visiting a subset T of m sites is asequence of triples

    (s1, t+1 , t

    1 ), (s2, t

    +2 , t

    2 ), . . . , (sm, t

    +m , t

    m)

    such that(1) for i {1,2, . . . ,m}, t+i , ti N {0};(2) T = {s1, s2, . . . , sm};(3) 0 = t+1 t1 t+2 t+m tm = t ;(4) for each i {1,2, . . . ,m 1}, t+i+1 ti = l(si ,

    si+1, ti ).It is useful to interpret the moment ti as the momentwhen salesperson leaves the site si and t+i as themoment when salesperson enters si . So t+i+1 ti isthe time spent in travel from si to si+1 and ti t+iis the time the salesperson stays in si (importantly, thetraveler is allowed to stay at any site any time). Thepath (or cycle) time-dependent orienteering problem

    is to find an open (closed, respectively) tour visitingmaximum number of sites within the time t .

    Note that the classical orienteering problem [1] isa special case of TDO where for any sites a, b,the travel time from a to b is time-independent, i.e.,l(sa, sb, t

    )= l(sa, sb, t ) for any t , t [0, t].

    3. Main procedure and algorithms

    We may assume without loss of generality thatthe minimum travel time between two sites,mins,s S,t [0,t ] l(s, s, t ), is 1.

    For a nonnegative integer q and a positive integeri t/q, we shall denote by Ii(q) the subinterval[q(i 1),min{qi 1, t}] of [0, t]. For a given set S ofsites, a q-partial salespersons tour is a sequence Qof triples (sl, t+l , t

    l ), sl S, t+l , tl [0, t], such that

    for every time interval Ii(q), 1 i t/q, the subse-quence (sp, t+p , tp )t+p ,tp Ii (q) of Q is a (salespersons)path-tour, i.e., an open tour visiting all sites in the in-terval. In other words, q-partial tour induces a path-tour for each time interval Ii(q), i {1,2, . . . , t/q},but in general it is not a tour.

    The following simple procedure is the heart of ouralgorithms for TDO.

    Procedure Greedy(S, q, t)INPUT: Set S of n sites, integer q , deadline t ;OUTPUT: q-partial tour.(1) T S;(2) for i = 1,2, . . . , t/q do

    let Ti be a maximum cardinality subset of T thatcan be visited in the time interval Ii(q) by apath-tour;

    compute a path-tour visiting Ti in the timeinterval Ii(q);T T \ Ti

    Obviously, the set Ti can be found in O(q!nqq)time by considering all possible choices of a subsetof T containing at most q elements and then applyingthe straightforward O(q!q)-time brute force methodfor the time-dependent traveling salesperson problemon the subset. Hence the overall time complexity ofGreedy(S, q, t) is O(q!tnq).

    Let k be the maximum time needed to travelbetween two sites, i.e., k =maxs,s S,t [0,t ] l(s, s, t ).

  • 60 F.V. Fomin, A. Lingas / Information Processing Letters 83 (2002) 5762

    Note that by our initial assumption k is also the ratiobetween the maximum and minimum time required fortraveling between two sites.

    Algorithm GreedyPath(S, q, t)INPUT: Set S of n sites, integer q , deadline t ;OUTPUT: path-tour.(1) Run procedure Greedy(S, q, t);(2) for i = 1, . . . , t/q 1 do

    Remove from the set of visited sites all sitesvisited in the time interval [qik/2, qi+k/2];

    Glue the obtained subtours by forcing the sales-person to go from the last visited site in thetime interval [q(i 1)+ k/2, qi k/2] to thefirst visited site in the time interval [qi + k/2,q(i + 1) k/2]

    Algorithm GreedyCycle(S, q, t) is obtained fromGreedyPath(S, q, t) by closing the path-route near itsendpoints, i.e., by forcing the salesperson to go fromthe last visited site in the time interval st/q1 to thefirst visited site from the time interval [k, q k/2].

    4. Approximation analysis

    Lemma 1. Let p be the maximum number of differentsites that can be visited by a q-partial tour. Thenthe number of sites participating in the q-partial tourproduced by Greedy(S, q, t) is at least p/2.

    Proof. Let W S be a set of p sites that can bevisited by a q-partial tour Qopt and let V be the set ofsites returned by Greedy(S, q, t). Denote by Vi the setof sites visited by Greedy(S, q, t) in the time interval[0, iq) and by Wi the subset of W visited by Qoptin the time interval [0, iq). For 1 i t/q, letXi :=W \ (Vi Wi), i.e., Xi consists of the sites in Wthat have been visited by neither Greedy(S, q, t) norQopt in the time interval [0, qi).

    We claim that for each 1 i t/q,|Xi | p 2|Vi|. (1)Since Xt/q = , this claim implies the lemma. Weprove it by induction on i .

    The procedure Greedy(S, q, t) finds the maximumnumber of sites than can be visited within time intervalI1(q). Hence |V1| |W1| and |V1 W1| 2|V1| hold.

    By definition, |X1| = p|V1W1| p2|V1| holds.Thus, for i = 1 the inequality (1) is true. Suppose that(1) is true for all r {1, . . . , i 1}.

    We claim that

    Xi1 \Xi = (Vi Wi) \ (Vi1 Wi1). (2)In fact, for every u W , we haveu Xi1 \Xi u Xi1 u /Xi

    u W \ (Vi1 Wi1)u /W \ (Vi Wi)

    u (Vi Wi) \ (Vi1 Wi1).By (2),

    (Xi1 \Xi) Vi Vi \ Vi1. (3)The set of sites Vi \Vi1 is chosen by Greedy(S, q, t)

    at the ith step. Therefore, |Vi \ Vi1| is the maximumnumber of sites from S\Vi1 that can be visited withinthe time interval Ii(q). Eq. (2) implies((Xi1 \Xi)Wi

    ) Vi1 = and we conclude that(Xi1 \Xi)Wi |Vi \ Vi1|. (4)

    It follows from (2), (3) and (4) that|Xi1 \Xi |(Xi1 \Xi) Vi+ (Xi1 \Xi)Wi

    2|Vi \ Vi1|. (5)Since Xi Xi1 and Vi Vi1, we have that

    |Xi | = |Xi1| |Xi1 \Xi | (6)and

    |Vi| = |Vi1| + |Vi \ Vi1|. (7)Combining (5), (6), (7) with the induction assump-

    tion |Xi1| p 2|Vi1|, we obtain|Xi | = |Xi1| |Xi1 \Xi |

    p 2|Vi1| 2|Vi \ Vi1|= p 2|Vi |,

    and (1) follows. Theorem 2. For any > 0, if the path- and cycle-TDO for n sites admit (2 + )-approximation algo-rithms running in time O((2k2 2+

    )!kn2k2( 2+ )+1).

  • F.V. Fomin, A. Lingas / Information Processing Letters 83 (2002) 5762 61

    Proof. Let S be a set of n sites. Suppose that anoptimal solution to the path- (cycle-) TDO visits msites. We may assume without loss of generality. thatm t/k (otherwise there is a trivial algorithm findingthe exact solution).

    Take q = 2k2((2 + )/). By Lemma 1, the pro-cedure Greedy(S, q, t) outputs a q-partial tour visit-ing at least m/2 sites. Consequently, each of the algo-rithms GreedyPath(S, q, t), GreedyCycle(S, q, t) out-puts a tour visiting at least

    m

    2 kt

    q= 1

    2

    (m 2kt

    q

    ) m

    2

    (1 2k

    2

    q

    )= m

    2+ sites. Hence, GreedyPath(S, q, t) and GreedyCycle(S,q, t) are (2+ )-approximation algorithms.

    By the remark on the time complexity of pro-cedure Greedy(S, q, t), both algorithms run in timeO((2k2 2+

    )!tn2k2( 2+ )) and the assumption n

    m t/k implies the complexity bound in the theoremthesis.

    We can trivially model the time-independent path-and cyc...

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