Dial a Ride from k-forest

Dial a Ride from k -forest Anupam Gupta ∗ MohammadT aghi Hajiaghayi † V iswanath Nagarajan ‡ R. Ra vi ‡ October 27, 2018 Abstract The k -for est pr oblem is a common generalizatio n of b oth the k -MST and the d ense- k -su bgraph problems. Formally , giv en a metric space on n vertices V , with m demand p airs ⊆ V × V and a “target” k ≤ m , the goal is to find a min imum cost su bgraph tha t con nects at least k dem and pairs. In this paper, we giv e an O (min { √ n, √ k } ) -app roximatio n alg orithm for k -fo rest, improving on the previous be st ratio of O ( n 2 / 3 log n ) by Se gev & Se gev [ SS06]. W e then ap ply our a lgorithm for k -forest to ob tain appro ximation alg orithms for several Dial- a-Ride problem s. The basic Dial-a-Ride p roblem is the following: giv en an n point metric spac e with m objects ea ch with its own source and d estination, and a vehicle capa ble of carrying at most k ob - jects at any time , find the minimu m length to ur that uses this vehicle to move each o bject f rom its source to d estination. W e prove that an α -app roximatio n algorithm fo r the k -forest pro blem im plies an O ( α · log 2 n ) -appro ximation a lgorithm for Dial- a-Ride. Using our results f or k -fo rest, we get an O (min { √ n, √ k } · log 2 n ) -appro ximation alg orithm for Dial-a-Ride . Th e only previous result known for Dial-a-Ride was an O ( √ k log n ) -appr oximation by Charik ar & Rag hav achari [CR98 ]; ou r r esults giv e a different p roof of a similar appro ximation g uarantee—in fact, when the vehicle capacity k is large, we gi ve a slight improvement on their results. The reductio n fro m Dial-a- Ride to th e k -fo rest pr oblem is fairly ro bust, and allows us to ob tain ap- proxim ation algo rithms (with the same g uarantee ) for the following gen eralizations: (i) N on-un iform Dial-a-Ride, where the cost of trav ersing ea ch edge is an arbitrary no n-decre asing function of the num - ber of objects in the vehicle; and (ii) W eighted Dial-a-Ride, where de mands are allowed to have d if- ferent weights. The reduction is e ssential, as it is u nclear how to extend th e tec hniques of Char ikar & Raghav achari to these Dial-a-Ride generalizations. 1 Introd uction In the S teiner forest problem, we are gi ven a set of verte x-pairs, and the goal is to find a forest such that each vertex pair lies in the same tree in the forest. This is a generaliz ation of the S teiner tree problem, where all the pairs contain a common vertex called the root; both the tree and forest vers ions are well- unders tood fundamen tal problems in netwo rk desig n [AKR91, GW92]. An important extensi on of the Steiner tree problem studied in the late 1990s was the k -MST problem, where one sought the least-c ost tree that connecte d any k of the terminals: sev eral approximatio ns algorith ms were giv en for the problem, ∗ Computer Science Department, Carneg ie Mellon University . Supported in part by an NSF C AREER award CCF-04480 95, and by an Alfred P . Sloan Fello wship. † Computer S cience Department, Carnegie Mellon University . Supported in part by N SF ITR grant CCR-0122581 (T he AL- ADDIN project). ‡ T epper S chool of Business, Carnegie Mellon University . S upported in part by NS F grants CCF-0430751 and IT R grant CCR- 012258 1 (T he ALADDIN project). 1 culminat ing in the 2 -approximati on of Garg [Gar05]; the k -MST problem pro ved crucial in many subseque nt de velo pments in netwo rk design and veh icle routing [CG R T 03, FHR03, BC K + 03, B BCM04]. One can analog ously define the k -forest problem where one needs to connect only k of the pairs in some Steiner forest instance: surp risingly , very little is kno wn about this problem, which was first studied formally as recent ly as last year [HJ06, S S06]. In this paper , we giv e a simpler and improve d approx imation algorithm for the k -forest problem. Moreo ver , just like the k -MST v ariant, the k -forest problem seems to be useful in applicatio ns to network design and ve hicle routing. In the secon d half of the paper , w e show a (some what surpris ing) reduction of a well-studie d vehic le routing probl em called the Dial-a-Ride problem to the k -fores t problem. In the D ial- a-Ride problem, w e are giv en a metric space with people havi ng sources and destinatio ns, and a b us of some capacity k ; the goal is to find a route for this bus so that each person can be taken from her sourc e to destinatio n without exc eeding the capacity of the b us at any point, such that the length of the bus route is minimized . W e sho w how the results for the k -fores t problem slightly improv e upon existin g results for the D ial-a-Ride problem; in fact, they gi ve the first approximatio n algorithms for some generalizati ons of Dial-a-Ride which do not seem amenable to pre viou s techniq ues. 1.1 The k -For est Pr oblem Our starting point is the k -fore st probl em, which general izes both the k -MST and the dense- k -subgra ph proble ms. Definition 1 (The k -Fo res t Pr oblem) Given an n -verte x metric space ( V , d ) , and demands { s i , t i } m i =1 ⊆ V × V , find the least- cost subgraph that connects at least k demand-pa irs. Note th at the k -forest prob lem is a general ization of the (minimiza tion versi on of th e) well-studi ed dense- k - subgra ph problem, for which nothing bette r than an O ( n 1 / 3 − δ ) approximati on is kn own. T he k -forest prob- lem was first defined in [HJ06 ], and the first non-tri vial approx imation was gi ven by Sege v and S ege v [SS06], who ga ve an algorit hm wit h an approximatio n guara ntee of O ( n 2 / 3 log n ) for the case when k = O ( poly ( n )) . W e impr ov e the approximat ion guarante e of the k -forest problem to O (min { √ n, √ k } ) ; formally , we prov e the follo wing theorem in Section 2. Theor em 2 (Appro ximating k -for est) Ther e is an O (min { √ n · log k log n , √ k } ) -appr oximation algorithm for the k -for est pr oblem. F or the case when k is less than a polynomial in n , the appr oximation guarante e impr oves to O (min { √ n, √ k } ) . Apart from gi ving an improv ed approximati on guarantee, our algorithm for the k -forest problem is ar guably simpler and more direct than that of [SS06] (w hich is based on Lagrangian relaxations for the proble m, and combining solut ions to this relaxati on). Indeed, we gi ve two algo rithms, both reducin g the k - forest p roblem to the k -MST pro blem in dif ferent wa ys and achie vin g diffe rent approxi mation guarant ees— we then return the better of the two answers. The first algorithm (gi ving an approx imation of O ( √ k ) ) uses the k -MST algorit hm to fi nd good solutions on the sources and the sinks indepen dently , and then uses the Erd ˝ os-Szek eres theorem on monoton e subsequence s to find a “good” subset of these source s and sinks to conne ct cheaply; details are giv en in Section 2.1. The secon d algorithm starts off with a single verte x as the initial solution, and uses the k -MST algorithm to repeated ly find a lo w-cost tree that satisfies a lar ge number of d emands which ha ve one en dpoint in the current so lution and the o ther en dpoint outsi de; this tree is th en used to greedily aug ment the curre nt solution and pr oceed. Choosing the parameters (as de scribed in Section 2.2) gi ves us an O ( √ n ) approx imation. 2 1.2 The Dial-a-Ride Problem In this paper , we use t he k -forest p roblem to gi ve approximation alg orithms f or the follo wing vehicl e ro uting proble m. Definition 3 (The Dial-a-Ride Proble m) Given an n -verte x metric space ( V , d ) , a starting vertex (or root ) r , a set o f m demands { ( s i , t i ) } m i =1 , and a ve hicle of capacity k , fi nd a m inimum length tour of the vehicl e sta rt- ing (and ending ) at r that moves each object i fr om its sour ce s i to its destina tion t i suc h that the vehicle carrie s at most k objec ts at any point on the tour . W e say that an object is pre empted if, after being picked up from its source, it can be left at some interme diate ver tices before being deliv ered to its destina tion. In this paper , we will not allo w this, and will m ainly be concer ned with the non- pr eemptive D ial-a-Ride proble m. 1 The approximabi lity of the Dial-a-Ride problem is not very well understoo d: the previo us best upper bound is an O ( √ k log n ) -approximat ion algorithm due to Charikar and Raghav ach ari [CR98], w hereas the best lo wer bound that we are aware of is APX-hardness (from TSP , say). W e establish the follo wing (some what surpris ing) connect ion between the Dial-a-Ride and k -forest problems in Section 3. Theor em 4 (Redu cing Dial- a-Ride to k -for est) Given a n α -appr oximatio n algorit hm for k -for est, the r e is an O ( α · log 2 n ) -appr oximation algorithm for the Dial-a-Ride pr oble m. In parti cular , combining Theorems 2 and 4 gi v es us an O (min { √ k , √ n } · log 2 n ) -appro ximation guara ntee for Dial-a-Ride . Of course, improving the approximatio n guaran tee for k -forest would improve the result for Dial-a-Ride as well. Note that our results match the results of [CR98] up to a logarit hmic term, and e ven giv e a slight im- pro vement when the vehicl e capac ity k ≫ n , the number of nodes. Much m ore interes tingly , our algorith m for Dial-a-Ride easily extend s to genera lization s of the Dial-a-Ride problem. In particular , we consider a substa ntially more general veh icle routing problem where the vehicle has no a priori capacity , and instead the cost of tra vers ing each edge e is an arbitrary non-decr easing function c e ( l ) of the number of objects l in the vehicle; setting c e ( l ) to the edge-length d e when l ≤ k , and c e ( l ) = ∞ for l > k gi ves us back the classic al Dial-a-Ride setting. In S ection 3.2, we show that this general non-uni form Dial-a-Ride problem admits an approx imation guarantee that m atches the best kno wn for the classical Dial-a-Ride probl em. An- other exten sion w e consid er is the weighted Dial-a-Ride problem. In this, each object m ay ha ve a dif ferent size, and total size of the items in th e vehicle must be bound ed by the vehicl e capacity; this has been earlier studie d as the pic kup and delive ry proble m [SS95]. W e sho w in Section 3.3 that this problem ca n be reduced to the (unweighted) Dial-a-Ride proble m at the loss of only a constant factor in th e approximat ion guaran tee. As a n aside , we co nsider the e ffe ct of preemptions in the Dial-a -Ride proble m (Section 4). It wa s sh own in Charika r & Raghav achari [CR98] that the ga p between the optimal preemp tiv e and non-pree mpti ve tours could be as larg e as Ω ( n 1 / 3 ) . W e sho w that the real dif ference arises between zer o and one preemptio ns: allo wing multiple preempti ons does not giv e us much added po wer . In particu lar , we show in Section 16 that for any instance of the Dial-a-Rid e problem, there is a tour that preempts each object at m ost once and has l ength at most O (log 2 n ) times an op timal pre empti ve tou r (which may preemp t each ob ject an a rbitrary number of times). Moti v ated by obtainin g a better guarantee for Dial-a-Rid e on the E uclidea n plane, we 1 A note on the parameters: a feasible non-preemptiv e t our can be short-cut over vertices that do not partici pate in any demand, and we can assume that e very vertex is an end point of some demand, and n ≤ 2 m . W e may also assume, by preprocessing some demands, that m ≤ n 2 · k . Howe ver in general, the number of dem ands m and the vehicle capac ity k may be much larger than the number of vertices n . 3 also study the preemption gap in such instances. W e show that eve n in this case, there are insta nces ha ving a gap of ˜ Ω( n 1 / 8 ) between opt imal preempti ve and non-preemp tiv e tours. 1.3 Related W ork The k -for est pr oblem: T he k -forest prob lem is re lati vely ne w: it was defined by Haj iaghayi & Jain [HJ06]. An ˜ O ( k 2 / 3 ) -appr oximation algorith m fo r ev en the directed k -fores t proble m ca n be inferred from [CCyC + 98]. Recently , Sege v & Sege v [SS06] ga ve an O ( n 2 / 3 log n ) app roximation algorithm for k -forest. Dense k -subgraph: The k -for est probl em is a g eneraliza tion of the dense - k -subgrap h problem [FPK01], as sho wn in [HJ06]. The best kno wn appr oximation guar antee for th e dens e- k -subgrap h problem is O ( n 1 / 3 − δ ) where δ > 0 is some con stant, due to Feige et al. [FPK01], a nd obtain ing an i mprov ed guarante e has been a long stand ing open problem. Strictly speaki ng, F eige et al . [FPK01] study a po tentially harder proble m: the maximizatio n ver sion of dense - k -subgrap h, w here one wants to pick k vertic es to maximize the number of edges in the induc ed graph. Howe ver , nothing better is kno wn e ve n fo r the minimization v ersion o f de nse- k - subgra ph (where one wants to pick the minimum numb er of v ertices th at induce k edges ), which is a specia l case of k -forest. The k -fore st problem is also a generaliza tion of k -MST , for w hich a 2-app roximation is kno wn (Garg [Gar05]). Dial-a-Ride: While the D ial-a-Rid e problem has been studi ed exte nsi vely in the opera tions researc h litera- ture, relati vely little is kno wn about its approximab ility . The curren tly best kno wn approximat ion ratio for Dial-a-Ride is O ( √ k log n ) due to Charikar & Ragha v achari [CR98]. W e note that their algorith m assumes instan ces with unweighted demands. Krumke et al. [KR W00] gi ve a 3-approximati on algo rithm for the Dial-a-Ride problem on a line m etric ; in fact , their algorithm fi nds a non-preempti v e tour that has length at most 3 times th e preempti ve lower b ound. (Clearly , the cost of an optimal preemp tiv e tour is at most that of an optimal non-pre empti ve tour .) A 2 . 5 -ap proximati on algorit hm for single sour ce vers ion of Dial-a-Ride (also called the “capaci tated vehic le routi ng” proble m) was gi ven by Haimovich & Kan [HK85]; again, their algorith m output a non-pre empti ve tour with length at most 2.5 times the preempti ve lower bound . For the pr eemptive Dial-a-Ride problem, Charikar & Ragha v achari [CR98] ga ve the current-b est O (log n ) approx imation algorithm, and Gørtz [r tz06] sh owed that i t is hard to approximate this problem to bette r than Ω(log 1 / 4 − ǫ n ) . Recall th at no super -cons tant hardness res ults are kno wn fo r the non-pree mpti ve Dial-a-Ride proble m. 2 The k -f orest problem In this section, we study the k -fores t proble m, and giv e an appro ximation guarantee of O (min { √ n, √ k } ) . This result improve s upon the pre vious best O ( n 2 / 3 log n ) -approximatio n guar antee [SS06] for this problem. The algorithm in S ege v & S ege v [SS06] is based on a L agrang ian relaxation for this problem, and suitably combinin g soluti ons to this relaxation. In contrast, our algorith m uses a more direct approach and is much simpler in descrip tion. Our approac h is based on approximat ing the follo wing “density” varia nt of k -fores t. Definition 5 (Minimum-ratio k -fo res t) G iven an n -verte x m etric space ( V , d ) , m pairs of vertices { s i , t i } m i =1 , and a tar get k , find a tr ee T that conne cts at m ost k pairs, and minimizes the ratio of the length of T to the number of pair s connected in T . 2 W e present two differe nt algo rithms for minimum-ratio k -for est , obta ining approximatio n guarantees of O ( √ k ) (Section 2.1) and O ( √ n ) (Section 2.2); these are then combined to gi ve the claimed result for the 2 Even if we relax the solution to be an y forest, we may assume (by averag ing) that t he optimal ratio solution is a tree. 4 k -forest problem. Both our algorithms are based on subtle reduc tions to the k -MST problem, albeit in very dif ferent ways. As is usual, when we say that our algorithm guesses a parameter in the follo wing discussio n, it means that the algorithm is run for each possible v alue of that parameter , and the best solution found over all the runs is returned. As long as only a constant number of parameters are being guesse d and the number of possib ilities for e ach of these p arameters is polynomial, the algorith m is rep eated only a pol ynomial numbe r of times. 2.1 An O ( √ k ) appr oximation alg orithm In th is secti on, we gi ve an O ( √ k ) approximation algori thm for minimum ra tio k -forest, which is based on a simple reduction to the k -MST pro blem. The ba sic intu ition is to look at t he solu tion S to minimum-ratio k - forest a nd cons ider an Euler to ur of this tree S —a th eorem of Erd ˝ os & Szeke res on increasi ng subseque nces implies that there must be at least p | S | source s w hich are visited in the same order as the correspond ing sinks. W e u se t his existen ce result to combine the s ource-si nk pairs to create an instance of p | S | -MST from which we can obtain a good-ra tio tree; the detail s follow . Let S denote an optima l ratio tree, that cov ers q demands & has length B , and let D denote the largest distan ce between any demand pair that is cove red in S (note D ≤ B ). W e define a new metric l on the set { 1 , · · · , m } of demands as follo ws. The distance b etween demand s i and j , l i,j = d ( s i , s j ) + d ( t i , t j ) , where ( V , d ) is the original m etric. The O ( √ k ) approximati on algorit hm first guesse s the number of demands q & the lar gest demand-pa ir distance D in the optimal tree S (there are at most m choices for each of q & D ). The algorith m discar ds all demand pairs ( s i , t i ) such that d ( s i , t i ) > D (all the pairs cov ered in the optimal solution S still remain). T hen the algorithm ru ns the unroo ted k -MST a lgorithm [Gar05] with tar get ⌊ √ q ⌋ , in the metric l , to obtain a tree T on the demand pairs P . From T , we easily obtain trees T 1 (on all source s in P ) and T 2 (on all sinks in P ) in metric d such that d ( T 1 ) + d ( T 2 ) = l ( T ) . Finally the algorithm outpu ts the tree T ′ = T 1 ∪ T 2 ∪ { e } , where e is any edge joining a source in T 1 to its corres ponding sink in T 2 . Due to the pruning on demand pairs that ha ve lar ge distance, d ( e ) ≤ D and the length of T ′ , d ( T ′ ) ≤ l ( T ) + D ≤ l ( T ) + B . W e now argu e that the cost of the solution T foun d by the k -MST algorithm l ( T ) ≤ 8 B . C onside r the optimal ratio tree S (in m etric d ) that has q demands { ( s 1 , t 1 ) , · · · , ( s q , t q ) } , and let τ deno te an Euler tour of S . Suppose that in a tra versal of τ , the sour ces of deman ds in S are seen in the ord er s 1 , · · · , s q . Then in the same tra v ersal, the sinks o f de mands in S will be se en in the order t π (1) , · · · , t π ( q ) , fo r some permutati on π . The follo wing fact is well kno wn (see, e.g., [Ste95]). Theor em 6 (Erd ˝ os & S zeker es) Every per mutation on { 1 , · · · , q } has eit her an incr easing subs equence of length ⌊ √ q ⌋ or a decr easin g subseq uence of length ⌊ √ q ⌋ . Using T heorem 6, we obtain a set M of p = ⌊ √ q ⌋ demands such that (1) the sou rces in M appear in increas- ing o rder in a tra vers al of the Eul er tour τ , and (2 ) the sin ks in M appear in inc reasing order in a tr av ersal of either τ or τ R (the re v erse tra ve rsal of τ ). L et j 0 < j 1 < · · · < j p − 1 denote the demands in M in incr easing order . From statement (1) above , P p − 1 i =0 d ( s ( j i ) , s ( j i +1 )) ≤ d ( τ ) , where the indices in the summation are modulo p . S imilarly , statement (2) implies that P p − 1 i =0 d ( t ( j i ) , t ( j i +1 )) ≤ max { d ( τ ) , d ( τ R ) } = d ( τ ) . Thus we obtain: p − 1 X i =0 [ d ( s ( j i ) , s ( j i +1 )) + d ( t ( j i ) , t ( j i +1 ))] ≤ 2 d ( τ ) ≤ 4 B But this sum is p recisely the leng th of the tour j 0 , j 1 , · · · , j p − 1 , j 0 in metric l . In other word s, there is a tree 5 of length 4 B in metric l , that contains ⌊ √ q ⌋ vertic es. So, the cost of the solution T found by the k -MST approx imation algori thm is at most 8 B . No w the final solution T ′ has length at m ost l ( T ) + B ≤ 9 B , and ratio that at most 9 √ q B q ≤ 9 √ k B q . Thus we ha ve an O ( √ k ) approx imation algori thm for minimum ratio k -forest. 2.2 An O ( √ n ) approximation algorithm In this section, we show an O ( √ n ) approximation algorithm for the minimum ratio k -fores t problem. The approa ch is again to reduce to the k -MST proble m; the intuition is rather diffe rent: either we find a verte x v such that a large number of demand-p airs of the form ( v , ∗ ) can be satisfied using a small tree (the “high- deg ree” case); if no such v ertex exists, we sho w that a re peated greed y procedure would c ov er most v ertices without paying too much (and since we are in the “lo w-deg ree” case, cov ering most ve rtices implies cov ering most demands too). The details follo w . Let S denote an optimal soluti on to minimum ratio k -fores t, and q ≤ k the number of demand pairs cov ered in S . W e define th e de gr ee ∆ of S to be the maximum nu mber of demands ( among th ose cov ered in S ) that are incide nt at any verte x in S . The algor ithm first guesse s the follo wing parameters of the optimal soluti on S : its length B (within a facto r 2), the number of pairs cove red q , the degree ∆ , and the verte x w ∈ S that has ∆ demands inciden t at it. Alth ough, there may be an expon ential number of choices for the optimal length, a polynomial number of guesses within a binary-sea rch suf fice to get a B such that B ≤ d ( S ) ≤ 2 · B . The algorithm then returns the better of the two proced ures describ ed belo w . Pro cedure 1 (high-degr ee case): Since the degree of vertex w in the optimal solution S is ∆ , there is tree rooted at w of length d ( S ) ≤ 2 B , that contains at least ∆ demands ha ving one end point at w . W e assign a w eight to each verte x u , equal to the number of demands that hav e one end point at this vertex u and the other end point at w . T hen we run the k -MST algorithm [Gar05] with root w and a tar get w eight of ∆ . B y the preceding arg ument, this problem has a feasible solution of lengt h 2 B ; so we obtain a solution H of length at most 4 B (since the algorithm of [G ar05] is a 2-approx imation). The ratio of solution H is thus at most 4 B / ∆ = 4 q ∆ B q . Pro cedure 2 ( low-degr ee case): Set t = q 2∆ ; not e that q ≤ ∆ · n 2 and so t ≤ n/ 4 . W e maintain a curre nt tree T (initially just verte x w ), which is upda ted in iterations as follo ws: shrink T to a super node s , and run the k -MST algorit hm with root s and a target of t new vert ices. If the resultin g s -tree has length at most 4 B , includ e this tree in the current tree T and continu e. If the resulting s -tree has length more than 4 B , or if all the vertices hav e been included, the procedu re ends. Since t ne w vert ices are added in each iteration , the number of iterations is at most n t ; so the length of T is at most 4 n t B . W e now sho w that T contains at least q 2 demands . Consider the set S \ T (reca ll, S is the optimal solutio n). It is clear that | S \ T | < t ; otherwise the k -MST instanc e in the last iteration (with the current T ) would hav e S as a feasible solution of length ≤ 2 B (and hence would fi nd one of lengt h at m ost 4 B ). So the number of demands cov ered in S that hav e at least one e nd point in S \ T is at most | S \ T | · ∆ ≤ t · ∆ = q / 2 (as ∆ is the degree of sol ution S ). Thus there are at least q / 2 demands conta ined in S ∩ T , in particu lar in T . Thus T is a solutio n hav ing ratio at most 4 n t B · 2 q = 8 n t B q . The better ratio soluti on among H and T from the two procedure s has ratio at most min { 4 q ∆ , 8 n t } · B q = min { 8 t, 8 n t } · B q ≤ 8 √ n · B q ≤ 8 √ n · d ( S ) q . So this algorithm is an O ( √ n ) approximati on to the minimum ratio k -forest problem. 2.3 Ap proximation algorithm f or k -for est Giv en the two algorit hms for minimum ratio k -fores t, we can use them in a standard greedy fash ion (i.e., kee p picki ng approximately minimum-rati o solut ions until we obtain a forest connecting at least k pairs); 6 the standa rd set cover analysis can be used to sho w an O (min { √ n, √ k } · log k ) -appro ximation guarantee for k -forest. A tighter analysis of the greedy algorithm (as done, e.g., in Charikar et al. [CC yC + 98]) can be used to remov e the logari thmic terms and obta in the guarantee stated in T heorem 2. 3 A pplications to Dial-a-Ride pr oblems In this se ction, we study app lication s of the k -forest problem to the Dia l-a-Ride proble m (Definition 3), and some generaliz ations. A natural solution -structu re for Dial-a-Ride in volv es servicing demands in batches of at most k each, w here a batch consisting of a set S of demands is serv ed as follows: the vehicl e starts out being empty , picks up each of the | S | ≤ k objects from their sources , then drops off each object at its destin ation, and is again empty at the end. If we kne w that the optimal solution has this struc ture, we could obtain a greed y frame work for Dial-a-Ride by repeated ly finding the best ‘batch ’ of k demands. H o wev er , the optimal solutio n may in v olv e carrying almost k objects at e very point in the tour , in which case it can not be decomposed to be of the abov e structure. In T heorem 7, we sho w that there is alway s a near optimal soluti on ha ving this ‘pick-d rop in batches’ structure. Buildin g on Theorem 7, we obtain appro ximation algori thms for the class ical Dial-a-Ride problem (Secti on 3.1), and tw o interesti ng extension s: non-un iform Dial-a-Ride (Section 3.2) & weighted Dial-a-Ride (Section 3.3). Theor em 7 (Structure Theor em) Given any instan ce of Dial-a-Rid e, ther e e xists a feasible tour τ satisfy - ing the following condit ions: 1. τ can be split into a set of se gments { S 1 , · · · , S t } (i.e., τ = S 1 · S 2 · · · S t ) wher e each se gment S i servic es a set O i of at most k demands such that S i is a path that first pick s up each demand in O i and then dr op s each of them. 2. The length of τ is at most O (log m ) times the length of an optimal tour . Pro of: Con sider an optimal non-preempti ve tour σ : let c ( σ ) denote its length, and | σ | denot e the number of edge tra ver sals in σ . Note that if in some visit to a verte x v in σ there is no pick-up or drop-of f, then the tour can be short-cut ove r vertex v , and it still remains feasibl e. Further , due to triang le inequali ty , the length c ( σ ) does not increase by this opera tion. So we may assume that each ver tex visit in σ in v olv es a pick-u p or drop-of f of some object. Since there is ex actly one pick-up & drop- off for each object, we hav e | σ | ≤ 2 m + 1 . Define the str etch of a demand i to be the number of edge tra versa ls in σ between the pick-u p and drop-o ff of object i . T he demands are partition ed as follo ws: for each j = 1 , · · · , ⌈ log(2 m ) ⌉ , group G j consis ts of all the demands whose stretch lie in the interv al [2 j − 1 , 2 j ) . W e consider each group G j separa tely . Claim 8 F or each j = 1 , · · · , ⌈ log(2 m ) ⌉ , ther e is a tour τ j that serves all the demands in gr oup G j , satisfi es condit ion 1 of Theor em 7, and has leng th at most 6 · c ( σ ) . Pro of: Con sider tour σ as a line L , with eve ry edge tra versa l in σ represent ed by a distinct edge in L . Number the ver tices in L from 0 to h , where h = | σ | is the number of edge trav ersals in σ . Note that each ver tex in V may be represented multiple times in L . Each demand is associa ted w ith the numbers of the ver tices (in L ) where it is pick ed up & dropped off. Let r = 2 j − 1 , and partition G j as follo ws: for l = 1 , · · · , ⌈ h r ⌉ , set O l,j consis ts of all demands in G j that are pick ed up at a verte x numbered between ( l − 1) r and l r − 1 . Since ev ery demand in G j has stretch in th e interv al [ r , 2 r ] , e very demand in O l,j is dropped of f at a v erte x numbered b etween lr and ( l + 2) r − 1 . Note that | O l,j | equals the number of demands in G j carried over edge ( l r − 1 , l r ) by tour σ , w hich is at 7 most k . W e define segmen t S l,j to start at ver tex number ( l − 1) r and trav erse all edges in L until verte x number ( l + 2) r − 1 (servici ng all demands in O l,j by first picking u p ea ch d emand be tween v ertices ( l − 1) r & lr − 1 ; then dropping off each demand between vertic es l r & ( l + 2) r − 1 ), and then return (with the veh icle being empty) to v ertex lr . Clearly , the number of objects carri ed over an y edge in S l,j is at most the number carried ove r the correspon ding edge trav ersal in σ . Also, each edge in L partic ipates in at m ost 3 seg ments S l,j , and each edge is trav ersed at most twice in any seg ment. So the total length of all segments S l,j is at most 6 · c ( σ ) . W e define tour τ j to be the concatenati on S 1 ,j · · · S ⌈ h/r ⌉ ,j . It is clear that this tour satisfies conditi on 1 of Theorem 7.  Applying this claim to each group G j , and concatenat ing the resultin g tours, w e obtain the tour τ satis- fying conditi on 1 and ha ving length at most 6 log (2 m ) · c ( σ ) = O (log m ) · c ( σ ) .  Remark: The ratio O (log m ) in Theorem 7 is almost best p ossible. There are insta nces of Dial-a-Ride (e ven on an unweight ed line), where ev ery solution satisfying condition 1 of T heorem 7 has length at least Ω(max { log m log log m , k log k } ) times the optimal non-pree mpti ve to ur . So, if we only use solutions of this structure, then it is not possible to obtain an approximatio n factor (just in terms of capacity k ) for Dial-a-Ride that is better than Ω( k / log k ) . The solutio ns found by the algorithm for D ial-a-Ride in [CR98] also satisfy condit ion 1 of Theorem 7. It is interest ing to note that w hen the underlyi ng m etric is a hierarc hically well- separa ted tree, [CR98] obtain a solution o f su ch stru cture ha ving l ength O ( √ k ) times the optimum, whereas there is a lo wer bound of Ω( k log k ) e ven for the simple case of an unweig hted line. 3.1 Classical Dial-a-Ride Theorem 7 suggest s a greedy strate gy for Dial-a- Ride, based on repeatedly finding the best batch of k demands to service . This greedy subpr oblem turns out to be the minimum ratio k -forest probl em (Defini- tion 5), for whic h we already h av e an ap proximatio n algorithm. The next theore m sets up the reductio n from k -forest to Dial-a-Ride. Theor em 9 (Redu cing Dial- a-Ride to minimu m ratio k -for est) A ρ -appr oxi mation algorithm for mini- mum ra tio k -for est implies an O ( ρ log 2 m ) -appr oximation algorithm for Dial-a-Rid e. Pro of: The a lgorithm for Dial-a-Ride is as follo ws. 1. C = φ . 2. Until the re are no uncovered demand s, do: (a) Solve the minimum ratio k -fo rest problem, to obtain a tree C covering k C ≤ k new d emands. (b) Set C ← C ∪ C . 3. For each tree C ∈ C , obtain an Euler tour on C to locally s ervice all demands (p ick up all k C objects in the first trav ersal, and dr op them all in the seco nd traversal). Then use a 1.5-ap proxim ate TSP tou r on the source s, to connect all the local tours, and obtain a feasible non-p reemptive tou r . Consider the tour τ and its seg ments as in Theorem 7 . If the number of uncov ered demands in some iterati on is m ′ , one of the segments in τ is a solution to the minimum ratio k -forest problem of va lue at most d ( τ ) m ′ . S ince we ha ve a ρ -approximatio n algor ithm for this problem, we would find a seg ment of ratio at most O ( ρ ) · d ( τ ) m ′ . Now a standard set cover type argume nt sho ws that the total length of trees in C is at most O ( ρ log m ) · d ( τ ) ≤ O ( ρ log 2 m ) · O P T , where O P T is the optimal valu e of the Dial-a-Ride instan ce. F urther , the TSP tour on all sources is a lo wer bound on O P T , and we use a 1.5-approx imate soluti on [Chr77]. So the fi nal non-preempti v e tour output in step 5 abov e has length at m ost O ( ρ log 2 m ) · O P T .  8 This theor em is in fact stro nger than Theorem 4 clai med earlier: it is easy to se e that any ap proximatio n algori thm for k -fores t implies an algorithm with the same guarantee for minimum ratio k -fores t. Note that, m and k may be super -poly nomial in n . Howe ve r , w e sho w in Section 3.3 that with the loss of a constant fact or , the general Dial-a-Ride proble m can be reduce d to one where the number of demands m ≤ n 4 . Based on this and Theorem 9, a ρ approxi mation algori thm for minimum ratio k -forest actually implies an O ( ρ log 2 n ) approximati on algorithm for Dial-a-Ride. Using the approx imation algorithm for m inimum ratio k -fore st (Section 2), we obta in an O (min { √ n, √ k } · log 2 n ) approximati on algorithm for the Dial-a- Ride prob lem. Remark: If we use the O ( √ k ) approximatio n for k -fores t, the resulting non-preempti v e tour is in fact feasib le eve n for a √ k capacity vehicle ! As noted in [CR98], this prope rty is also true of their algorit hm, which is based on an entire ly dif ferent approac h. 3.2 Non-unif orm Dial-a-Ri de The greedy f ramew ork fo r Dial- a-Ride de scribed ab ov e is actually more general ly applica ble than to just the classic al Dial-a-Ride problem. In this section, we consider the Dial-a-Ride problem under a substantia lly more general class of cost functi ons, and sho w how the k -fores t problem can be used to obtain an approx- imation algorithm for this generaliza tion as well. In fac t, the approx imation guarantee we obtain by this approa ch matches the best known for the classical Dial-a-Ride problem. Our frame work for Dial-a-Rid e is well suited for such a generalizat ion since it is a ‘primal’ approach, based on directly appro ximating a near -opti mal solution; this approach is not too s ensiti ve t o th e cost function . On the oth er hand , the Cha rikar & Raghav ach ari [CR98] algorith m is a ‘dual’ approach, based on obtain ing a good lo wer bound, which depen ds hea vily on the cost funct ion. Thus it is unclear whether their techn iques can be extende d to handle such a gene ralization . Definition 10 (Non-unif orm Dial-a-Ride) Given an n verte x undir ected grap h G = ( V , E ) , a r oot vertex r , a set of m demands { ( s i , t i ) } m i =1 , and a non-decr easing cost function c e : { 0 , 1 , · · · , m } → R + on each edg e e ∈ E (wher e c e ( l ) is the cost incurr ed by the vehicle in tra versin g edge e while carryin g l objects), find a non-pr eempti ve tour (startin g & ending at r ) of m inimum total cost that move s each object i fr om s i to t i . Note that the classical Dial-a-Ride probl em is a specia l case when the edge costs are gi ven by: c e ( l ) = d e if l ≤ k & c e ( l ) = ∞ otherwise , where d e is the edge length in the underlying metric. W e may assume (without loss in generali ty) that for any fixed va lue l ∈ [0 , m ] , the edge costs c e ( l ) induce a metric on V . Similar to Theorem 7, we hav e a near optima l solution w ith a ‘batch’ structur e for the non-uniform Dial-a- Ride prob lem as well, which implies the algorithm in Theorem 12. T he proof of the follo wing corollary is almost identi cal to that of Theorem 7, and is omitted . Cor ollary 11 (Non-uniform Structur e Theore m) Given any instance of non-unifor m Dial-a-Rid e, ther e e xists a feasible tour τ satisfying the following condition s: 1. τ can be split into a set of se gments { S 1 , · · · , S t } (i.e., τ = S 1 · S 2 · · · S t ) wher e each se gment S i servic es a set O i of demands such that S i is a path that first pic ks up each demand in O i and then dr ops eac h of them. 2. The cost of τ is at m ost O (log m ) times the cost of an optimal tour . 9 Theor em 12 (Appr oximating non-unifor m Dial-a-Ride) A ρ -appr oximation algorithm for m inimum ra- tio k -for est implies an O ( ρ log 2 m ) -appr oximation algorithm for non-u niform D ial-a-Rid e. In partic ular , ther e is an O ( √ n log 2 m ) -appr oximation algorithm. Pro of: Coro llary 11 again sugges ts a gree dy algorit hm for n on-unifo rm D ial-a-Rid e based on the follo wing gr eedy subpr oblem : find a set T of uncov ered demands and a path τ 0 that first picks up each object in T and then drops off each of them, such that the ratio of the cost of τ 0 to | T | is m inimized . Ho wev er , unlik e in the classical Dial-a-Ride problem, in this case the cost of path τ 0 does not come from a single m etric. Nev erthe less, the m inimum ratio k -forest problem can be used to solve this sub problem as follows. 1. For ev ery k = 1 , · · · , m : (a) Define leng th function d ( k ) e = c e ( k ) on the edges. (b) Solve the m inimum ratio k -forest p roblem o n metric ( V , d ( k ) ) with target k , to obtain tree T ′ k covering n k ≤ k demand s. (c) Obtain an Euler tour T k of T ′ k that services these n k demand s, by picking up all deman ds in one tra versal and then drop ping th em all in a second trav ersal. 2. Return the tour T k having the s mallest ratio c ( T k ) n k (over all 1 ≤ k ≤ m ). Assuming a ρ -approx imation algorith m for m inimum ratio k -forest (for all va lues of k ), we no w sho w that the abov e algorithm obtains a 16 ρ -ap proximate soluti on to the greedy subproblem. The cost of tour T k in step 3 is c ( T k ) ≤ 4 · d ( k ) ( T ′ k ) , since T k in volv es trav ersing a tour on tree T ′ k twice and the ve hicle carries at most n k ≤ k objects at ev ery point in T k . S o the ratio of tour T k is c ( T k ) n k ≤ 4 d ( k ) ( T ′ k ) n k = 4 · ratio ( T ′ k ) . Let τ denote the optimal path for th e greedy subpro blem, T the set of demand s that it services , and t = | T | . Let T 1 denote the last 3 4 t demands that are pick ed up, and T 2 denote the first 3 4 t demands that are droppe d of f. It is clear that T ′ = T 1 ∩ T 2 has at least t/ 2 demands; let T ′′ ⊂ T ′ be any subset with | T ′′ | = t/ 4 . Let τ ′ denote the portion of τ between the t 4 -th pick up and the 3 t 4 -th drop off. Note that when path τ is trav ersed, there are at least t 4 object s in the vehicle while tra ver sing each edge in τ ′ . S o the cost of τ , c ( τ ) ≥ P e ∈ τ ′ c e ( t/ 4) . Since τ ′ contai ns the end points of all demand s in T ′ ⊃ T ′′ , it is a feasib le solution (co vering the demands T ′′ ) to minimum ratio k -forest with tar get k = t/ 4 in the m etric d ( t/ 4) , havin g ratio ( P e ∈ τ ′ c e ( t/ 4)) / t 4 ≤ 4 c ( τ ) t . So the ratio of tour T t/ 4 (obtai ned from the ρ -appr oximate tree T ′ t/ 4 ) is at most 4 · ratio ( T ′ k ) ≤ 4 ρ 4 c ( τ ) t = 16 ρ c ( τ ) t . Thus we ha ve a 1 6 ρ -appro ximation algorith m for the gr eedy subprob lem. Based o n Corolla ry 11, it can now be sho wn (as in Theo rem 9) that a ρ ′ -appro ximation algorith m for the greedy sub problem implies an O ( ρ ′ · log 2 m ) -appro ximation algor ithm for non- uniform Dial-a-Ride. Using the abo ve 16 ρ -app roximation for the greedy subpro blem, we ha ve the theor em.  3.3 W eighted Dial-a-Ride So far we work ed w ith the unweighte d version of Dial-a-Ride, where each object has the same weight. In this section, we extend our greedy framew ork for D ial-a-Rid e to the case when objects hav e differe nt sizes, and t he tota l size of objects in th e ve hicle must be b ounded by the v ehicle cap acity . Here we o nly e xtend th e classic al D ial-a-Rid e problem and not the g eneraliza tion of Section 3 .2. T he pr oblem studied in thi s section has been studi ed earlier as the pic kup and delivery problem [SS95]. Definition 13 (W eighted Dial-a-Ride) G iven a vehicle of capac ity Q ∈ N , an n -verte x metric space ( V , d ) , a r oot vertex r , and a set of m objec ts { ( s i , t i , w i ) } m i =1 (with object i havin g sour ce s i , desti nation t i & an inte ger size 1 ≤ w i ≤ Q ), find a minimum length (non-pr eemptive) tour of th e vehicle starting (and endin g) 10 at r that moves each objec t i fr om its sour ce to its destina tion suc h that the total size of objects carried by the vehicle is at most Q at any poin t on the tou r . The classic al Dial-a-Ride problem is a special case when w i = 1 for all demands and the v ehicle capaci ty Q = k . The follo wing are two lo wer bounds for weighted D ial-a-Rid e: a TSP tour on the set of all sour ces & destination s (Steiner lo wer bound ); and P m i =1 w i · d ( s i ,t i ) Q (flo w lo wer boun d). In fact , as can be seen easily , these two lower bounds are valid e ven for the preempti ve version of weighted Dial-a-Ride ; so the y are termed pre emptive lower bounds . The main result of this secti on (Theor em 15) reduces weighte d Dial-a-Ride to the classical D ial-a- Ride problem with the additional prope rty that the number of demands ( m ) is small (polyno mial in the number of v ertice s n ). This sho ws that in ord er to approx imate w eighte d Dial-a-Rid e, it suf fices to cons ider instan ces of the classical Dial-a-Ride problem with a small number of demand s. The next lemma sho ws that eve n if the ve hicle is allo wed to split each object ove r multiple deli verie s, the resultin g tour is not much shorter than the tour where each object is required to be serv ed in a single deli very (as is the case in weighted Dial-a-Ride ). This lemma is th e main ingredie nt in the pro of of Theorem 15. In the follo wing, for any instance of weighted Dial-a-Ride, we define the unweight ed instance correspo nding to it as a classi cal Dial-a-Ride instan ce with vehicle capacity Q , and w i (unweigh ted) demands each havi ng source s i and destin ation t i (for each 1 ≤ i ≤ m ). Lemma 14 G iven any instance I of weighted Dial-a-Ride , and a solutio n τ to the unweighted instance corr esponding to I , ther e is a polynomial time computable solution to I having length at most O (1) · d ( τ ) . Pro of: Let J denote the unweight ed instance correspon ding to I . Define line L as in the proof of Theo- rem 7 by trav ersing τ from r : for ev ery edge tra versa l in τ , add a new edge of the same length at the end of L . For each unweighted object in J correspon ding to demand i in I , there is a se gment in τ (corres pond- ingly in L ) w here it is moved from s i to t i . So each demand i ∈ I correspon ds to w i seg ments in τ (each being a path from s i to t i ). For each demand i in I , we assign i to one of its w i seg ments pick ed uniformly at random: call this segment l i . For an edg e e ∈ L , let N e = P i : e ∈ l i w i denote the random varia ble which equals the total weight of demands w hose assigned segments contain e . N ote that the expecte d valu e of N e is exa ctly th e nu mber o f unweighted o bjects carried by τ w hen tra vers ing the edge corresp onding to e . Since τ is a feasib le tour for J , E [ N e ] ≤ Q for all e ∈ L . Consider a random inst ance R of D ial-a-Rid e on line L with vehicl e capacity Q and demands as follo ws: for each demand i in I , a n object o f weight w i is to b e move d along segme nt l i (chose n rando mly a s abo ve). C learly , any feasible t our f or R correspon ds to a feasible tour for I of th e sa me len gth. Note th at the flow lo wer bound for instance R is F = P e ∈ L d e ⌈ N e Q ⌉ , and the Steiner lower bound is P e ∈ L d e = d ( τ ) . Using linearity of exp ectation , E [ F ] ≤ P e ∈ L d e ( E [ N e ] Q + 1) ≤ 2 · d ( τ ) . Let R ∗ denote the instance (on line L ) obtained by assigning each demand i in I to its shortest length segmen t (among the w i seg ments corres ponding to it). Clearly this assignment minimizes the fl o w lower boun d (over all assig nments of demands to se gments). S o R ∗ has flow bound ≤ E [ F ] ≤ 2 · d ( τ ) , and Steiner lower boun d d ( τ ) . Finally , we note that the 3-appr oximation algorith m for Dial-a-Ride on a line [KR W00] extends to a constant factor approximatio n algori thm for the case w ith weighted demands as well (this can be seen directl y from [KR W 00]). Addi tionally , this approximation guarantee is relati ve to the preempti ve lower bound s. Thus, us ing this algorithm on R ∗ , we obtain a feasible solutio n to I of leng th at mo st O (1) · d ( τ ) .  Theor em 15 (W eighted D ial-a-Ride to unweighted) Suppose ther e is a ρ -appr oxi mation algo rithm f or in- stance s of class ical Dial-a-Ride with at most O ( n 4 ) demands. Then ther e is an O ( ρ ) -appr oximation algo- 11 rithm for weighted Dial-a-Ride (with any number of demands). In particula r , ther e is an O ( √ n log 2 n ) appr oximation for w eighted Dial-a-Ride . Pro of: Let I denote a n insta nce of weight ed Dial-a-Ride with object s { ( w i , s i , t i ) : 1 ≤ i ≤ m } , an d τ ∗ an optimal tour for I . Let P = { ( s 1 , t 1 ) , · · · , ( s l , t l ) } be the distin ct pairs of vertices that ha ve some demand between them, and let T i denote the total size of all objects havin g source s i and destinati on t i . Note that l ≤ n ( n − 1) . Let P hig h = { i ∈ P : T i ≥ Q 2 } , P low = { i ∈ P : T i ≤ Q l } , and P ′ = P \ ( P hig h ∪ P low ) . W e no w sho w ho w to separate ly service objects in P low , P hig h & P ′ . Servic ing P low : The total size in P low is at most Q ; so we can service all these pairs by trav ersin g a single 1.5-app roximate tour [Chr77] on the sources and destinati ons. Note that the length of this tour is at most 1.5 times the Steiner lo wer bound, hence at most 1 . 5 · d ( τ ∗ ) . Servic ing P hig h : Let C be a 1.5-approximate minimum tour on all the sourc es. T he pairs in P hig h are servic ed by a tour τ 1 as follo ws. T rav erse along C , and when a source s i in P hig h is visited , tra vers e the direct e dge to th e correspo nding destination t i & ba ck, as fe w times as pos sible so as t o mov e all the ob jects between s i and t i , as describe d next . Note that ev ery object to be mov ed between s i and t i has size (the origin al w i size) at most Q , and the total size of such obje cts T i ≥ Q/ 2 . So thes e obje cts can be partitione d such that the size of each part (exc ept possibly the last) is in the interv al [ Q 2 , Q ] . S o the number of times edge ( s i , t i ) is tra v ersed to service the de mands between them is at most 2 ⌈ 2 T i Q ⌉ ≤ 2( 2 T i Q + 1) ≤ 8 T i Q . Now , the length of tour τ 1 is at most d ( C ) + P ( s i ,t i ) ∈P high 8 d ( s i , t i ) T i Q ≤ d ( C ) + 8 P m i =1 w i · d ( s i ,t i ) Q . Note that d ( C ) is at most 1.5 times t he min imum tou r on all sou rces (Steiner lo wer bo und), and the second t erm ab ov e is th e flo w lo w er bo und. So tou r τ 1 has l ength at most O (1) times the pr eempti ve lo wer b ounds for I , whic h is at most O (1) · d ( τ ∗ ) . Servic ing P ′ : W e know that the total size T i of each pair i in P ′ lies in the interv al ( Q/l , Q/ 2) . Let I ′ denote the instance of w eighte d Dial-a-Rid e with demands { ( s i , t i , T i ) : i ∈ P ′ } and vehicle capacity Q ; note that the number of demands in I ′ is at most l . The tour τ ∗ restric ted to the objects correspond ing to pairs in P ′ is a feasible soluti on to the unweigh ted insta nce correspo nding to I ′ (b ut it m ay not feasib le for I ′ itself) . Howe v er Lemma 14 implies that the optimal va lue of I ′ , opt ( I ) ≤ O (1) · d ( τ ∗ ) . Next w e reduce instance I ′ to an instance J of weighted Dial-a-Ride satisf ying the followin g conditio ns: (i) J has at most l demands, ( ii) each object in I has size at most 2 l , (i ii) any feasibl e solution to J is feasible for I ′ , and (iv) the optimal valu e opt ( J ) ≤ O (1) · opt ( I ′ ) . If Q ≤ 2 l , J = I ′ itself satisfies the require d condit ions. Suppose Q ≥ 2 l , then define p = ⌊ Q l ⌋ ; note that Q ≥ l · p ≥ Q − l ≥ Q 2 . Round up each size T i to the smallest integral multiple T ′ i of p , and round down the capaci ty Q to Q ′ = l · p . Since each size T i ∈ ( Q l , Q 2 ) , all siz es T ′ i ∈ { p, 2 p, · · · , l p } . Now let I ′′ denote the wei ghted Dial-a-Ride instanc e with demands { ( s i , t i , T ′ i ) : i ∈ P ′ } and vehicle capacity Q ′ = l p . O ne can obtain a feasible solution for I ′′ from an y feasible solution σ for I ′ by trav ersing σ a const ant number of times: this follo ws from Q ′ ≥ Q 2 & T ′ i ≤ max { 2 T i , Q ′ } . 3 So the optimal valu e of I ′′ is at most O (1) · opt ( I ′ ) . Now note that all sizes and the veh icle capacity in I ′′ are multiples of p ; scaling down each of these quantiti es by p , we get an instance J equivale nt to I ′′ where the vehicle capaci ty is l (and ev ery demand size is at most l ). This instance J satisfies all the four condi tions claimed abo ve. No w observe that the instance J can be solved using ρ -approximati on algorithm assumed in the the- orem. Since J has at most l demands (each of size ≤ 2 l ), the unweighted instance correspond ing to J 3 In parti cular , consider simulating a tr av ersal along σ of a capacity Q vehicle ( T 0 ) by 8 capacity Q ′ vehicles T ′ 1 , · · · , T ′ 8 , each running in parallel along σ . Whenev er ve hicle T 0 picks-up an object i , one of the vehicles { T ′ g } 8 g = 1 picks-up i : if w i ≤ Q 4 , an y vehicle { T ′ g } 4 g = 1 that has free capacity picks-up i ; if w i > Q 4 , any vehicle { T ′ g } 8 g = 5 that is emp ty picks-up i . It is easy to see that if at some point none of the vehicles { T ′ g } 8 g = 1 picks-up an object, there must be a capacity violation in T 0 . 12 has at most 2 l 2 ≤ 2 n 4 demands . Thus, this unweighted instance can be solved using the ρ -approximat ion algori thm for such instanc es, assumed in the theorem. Then using the algorithm in Lemma 14, we obtain a soluti on to J , of len gth at most O ( ρ ) · opt ( J ) ≤ O ( ρ ) · opt ( I ′ ) ≤ O ( ρ ) · d ( τ ∗ ) . Since an y feasi ble solution to J correspond s to one for I ′ , we ha ve a tour servici ng P ′ of lengt h at most O ( ρ ) · d ( τ ∗ ) . Finally , combini ng the tours servicing P low , P hig h & P ′ , we obtai n a feasib le tour for I havin g length O ( ρ ) · d ( τ ∗ ) , which gi ves us the desi red appro ximation algor ithm.  Theorem 15 also just ifies the assumption log m = O (log n ) made at the end o f S ection 3. This is important becaus e in gene ral m may be super- polynomia l in n . 4 The Effect of Pre emptions In th is secti on, we study the ef fect of the nu mber of preemption s in t he Dial-a- Ride problem. W e mentio ned two versions of the Dial-a-Ride problem (Definition 3): in the preempti v e versi on, an object may be pre- empted any number of times, and in the non-pr eempti ve versio n objects are not allowed to be preempted e ven once. Clearly the preempti v e version i s least re stricti ve and t he non -preempti v e ve rsion is most restric- ti ve. One may consider oth er versi ons of the Dial-a-Ride pro blem, where th ere is a speci fied upper bound P on the number of times an object c an b e pr eempted. Note th at t he c ase P = 0 is the non-p reempti ve v ersion, and the case P = n is the preempti ve version. W e sho w that for any instanc e of the D ial-a-Rid e prob- lem, there is a tour that preempts each object at most once (i.e., P = 1 ) and has length at m ost O (log 2 n ) times an optimal preempti v e tour (i.e., P = n ). This implies that the real gap between preempti v e and non-p reempti ve tours is between zero and one preemption per objec t. A tour that preempts each object at most once is called a 1-pr eemptive tour . Theor em 16 (Many preemptions to one pr eemption) Given any instance of the D ial-a-Ride pr ob lem, t her e is a 1-pr eempt ive tour of length at most O (log 2 n ) · O P T pmt , wher e O P T pmt is the length of an optimal pr eempti ve tour . Such a tour can be found in rand omized polynomial time. Pro of: Usin g the results on proba bilistic tree embedding [FR T03], we may assume that the gi ven metric is a hierar chically well-separ ated tree T . This only increa ses the exp ected length of the optimal solution by a facto r of O (log n ) . F urther , tree T has O (log d max d min ) lev els, where d max and d min denote the maximum and minimum distances in the original metric. W e first observ e that using standa rd scalin g ar guments, it suf fices to assume that d max d min is polynomia l in n . W ithou t loss of generality , any pree mptiv e tour in volv es at most 2 m · n edge tra versal s: each object is picked or dropped at most 2 n times (once at each verte x), and e very visit to a ver tex in v olves picking or dropping at least one object (otherwis e the tour can be shortcut ov er this verte x at no increa se in length). By retaining only vert ices within distan ce O P T pmt / 2 from the root r , we prese rve the opti mal preempti v e tour and ensu re that d max ≤ O P T pmt . Now con sider modifyin g the orig inal metric by s etting all e dges of len gth smaller th an O P T pmt / 2 mn 3 to le ngth 0; the n ew d istances are shortest paths under the modified edge lengths . So any pairwise distan ce decreases by at most O P T pmt 2 mn 2 . Clearly the length of the optimal preempt iv e tour only decreases under this modification. S ince there are at most 2 mn edge tra v ersals in an y pr eempti ve tou r , the increase in tour length i n go ing fro m the ne w metr ic to the origina l metric is at most 2 mn · O P T pmt 2 mn 2 ≤ O P T pmt n . Thus at the loss of a c onstant f actor , we may assume that d max /d min ≤ 2 mn 3 . F urther , the reductio n in Theorem 14 also holds for preempti ve Dial-a-Ride; so we may assume (at the loss of an addition al constant factor) that the number of demands m ≤ O ( n 4 ) . So we ha ve d max /d min ≤ O ( n 7 ) and hence tree T has O (log n ) le v els. The tree T resulting from the probab ilistic embedding has se ve ral Steiner v ertices t hat are n ot pres ent in the original metric; so the tour that we fi nd on T may actual ly preempt objects at S teiner vertices , in which 13 case it is not feasible in the original m etric. H o wev er as sho wn by Gupta [Gup01], these S teiner vertice s can be simulated by vertices in the original metric (at the loss of a constant factor). Based on the preceding observ ations, we assume that the metric is a tree T on the origina l ver tex set ha ving l = O (log n ) leve ls, such that the ex pected length of the optimal preempti ve tour is O (log n ) · OP T pmt . W e now partition the demands in T into l sets with D i (for i = 1 , · · · , l ) consistin g of all demands ha ving their least common ancestor (lca) in le vel i . W e service each D i separa tely using a tour of length O ( O P T pmt ) . Then concate nating the tours for each le vel i , we obta in the theorem. Servic ing D i : For each verte x v at leve l i in T , let L v denote the demands in D i that hav e v as their lca. Consider an optimal pre emptive tour that services the demands D i . Since the subtrees under any two diffe rent lev el i vertices are disjoi nt and there is no demand in D i across such subtre es, we may as- sume that this optimal tour is a concate nation of disjoint preempti ve tours servicin g each L v separa tely . If O P T pmt ( v ) denotes the length of an optimal preempti ve tour servicing L v with v as the starting vertex , P v O P T pmt ( v ) ≤ O P T pmt . No w co nsider an op timal pre empti ve t our τ v servic ing L v . Since the s j − t j path of each de mand j ∈ L v crosse s vertex v , at s ome point in tour τ v the v ehicle is at v with object j in it. Consider th e tour σ v obtain ed by modifying τ v so that it drops each object j at v when the vehicle is at v with object j in it. Clearly d ( σ v ) = d ( τ v ) = O P T pmt ( v ) . Note that σ v is a feasible preempti v e tour for the single sour ce Dial-a- Ride problem with sink v and all sources in L v . Thus the algor ithm of [HK85] giv es a non-preempti ve tour σ ′ v that mov es all objects in L v from their sources to v , ha ving length at most 2 . 5 d ( σ v ) = 2 . 5 O P T pmt ( v ) . Similarly , we can obtain a non-p reempti ve tour σ ′′ v that mov es all objects in L v from v to their destination s, ha ving length at most 2 . 5 O P T pmt ( v ) . Now σ ′ v · σ ′′ v is a 1-preempti v e tour servic ing L v of length at most 5 · O P T pmt ( v ) . W e no w run a DFS on T to visit all vertic es in lev el i , and use the algor ithm descr ibed abo ve for servic ing demands L v when v is visited in the DFS. This results in a tour servici ng D i , ha ving length at most 2 d ( T ) + 5 P v O P T pmt ( v ) . Here 2 d ( T ) is the Steiner lo wer bound, and P v O P T pmt ( v ) ≤ O P T pmt . Thus the tour servicin g D i has length at most 6 · O P T pmt . Finally concatenat ing the tours for each le vel i = 1 , · · · , l , we obtain a 1-preempti ve tour on T o f length O (log n ) · O P T pmt , which translates to a 1-preempti ve tour on the original metric ha ving length O (log 2 n ) · O P T pmt .  Moti v ated by obtainin g an improv ed appro ximation for D ial-a-Ride on the Euclidean plane, w e next consid er the worst case gap between an optimal non-preempti v e tour and the preempt iv e lower bound s. A s mentione d earlier , [CR98] sho wed th at th ere are instances of Dial-a-Ride where the ratio of the optimal non- preempti v e tour to the optimal preempti v e tour is Ω( n 1 / 3 ) . Howe ver , the metric in volv ed in this exampl e was the uniform metric on n points, which can not be embedded in the E uclidea n plane. The follo wing theore m sho ws that ev en in this special case, there can be a polynomia l gap between non-pr eempti ve and preempti v e tours, and implies that just preempti ve lower bound s do not suffice to obtain a poly-l ogarithmic approx imation guaran tee. Theor em 17 (Preemption gap in Euclidean plane) Ther e are instances of Dial-a-Ride on the Euclidean plane wher e the optimal non-p r eemptive tour has leng th Ω( n 1 / 8 log 3 n ) times the optimal pr eemp tive tour . Pro of: Con sider a square of side 1 in the Euclidean plane, in which a set of n demand pairs are distri bu ted unifor mly at r andom (each deman d point is gene rated independen tly and is di strib uted uniformly at random in the square). T he ve hicle capaci ty is set to k = √ n . Let R denote a random instance of Dial-a-Ride ob- tained as abov e. W e sho w that in this case, the optimal non-pr eempti ve tour has length ˜ Ω( n 1 / 8 ) with high probab ility . W e first sho w the follo wing claim. 14 Claim 18 The m inimum lengt h of a tr ee containin g k pairs in R is Ω( n 1 / 8 log n ) , w .h.p. Pro of: T ake an y set S of k = √ n demand pai rs. N ote that the nu mber of such sets S is  n k  . This set S has 2 k poin ts each of them generate d uniformly a t random. It is kno wn tha t th ere a re p p − 2 dif ferent la beled trees on p vertice s (see e.g. [v L W92], Ch.2). T he ter m labeled empha sizes that we a re not id entifying isomorphic graphs , i.e., two trees are counted as the same if and only if exactly the same pairs of vertices are adjace nt. Thus there are at most (2 k ) 2 k − 2 such trees just on set S . Consider any tree T among these trees and root it at the source point with minimum label. Here we assume that T has been generate d using the “Princip le of Deferred Dec isions”, i.e., n odes will b e gener ated one by one acco rding to so me bread th-first ordering of T . W e say that an edg e is short if its length is at most c αk ( c and α ∈ (0 , 1 2 ) will be fix ed later). If T has length at most c , it is clear that at most an α fraction of its edges are not short. So P r [ l eng th ( T ) ≤ c ] ≤ P H P r [ edges in H ar e shor t ] , where H in the summation ranges o ver all edge-subse ts in T with | H | ≥ (1 − α )2 k . For a fixed H , we bound P r [ edg es in H ar e shor t ] as follo ws. For any edge ( v , parent ( v )) (note parent ( v ) is w ell-define d since T is roote d), assuming that parent ( v ) is fixed, the prob- ability that this edge is short is p = π ( c αk ) 2 . So we can upper bound the probab ility that edges H are short by p | H | ≤ p (1 − α )2 k . So w e ha ve P r [ l eng th ( T ) ≤ c ] ≤ 2 2 k · p (1 − α )2 k , as the number of dif ferent edge sets H is at most 2 2 k . By a union boun d over all suc h labeled tree s T , the probab ility that the le ngth of the min imum spann ing tree on S is less than c is at m ost (2 k ) 2 k · 2 2 k · p (1 − α )2 k . Now taking a union b ound ov er all k -sets S , t he prob- ability that the minimum length of a tree containing k pairs is less than c is at most  n k  (2 k ) 2 k 2 2 k p (1 − α )2 k . Since k = √ n , this term can be bound ed as follo ws: ( ek ) k (4 k ) 2 k π (1 − α )2 k ( c αk ) (1 − α )4 k ≤ 500 k k 3 k ( c αk ) (1 − α )4 k = [500 · ( c α ) 4 − 4 α ( 1 k ) 1 − 4 α ] k ≤ 2 − k The last inequ ality above holds when c ≤ α 1000 · k 1 / 4 − 3 α/ (1 − 4 α ) . Setting α = 1 log k , we get P r [ ∃ k 1 / 4 8000 · log k length tree contai ning k pairs in R ] ≤ 2 − k So, with probabili ty at least 1 − 2 − √ n , the minimum length of a tree containin g k pairs in R is at least Ω( n 1 / 8 log n ) .  From T heorem 7 , we obtain that there is a near optimal non-pre empti ve tour servicin g all the demands in segmen ts, where each segment (except possibly the last) in volv es servic ing a set of k 2 ≤ t ≤ k demands. Although the lower bou nd of k / 2 is not stated in Theorem 7, it is easy to extend the statement to include it. This implies that any solut ion of this structure has at least n k = k segments . Since each segmen t cov ers at least k / 2 pairs, Claim 18 i mplies th at ea ch o f th ese s egment s has length Ω( n 1 / 8 / log n ) . S o the best solution of the structure giv en in T heorem 7 has length Ω( n 1 / 8 log n k ) . But since there is a near -optimal solution of this structu re, the optimal non-p reempti ve tour on R has length Ω( n 1 / 8 log 2 n k ) . On the othe r hand, the fl o w lo wer boun d for R is at most n k = k , and the Steiner lo wer bound is at mo st O ( √ n ) = O ( k ) (an O ( √ n ) length tree on the 2 n points can be construct ed using a √ 2 n × √ 2 n griddi ng). So the pr eempti ve lo wer bound s are both O ( k ) ; now using the algo rithm of [CR98], we see tha t the optimal preempti v e tour has len gth O ( k log n ) . Combined with the lower bound fo r no n-preempt iv e tours, we o btain the Theorem.  Acknowledgements: W e thank Alan Frieze for his help in prov ing T heorem 17. 15 Refer ences [AKR91] Ajit Agraw al, Philip Klein, and R. Rav i. When trees collid e: an appro ximation algorit hm for the generalized st einer problem on networks . Pr oceedin gs of the 23r d Annu al A CM Symposiu m on Theory of Computing , pages 134–14 4, 1991. [BBCM04] N. Bansal, A. Blum, S . 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