PTAS for k-tour cover problem on the plane for moderately large values of k

PT AS for k -tour co v er problem on the plane for mo derately large v alues of k ∗ Anna Adamaszek DIMAP and Departmen t of Computer Science Univ ersit y of W arwic k A.M.Adamaszek@w arwic k.ac.uk Artur Czuma j DIMAP and Departmen t of Computer Science Univ ersit y of W arwic k A.Czuma j@w arwic k.ac.uk Andrzej Lingas Departmen t of Computer Science Lund Univ ersit y Andrzej.Lingas@cs.lth.se Abstract Let P b e a set of n p oin ts in the Euclidean plane and let O b e the origin p oin t in the plane. In the k -tour c over pr oblem (called frequently the c ap acitate d vehicle r outing pr oblem ), the goal is to minimize the total length of tours that cov er all p oints in P , such that eac h tour starts and ends in O and co v ers at most k p oin ts from P . The k -tour cov er problem is kno wn to b e N P -hard. It is also kno wn to admit constan t factor approximation algorithms for all v alues of k and even a p olynomial- time approximation scheme (PT AS) for small v alues of k , i.e., k = O (log n/ log log n ). W e significantly enlarge the set of v alues of k for whic h a PT AS is prov able. W e presen t a new PT AS for all v alues of k ≤ 2 log δ n , where δ = δ ( ε ). The main tec hnical result pro ved in the pap er is a nov el reduction of the k -tour cov er problem with a set of n p oin ts to a small set of instances of the problem, eac h with O (( k /ε ) O (1) ) p oints. 1 In tro duction The k -tour c over pr oblem ( k -TC), is a very natural and w ell known generalization of the tra v eling salesp erson problem (TSP) to include sev eral tours [3, 4, 9, 13]. Namely , we are giv en a set P of p oin ts (sites), a distinguished p oint O outside P , called the origin as w ell ∗ Researc h supp orted in part by the Centre for Discrete Mathematics and its Applications (DIMAP), EPSR C a w ard EP/D063191/1, and b y VR gran t 621-2005-408. 1 as a distance function defined on P ∪ { O } . A tour is a cycle whose vertices are in P ∪ { O } . The length of a tour is the sum of distances b et w een the adjacent p oints on the tour. The ob jectiv e is to find a set of tours, eac h including the origin and at most k points in P , which co v ers all p oin ts in P and achiev es the minimum total length. In Op erations Researc h, the k -TC problem is w ell known as the c ap acitate d vehicle r outing pr oblem [13]. The name comes from its standard application when the p oin ts in P represent customer locations, and the origin O stands for a dep ot. Then, a fleet of v ehicles lo cated at the dep ot m ust serv e all the customers, so that eac h v ehicle can serv e at most k customers. The ob jective is to minimize the total distance trav eled b y the fleet. The k -TC problem (capacitated vehicle routing problem) is one of the cen tral sp ecial cases of a more general v ehicle routing problem, introduced b y Dantzig and Ramser [6] fift y years ago, and studied v ery extensiv ely in the literature ev er since (cf. [10, 13]). The k -TC problem con tains the TSP problem as a special case and it is kno wn to be N P - hard for all k ≥ 3. F or this reason, the research on k -TC has fo cused on heuristic algorithms and appro ximation algorithms. The most extensiv ely studied v arian ts of k -TC are the metric one, when the distance function is symmetric and satisfies the triangle inequality , and in particular the two-dimensional Euclide an one, when the p oints are placed in the plane and the distance is Euclidean. The general metric case of k -TC for k ≥ 3 has been shown to b e APX-complete [3], i.e., complete for the class of optimization problems admitting constan t factor approximations. Ho w ev er, the approximabilit y status of the tw o-dimensional Euclide an k -TC problem, in particular, the problem of the existence of a PT AS, has not b een completely settled yet. One of the first studies of tw o-dimensional Euclidean k -TC has b een due to Haimovic h and Rinno o y Kan [9], who presen ted sev eral heuristics for the metric and Euclidean k -TC, including a PT AS for the tw o-dimensional Euclidean k -TC with k < c log log n , for some constan t c [9, Section 6]. Asano et al. [4] substan tially subsumed this result by designing a PT AS for k = O (log n/ log log n ). They also observed that Arora’s [1, 2] or Mitc hell’s [11] PT AS for the t w o-dimensional Euclidean TSP implies a PT AS for the corresp onding k -TC where k = Ω( n ). There has not b een an y significan t progress since the pap er by Asano et al. [4] until v ery recently , when Das and Mathieu [7] show ed a quasi-p olynomial time appro ximation scheme (QPT AS) for the tw o-dimensional Euclidean k -TC for every k . Their algorithm com bines the approac h dev elop ed b y Arora [1] for Euclidean TSP with some new ideas to deal with k -TC (in particular, how to handle a large num b er of p ossible v alues of the lengths of the subtours arising in the subproblems of the original k -TC), and gives a (1 + ε )-appro ximation for the t w o-dimensional Euclidean k -TC in time n log O (1 /ε ) n (this bound holds for an y v alue of k ). In this pap er w e fo cus on the tw o-dimensional Euclidean v arian t of k -TC. (T o simplify the notation, w e shall further refer to this v arian t as to k -TC). Our main result is a new PT AS for k -TC for all v alues of k ≤ 2 log δ n , where δ = δ ( ε ). This significan tly enlarges the set of v alues of k for which a PT AS is know n. Our PT AS relies on a nov el reduction of an instance of k -TC with a set of n p oints to an instance or a small num b er of indep endent instances of the problem with a small num b er of p oin ts. 2 Our first reduction tak es any instance of k -TC on n p oin ts and reduces it to an instance of the problem with O (( k /ε ) O (1) log 2 ( n/ε )) p oin ts. Then we present a refinement, where the instance of k -TC is reduced to a small set of instances of k -TC, each with O (( k /ε ) O (1) ) p oin ts. These results, when com bined with the recent QPT AS due to Das and Mathieu [7], giv e the aforemen tioned PT AS for k -TC for all v alues k ≤ 2 log δ n , where δ = δ ( ε ). Our pap er is structured as follows. In the next section, we introduce useful notation and facts regarding k -TC. In Section 3, w e sho w the first reduction yielding our PT AS. In Section 4, w e presen t the refined reduction. W e conclude with final remarks. F or simplicit y of the presen tation, we will present (1 + O ( ε ))-appro ximation algorithms; reduction to (1 + ε )-approximation is straigh tforward. 2 Preliminaries W e assume a fixed origin in the plane and denote it b y O . F or a tour T , its (Euclidean) length is denoted b y |T | . F or a set U of tours, w e set | U | to P T ∈ U |T | . F or a set P of p oints in the plane, we denote b y T S P ( P ) the minim um length of a TSP-tour through P and by opt ( P ) the minim um length of a solution to k -TC (i.e., the minim um length of a set of tours, each through the origin and containing at most k p oints of P , which co v ers all p oin ts in P ). When P is clear from the context, we shall simply use the notation opt . F or a p oin t p ∈ P , we denote b y r ( p ) the distance of p from the origin O . The following simple low er b ound plays a very imp ortan t role in the previous approaches to k -TC, see [4, Prop osition 2] and [9, Lemma 1]. F act 1 opt ( P ) ≥ 2 k P p ∈ P r ( p ) . F ollo wing [4], we shall term 2 k P p ∈ P r ( p ) as the r adial c ost of P , and denote by r ad ( P ). Among other things, Haimovic h and Kan considered the so called iter ate d tour p artitioning heuristic for k -TC in [9]. The heuristic starts from constructing a TSP-tour T through P . Then, it considers all k -tour cov ers resulting from partitioning T into paths visiting exactly k p oints (assuming that n is divisible b y k ), and connecting the endp oin ts of the paths with O . The heuristic outputs the shortest among these solutions. F act 2 [4] If the iter ate d tour p artitioning heuristic uses a TSP tour U , then it r eturns a k -tour c over of total length not exc e e ding (1 − 1 k ) · | U | + r ad ( P ) . Note that given a TSP tour, the iterated tour partitioning heuristic can b e implemen ted in time O ( k n k + n ) by rep eatedly up dating the previous partition and k -tour cov er to the next one in time O ( n k ). Using the minim um spanning tree heuristic for TSP we can find a 2-appro ximation of the TSP in time O ( n log n ). Hence, w e obtain the follo wing. Corollary 3 If the iter ate d tour p artitioning heuristic uses the minimum sp anning tr e e heuristic for TSP then it r eturns a (3 − 2 k ) -appr oximation of an optimal k -tour c over of an n -p oint set and it c an b e implemente d in time O ( n log n ) . 3 c i 2 π /s locations O O Figure 1: The structure of circles, rays, and lo cations. The p oin t lab eled O is the origin. Other fat dots represen t the p oin ts from P . In the righ t picture eac h p oint has b een mo v ed to its nearest lo cation. 3 PT AS for mo derate v alues of k In this section we presen t a reduction that takes as an input any instance of the k -tour problem on a set of n p oints in the Euclidean plane and reduces it to an instance of the problem with O (( k /ε ) O (1) log 2 ( n/ε )) p oin ts. Then, we apply this reduction to obtain a PT AS for the k -tour problem for all k ≤ 2 log δ n , where δ is some p ositiv e constan t, δ = δ ( ε ). Our construction uses a series of transformations that eliminate most of the input p oin ts and reduce the input problem instance to a significan tly smaller one. 3.1 Remo ving close p oin ts Let L b e the maxim um distance from a p oin t in P to the origin O , that is, L = max { p ∈ P : r ( p ) } . Since opt ≥ 2 L , w e can ignore any p oint that is at a distance at most Lε/n from the origin: co v ering all suc h points with 1-tours will giv e us additional cost not greater than n · 2 Lε n ≤ ε · opt . Therefore, from no w on, we will consider only the p oints p with r ( p ) ≥ Lε/n . 3.2 Circles, ra ys, and lo cations Let us create cir cles around the origin, the i -th circle with a radius c i = Lε n ·  1 + ε k  i , for 0 ≤ i ≤ l log (1+ ε/k ) n ε m . Let us draw r ays from the origin with the angle b etw een an y pair of neigh b oring ra ys equal to 2 π /s (that is, partition the space into s sectors) with s = d 2 π k ε e . 4 Define a lo c ation to b e any p oint on the plane that is the intersection of a circle and a ra y . Since log (1+ ε/k ) n ε = log n ε log(1 + ε/k ) = Θ  k ε · log n ε  , there are Θ  k ε log( n/ε )  circles and Θ  k ε  ra ys. Therefore we obtain: Claim 4 The total numb er of lo c ations T satisfies T = Θ( k 2 ε − 2 log( n/ε )) . No w, w e transform the input set P and mov e each point from P to its nearest lo cation. Claim 5 The op er ation of moving e ach p oint to its ne ar est lo c ation c an change the c ost of a k -tour by at most ε · opt . Pro of . Let p b e a p oint in P . Supp ose that p lies b etw een the circles with radius c i and c i +1 (the distance b etw een p and the origin is in the in terv al [ Lε/n, L ], so w e kno w such circles exist). The distance b et ween these circles equals c i +1 − c i = ε k · c i . The distance b et ween t w o consecutiv e lo cations at the i -th circle is less than 2 π c i /s ≤ ε k · c i . Therefore the distance b et w een p and its nearest lo cation is at most √ 2 · ( 1 2 · ε k c i ) < ε k · c i ≤ ε k · r ( p ). If we mov e a p oint p ∈ P b y a distance at most ε k · r ( p ), the cost of a tour can c hange by at most 2 ε k · r ( p ). If w e add up the c hanges of the cost generated b y mo ving all p oints in P , then this total c hange is upp er b ounded b y P p ∈ P 2 ε k · r ( p ). Next, we use F act 1 to conclude that the total cost of mo ving all the p oin ts is at most ε · opt . u t F rom a k -tour U 0 for a mo dified instance of the problem (where all p oints ha v e b een mo v ed to their nearest lo cations) we can easily get a k -tour U for the original v ersion of the problem such that | U | ≤ | U 0 | + ε · opt . So a PT AS for the mo dified version yields a PT AS for the original version. In the rest of this pap er we will consider the modified v ersion of the problem. 3.3 T rivial and non trivial tours W e sa y that a tour visits a lo cation if it contains at least one p oin t from that lo cation. (If an edge of a tour passes trough a lo cation, but the tour do es not con tain any p oint from that lo cation, then the tour do es not visit that lo cation.) W e call a tour trivial if it visits only a single lo cation in P ; a tour is nontrivial otherwise. Theorem 6 Ther e is an optimal solution in which ther e ar e at most T nontrivial tours. Pro of . W e sa y that a set of tours t 1 , t 2 , . . . , t m ( m ≥ 2) forms a cycle if there is a set of lo cations ` 1 , ` 2 , · · · , ` m , ` m +1 = ` 1 suc h that eac h tour t i visits lo cations ` i and ` i +1 . Note that the origin is not considered as a lo cation. T o pro v e our theorem w e will need the follo wing: Lemma 7 Ther e is an optimal solution in which ther e ar e no cycles. 5 Pro of . Let U b e suc h an optimal solution which minimizes the sum o v er all its non trivial tours of the n um b er of lo cations visited b y that tour. Let us supp ose that U has a cycle, and let t 1 , t 2 , . . . , t m b e a minimal cycle ( m is minimal). Let ` 1 , ` 2 , . . . , ` m b e the locations in which the consecutiv e tours meet. F rom the minimality of the cycle w e kno w that b oth tours and lo cations are pairwise distinct. Let v ( t, ` ) denote the num b er of p oin ts from a lo cation ` visited by a tour t . Let min = min i ∈{ 1 ,...,m } { v ( t i , ` i ) } . Now we are ready to swap p oints b etw een the tours: the i -th tour, instead of visiting v ( t i , ` i ) points in the lo cation ` i and v ( t i , ` i +1 ) points in the lo cation, ` i +1 will now visit ( v ( t i , ` i ) − min) p oints in ` i and ( v ( t i , ` i +1 ) + min) p oints in ` i +1 . Here ` m +1 denotes ` 1 . Observ e that the mo dification do es not change the num b er of p oints visited by each tour. It also do es not increase the length of any tour. Therefore, w e obtain another optimal solution, in which the sum o ver all non trivial tours of the num b er of lo cations visited by that tour is smaller than in U (we managed to remov e one lo cation from eac h tour t i for which v ( t i , ` i ) = min). This is a con tradiction with the minimalit y of that sum in U . Therefore the optimal solution U has no cycles. u t Consider an optimal solution without cycles. Note that the lack of 2-cycles means that no tw o tours visit the same pair of locations. T o eac h nontrivial tour we can assign a pair of distinct lo cations visited b y this tour. The c hosen pairs are in one-to-one corresp ondence with the non trivial tours and they induce an acyclic undirected graph on the lo cations. Hence, we can ha v e at most T − 1 non trivial tours in an acyclic solution, so using Lemma 7 w e ha v e pro v ed the theorem. u t 3.4 Reduction to an instance of k -TC with ( k log n/ε ) O (1) p oin ts Observ e that Theorem 6 implies that there is an optimal solution in whic h at most T k p oints are co v ered b y non trivial tours. Therefore it is enough to consider only solutions which fulfill that prop ert y . If the n umber of points in a lo cation ` is greater than T k , some of the p oints will hav e to b e co vered b y trivial tours. W e ma y assume, without loss of generalit y , that among all trivial tours visiting a giv en lo cation there is at most one that visits less than k p oints. Moreo v er, if at least one point from some lo cation is visited b y a non trivial tour, w e can assume that all trivial tours visiting that lo cation con tain exactly k elements. Therefore, for each lo cation ` con taining c ` p oin ts, w e only ha ve to consider at most min { c ` , c ` − k · d c ` − T k k e} ≤ T k p oints for nontrivial tours. After finding a (1 + ε )-appro ximation for such reduced case, w e will add trivial tours co v ering all remaining p oin ts. That will give us (1 + ε ) − appro ximation for the original problem. Corollary 8 One c an r e duc e the k -TC pr oblem on n p oints to one on at most T 2 k p oints. 6 3.5 PT AS for k -TC with k ≤ 2 log δ n W e use Corollary 8 to reduce an y instance of k -TC with the input set of n p oin ts P to an instance of k -TC with N = T 2 k = Θ( k 5 ε − 4 log 2 ( n/ε )) input p oints. F or such input instance, w e apply the quasi-p olynomial time appro ximation sch eme for k -TC due to Das and Mathieu [7]. The obtained algorithm returns a (1 + ε )-approximation in time N log O (1 /ε ) N . This gives p olynomial time for all k ≤ 2 log δ n for some constant δ = δ ( ε ) > 0. Hence, w e ha v e the follo wing main theorem. Theorem 9 Ther e is a PT AS for the k -TC pr oblem pr ovide d that k ≤ 2 log δ n for some p ositive c onstant δ = δ ( ε ) . 4 Refinemen t: reduction to ( k /ε ) O (1) p oin ts In the preceding section, we ha v e demonstrated that the problem of close approximation of the k -TC problem on the input set of n p oints in the plane reduces to that for a multi- p oin t-set of size p olynomial in k /ε and polylogarithmic in n in the relev ant locations. In this section, w e shall eliminate the p olylogarithmic dep endency of n in the reduction. This will ha v e only a relatively small effect on the asymptotics for the size of the largest k in terms of n for whic h w e can attain a PT AS and w e will obtain a PT AS for all k ≤ 2 log δ 0 n , where comparing to the b ound in Theorem 9, we will hav e δ 0 > δ . Ho w ev er, for small v alues of k this will lead to a faster PT AS. Hop efully , because it remo ves completely the dep endency on n from the size of the reduced instance, it also might be a step tow ards a PT AS for arbitrary v alues of k . The idea of our refinement resem bles Baker’s metho d [5] of closely approximating sev eral hard problems on planar graphs. It relies on the following separation lemma. Lemma 10 L et P b e a set of p oints situate d in the lo c ations and let ε > 0 . Ther e is a clustering of the cir cles into rings of d log 1+ ε k (6 /ε ) e c onse cutive cir cles and ther e ar e p ositive inte gers a = O ( ε − 1 ) and b ∈ { 1 , . . . , a } such that if we mark e ach ( b + j a ) -th ring then any k -tour c over U of P c an b e tr ansforme d to a k -tour c over U 0 of the p oints in the unmarke d rings such that 1. no tour in U 0 visits two p oints in P sep ar ate d by a marke d ring, and 2. | U 0 | ≤ (1 + ε 2 ) | U | . F urthermor e, the p oints in the marke d rings c an b e c over e d with k -tours of total length at most ε 2 | U | pr o duc e d by the iter ate d tour p artitioning heuristic fr om [9] (cf. Se ction 2). Pro of . Let t denote a tour obtained b y remo ving its edges incident to O . Supp ose that t crosses one of the marked rings. Let i b e the num b er of the most inner circle of the ring. Denote the circle b y C i . It follo ws by straigh tforw ard calculation and the definition of the circles that each minimal fragment of t crossing the aforemen tioned ring is at least 2 ε times 7 O C i O C i Figure 2: Splitting t in to smaller tours. The grey area is the mark ed ring. In the left picture dotted lines represen t the lines whic h will be added to our solution. The righ t picture shows t w o separate tours obtained from the original tour (one is mark ed with a dashed line, and the other with a solid one), b efore the short-cutting. longer than the doubled radius of C i . W e can appropriately split the tour t along C i in to smaller ones by connecting pairs of crossing p oin ts on C i with O or just with themselves, see Figure 2. The total length of the smaller tours is longer than | t | b y at most ε 2 of the total length of the aforemen tioned fragmen ts of t . W e may assume, without loss of generality , that the aforementioned mark ed ring is the outermost among those crossed by t . W e can iterate the elimination of the crossings of the smaller resulting tours but for their edges incident to O with more inner marked rings. Note that then other disjoint fragmen ts of t will b e charged with the increase of the length of the union of the resulting smaller tours. Finally , b y applying short-cutting, w e can drop the p oin ts in the mark ed rings from the resulting tours. W e conclude that w e can transform U in to a k -tour cov er U 0 of the p oin ts in P in the unmark ed rings such that no tour in U 0 crosses any marked ring (but for its edges incident to O ) and | U 0 | ≤ (1 + ε 2 ) | U | . It remains to sho w that w e can set a and b ∈ { 1 , . . . , a } suc h that one can easily co v er the p oin ts in P contained in the mark ed rings with k -tours of total length not exceeding ε | U | 2 . Let R j denote the set of p oin ts from P lying in the j -th ring. Set a to d 24 ε e . F or each b ∈ { 1 , . . . , a } , let P b b e the set of p oin ts in P in the marked rings, P b = P j ≡ b mod a R j . W e shall sho w that there is some b ∈ { 1 , . . . , a } suc h that b y applying the k -TC heuristic given in Corollary 3 for P b , w e can co ver P b with k -tours of length at most ε | U | 2 . F or this purp ose, w e shall observ e that P j T S P ( R j ) ≤ 3 · T S P ( P ). Supp ose for the sake of this observ ation that the tour t considered in the first part of the pro of is an n -tour, i.e., an optimal TSP tour of P ∪ { O } . Apply almost the same transformation to the tour t as b efore with the exception that instead of connecting the 8 outer cut part b y tw o ra ys to O , w e connect the cutting p oints directly . By the triangle inequalit y , the total length of the so mo dified TSP tour t is at most (1 + ε 2 ) · T S P ( P ). The mo dified TSP tour t can b e easily reduced to the non-necessarily optimal TSP tours of the unmark ed regions by short-cutting. Assuming first for a momen t that the unmark ed rings are the ev en ones, and then con versely , that the unmarked rings are the o dd ones, and that ε < 1 2 , w e conclude that P j T S P ( R j ) ≤ 3 · T S P ( P ). Using F act 2 w e get that X b ∈{ 1 ,...,a } opt ( P b ) ≤ X b ∈{ 1 ,...,a } X j ≡ b ( mod a ) opt ( R j ) = X j opt ( R j ) ≤ X j ( r ad ( R j ) + T S P ( R j )) ≤ r ad ( P ) + 3 · T S P ( P ) ≤ 4 | U | . There must b e some b ∈ { 1 , . . . , a } such that opt ( P b ) ≤ 4 a | U | ≤ ε | U | 6 . Thus, if w e apply the 3-approximation algorithm for the k -tour of P b , whic h is a comp osition of the iterated tour partitioning heuristic with the minimum spanning tree heuristic for TSP , w e obtain a k -tour cov er of P b of length not exceeding ε | U | 2 . u t Theorem 11 The k -TC pr oblem on a set P of n p oints on the plane c an b e r e duc e d to a c ol- le ction of O ( ε − 1 log( n/ε ) / log(1 /ε )) disjoint k -tour c over pr oblems, e ach on O ( k 5 ε − 6 log 2 (1 /ε )) - p oint set and e ach having the maximum distanc e to the origin at most (1 /ε ) O (1 /ε ) lar ger than the minimum one, such that (1 + ε ) -appr oximate solutions to e ach of the latter pr oblems yield a (1 + O ( ε )) -appr oximation to the original k -tour c over pr oblem. The r e duction c an b e done in time O ( n log n ) for a fixe d ε . Pro of . Mov e the p oints to the lo cations and compute the sets R j of input p oints lying in the rings for a fixed ε . This all can b e easily done in time O ( n log n ) by using standard data structures for p oin t lo cation [12]. Next, compute the v alue a (the distance b et w een mark ed rings) and for each b ∈ { 1 , . . . , a } , compute a 3-appro ximate k -tour co ver of the set P b of p oints contained in the marked rings. All the a computations tak e O ( an log n ) = O ( n log n ) time b y Corollary 3. Fix b to that minimizing the length of the aforementioned tour. It follows from Lemma 10 that the pro duced co v er of P b has length at most ε 2 opt . No w w e will hav e to compute appro ximate solutions for each maximal sequence of consecutiv e not marked rings. Let us de- note the n umber of suc h sequences b y q . W e can easily compute that q = O ( ε − 1 log n ε / log 1 ε ). F or i = 1 , . . . , q , let I i denote the set of p oints contained in such i -th sequence. Note that these p oin t sets can b e also easily computed in time O ( n log n ). It follo ws from Lemma 10 that if w e compute separately (1 + ε )-approximation of the optimal cov er with k -tours for each set I i , then the union of these cov erings will hav e length at most (1 + O ( ε )) opt . 9 Note that for a giv en i , the n um b er of lo cations in I i is O ( a · k ε · log (1+ ε k ) 1 ε ) = O ( k 2 ε − 3 log 1 ε ). Hence, b y the discussion in Section 3, we can accoun t to the in tended (1 + ε )-appro ximation of opt ( I i ) the trivial tours decreasing the p oin t-m ultiplicit y in eac h lo cation to O ( k 3 ε − 3 log 1 ε ). Th us, for eac h I i w e can reduce the problem to one with O ( k 5 ε − 6 (log 1 ε ) 2 ) p oin ts. Eac h I i consists of O ( ε − 1 ) consecutiv e rings and for a p oin t in a ring the maximum distance to the origin is at most O ( ε − 1 ) times larger than the minim um one. Hence, for a p oin t in I i the maxim um distance to the origin is at most (1 /ε ) O (1 /ε ) times larger than the minim um one. The appropriate q sets of p oints can b e computed in time O ( n log n ) and they sp ecify the problems to whic h w e appro ximately reduce the original k -tour cov er problem. u t 5 Final remarks In this pap er, w e hav e considered the problem of approximating t wo-dimensional Euclidean k - TC. Prior to our w ork, a PT AS has b een known only for the v alues of k ≤ O (log n/ log log n ) and for k = Ω( n ) [4], and in this pap er we significantly enlarge the set of v alues of k to k ≤ 2 log δ n for some p ositiv e constant δ = δ ( ε ). The main tec hnical con tribution is a reduction of the k -TC problem on n p oints to either that on ( k log n/ε ) O (1) p oin ts, or to a small num b er of indep enden t instances of the k -TC problem on ( k /ε ) O (1) p oin ts. When com bined with a QPT AS for k -TC due to Das and Mathieu [7], this giv es a PT AS for k ≤ 2 log δ n for some p ositiv e constan t δ = δ ( ε ). The cen tral op en question left is whether there is a PT AS for the k -TC problem for all v alues of k . While w e hav e enlarged the set of v alues of k for which a PT AS exists, w e still do not know how to reach p olynomial v alues for k , even k = n 0 . 001 . In particular, a PT AS k -TC for k = Θ( √ n ) is elusive. F or arbitrary v alues of k , the b est curren tly kno wn result is either a quasi-p olynomial time appro ximation scheme b y Das and Mathieu [7] that runs in time n log O (1 /ε ) n , or the p olynomial-time constant-factor approximation algorithm due to Haimo vic h and Rinno oy Kan [9]. Similarly as in [4], w e b elieve that the case k = Θ( √ n ) is the hardcore of the difficult y in obtaining a PT AS for all v alues of k . F ollo wing [9], let us observe that if we divide the range of k , i.e., the in terv al { 1 , . . . , n } , in to a logarithmic n um b er of interv als of the form [ ε − 2 i , ε − 2( i +1) ), then for k in at most one of the interv als none of the inequalities T S P ( P ) ≤ ε · rad ( P ), r ad ( P ) ≤ ε · T S P ( P ) hold. Note that if any of the inequalities holds then by plugging any PT AS for TSP in the iterated tour partitioning heuristics, w e obtain an (1 + O ( ε ))-appro ximation of k -TC. Thus, the aforemen tioned heuristic is in fact a PT AS for a substan tial range of k dep ending on P : for ev ery set of p oints P there is k 0 suc h that there is a p olynomial-time (1 + O ( ε ))- appro ximation algorithm for k -TC for ev ery k ≤ εk 0 and for every k > k 0 /ε . 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