Euclidean Prize-collecting Steiner Forest

Euclidean Prize-collecting Steiner F orest ∗ MohammadHossein Bateni † MohammadT a ghi Ha jiaghay i ‡ Abstract In this pap er, we consider Steiner for est and its gener a lizations, prize-c ol le ct ing Steiner for est and k - Steiner for est , when the vertices of the input gr a ph are p oints in the Euclidean plane and the leng ths are Euclidean distances . First, we present a simpler analysis of the p o lynomial-time approximation scheme (PT AS) of Bo rradaile et al. [1 2] for the Euclide an St einer for est pr o blem. This is done by proving a new structural pr op erty and modifying the dynamic progr a mming by adding a new piece of information to eac h dynamic programming sta te. Next we develop a PT AS for a well-motiv ated ca se, i.e., the mult iplicative cas e, of prize-c ollecting and budgeted Steiner for est. The ideas us ed in the algorithm may have applica tio ns in design o f a broa d class of bicriteria PT ASs. At the end, we demonstr ate why PT A Ss for these problems c a n b e hard in the ge ne r al Euclidean c ase (and t hus for PT ASs we ca nnot go beyond the m ultiplica tive cas e ). 1 In tro duction Prize-colle cting Steiner problems are wel l-known n et wo rk design problems with several applica- tions in expanding telecommunicat ions n etw orks (see e.g. [25, 32]), cost sh aring, and Lagrangian relaxation tec hniqu es (see e.g. [24, 15]). The most general version of these p roblems is called the prize-c ol le cting Steiner for est (PCSF) problem 1 , in which, give n a graph G = ( V , E ), a set of (commod it y) pairs D = { ( s 1 , t 1 ) , ( s 2 , t 2 ) , . . . } , a non-negativ e cost fun ction c : E → Q ≥ 0 , an d finally a non-negativ e p en alt y f unction π : D → Q ≥ 0 , our goal is a minimum-cost way of bu ying a set of edges and paying the p enalt y for those pairs w h ic h are n ot connected v ia b ought edges. When all p enalties are ∞ , the problem is the classic APX-hard Steiner for est problem f or wh ich the b est ap p roximat ion factor is 2 − 2 n ( n is the number of v ertices of the graph) d ue to Go e- mans and Williamson [19]. When all sinks are identical in the PCSF problem, it is the classic prize-collec ting S teiner tree p roblem. Bienstock, Goemans, Sim chi-Levi, and Williamson [8] fir st considered th is problem (based on a problem earlier pr op osed by Balas [3]) for which they gav e a 3-appro ximation algorithm. The current b est ap p roximat ion algorithm for this problem is a recent 1.992- approximatio n alg orithm of Archer, Bateni, Ha jiaghayi , and Karloff [1] improving u p on a primal-dual  2 − 1 n − 1  -approximat ion algorithm of Go emans and Williamson [19]. When in ad- dition all p enalties are ∞ , the p roblem is th e classic Steiner tr e e problem, wh ich is kn own to b e APX-hard [7] and for whic h the b est known app roximat ion factor is 1 . 5 5 [31]. ∗ A short version of this pap er app ears in Pro ceedings of LA TIN 2010 [5]. † Department of Computer Science, Princeton Universit y , Princeton, N J 08540; Email: mbateni@cs.pri nceton.edu . The author w as sup p orted by a Gordon W u fello wship as w ell as N SF ITR gran ts CCF-0205594, CCF-0426582 and NSF C CF 0832 797, NSF CAREER aw ard CCF-023711 3, MSP A-MCS a ward 0528 414, NS F expeditions a ward 0832797 . ‡ A T&T Labs—Researc h, Florham Park, NJ 07932; Email: hajiagha@research.att.com . 1 It is sometimes called prize-c ol le cting gener alize d St einer tr e e (PCGST) in the literature. 1 There are several 3-appro ximation algorithms for the prize-c ol le cting Steiner for est p roblem u s - ing L P round ing, prim al-du al, or iterativ e rou n ding metho ds w hich are firs t in itiated by Ha jiagha yi and Jain [22] (see [8, 23]). Cur r entl y the b est appr o ximation factor for th is problem is a randomized 2 . 54-a pproximation algorithm [22]. The ap p roac h of Ha j iagha yi and Jain has b een generalized by Sharma, S wa my , and Williamson [33] for n et work d esign p roblems w here violating arbitrary 0-1 connectivit y constr aints are allo wed in exc hange for a ve ry general p enalt y fu nction. Lots of attentio n h as b een paid to budgeted versions of Steiner problems as we ll. In the k - Steiner for est (or just k -forest f or abbreviatio n), give n a graph G = ( V , E ) and a set of (commo dity) pairs D , the goal is to fin d a minimum-cost forest that connects at least k pairs of D . The b est cu r rent approximati on factor for this problem is in O (min { √ k , √ n } ) [21]. On the other han d , Ha jiagha yi and Jain [22] could transform notorious dense k -sub gr aph to this problem, for w hich th e current b est approximation f actor is O ( n 1 / 3 − ǫ ) [16]. The sp ecial case in wh ic h we h a ve a ro ot r and D consists of all pairs ( r , v ) for v ∈ V ( G ) − { r } is the well-kno wn NP-h ard k -MST problem. The first non-trivial app roximat ion algorithm for the k -MST problem was give n by Ravi et al. [30], who achiev ed an appr o ximation ratio of O ( √ k ). Later th is approximati on r atio is impr ov ed to a constan t by Blum et al. [9]. C urrently the b est app roximat ion factor f or this p roblem is 2 due to Garg [17]. In this pap er, we consider Euc lide an prize-c ol le cting Steiner for est and Euclide an k -for est in which the vertices of the inp u t graph are p oints in the Euclidean plane (or lo w-dimensional Eu - clidean sp ace) and the lengths are Euclidean distances. F or the Eu clide an Steiner tr e e problem, Arora [2] and Mitc hell [29] gav e p olynomial-time appr o ximation sc hemes (PT AS s). Recently Bor- radaile, Klein and K eny on-Mathieu [12] claim a PT AS for the more general p roblem of E uclide an Steiner for est . 1.1 Problem definition Motiv ated by the settings in which th e demand of eac h pair is the prod uct of the wei ght of th e origin ve rtex and the weigh t of the destination vertex in the p air an d thus in a sen se contributions of each ve rtex to all adjacent pairs are the same (e.g., see pr o duct multi-c ommo dity flow in Leighto n and Rao [27] or [10, 26], and its app licatio ns in wireless netw orks [28] or r outing [13, 14]), we consider the following multiplicativ e version of pr ize-coll ecting S teiner forest for the Euclidean case. In the Multiplic ative prize-c ol le cting Steiner for est (M PCSF) problem, giv en an und irected graph G ( V , E ) with non-n egative edge lengths c e for eac h edge e ∈ E , and also give n weig hts φ ( v ) for eac h vertex v ∈ V , our goal is to fin d a forest F which m in imizes the cost X e ∈ F c e + X u,v ∈ V : u and v are not connect ed via F φ ( u ) φ ( v ) . Indeed, this is an instance of PC SF in which each ordered vertex pair ( u, v ) forms a request with p enalt y φ ( u ) φ ( v ). 2 W e may b e asked to c ol le ct a c ertain prize S , in wh ic h case the goal is to find the forest F of min imum cost for w hich X u,v ∈ V : u and v are connect ed via F φ ( u ) φ ( v ) ≥ S. 2 W e can change the d efinition to u nordered pairs whose treatment requires only a slight mo difications of th e algorithms. Currently , each u nordered pair ( u, v ) has a prize of 2 φ ( u ) φ ( v ) if u 6 = v . 2 Let u s call this pr ob lem S -MPCSF. W e show that this is a generalization of th e k -MST problem (see App endix A.2) and thus cur rently ther e is no approximati on b etter than 2 for this p roblem either. When working on th e Euclidean case, the inpu t d oes not include any Steiner vertic es, as all the p oints of the plane are p otent ial S teiner p oints. A bicriteria ( α, β )-approximate solution for th e the S -MPCSF problem is one whose cost is at most α OPT , yet collects a prize of at least β S . O ur main con tribution in this pap er is a bicriteria (1 + ǫ, 1 − ǫ ′ )-approxima tion algorithm that runs in time exp onential in 1 /ǫ b ut p olynomial in n and 1 /ǫ ′ . W e then use th is algorithm to obtain a PT AS for MPCSF. 1.2 Our con tribution First of all, we pr esent a simp ler analysis for the algorithm of Borradaile et al. [12] for the Euclide an Steiner for est pr oblem and reprov e th e follo w ing theorem. Theorem 1. F o r a n y constant ǫ > 0 , there is an algorithm that runs in p olynomial t ime and appro xi- mates the Euclidean Steiner forest problem wi thin 1 + ǫ of the optimal solution. This is done by mo difying th e dyn amic programming (DP) algorithm s o that instead of storing paths enclosing th e zones in the algorithm b y Borradaile et al., we u se a bitmap to iden tify a zone. The mo dification results in simplification of th e structural prop erty requir ed for the p roof of correctness (See Section 3). W e pr o ve th is structural pr op ert y in Theorem 6. The pro of h as some ideas similar to [12], bu t we p resent a simpler c harging scheme that has a universal treatment throughout. Next we giv e an o ve rview of the dynamic pr ogramming algorithm in S ection 4. W e hav e recent ly come to know that similar simplifications hav e b een indep endently d iscov ered by the authors of [12], to o. Next we extend the algorithm for Eu clidean S -MPCS F and MPCS F pr oblems in Section 5. Theorem 2. F or any ǫ, ǫ ′ > 0 , there is a bicriteria (1 + ǫ, 1 − ǫ ′ ) -appro ximation a lgo rithm for the Euclidean S -MPCSF p roblem, tha t runs in time p olynomial in n, 1 /ǫ ′ and exponentia l i n 1 /ǫ . Notice that ǫ ′ need not b e a constant. In particular, if all weigh ts are p olynomially b oun ded inte gers, we can fi nd in p olynomial time a (1 + ǫ )-approximat e solution that collect s a prize of at least S ; this can b e d one by picking ǫ ′ to b e su fficient ly small ( ǫ ′− 1 is still p olynomial). Next we present a PT AS for Euclide an MPCSF . Theorem 3. Fo r any constant ǫ , there is a (1 + ǫ ) -appro ximation algo rithm for the Euclidean MPCSF pro blem, that runs in p olynomial time. W e also stu dy the case of asymmetric prizes for vertice s in which each vertex v h as tw o types of wei ghts (type one and type tw o) and the prize for an ordered pair ( u, v ) is the pr odu ct of the first type weig ht of u , i.e., φ s ( u ), and the second type wei ght of v , i.e., φ t ( v ). This case is esp ecially inte resting b ecause it generalizes the multiplicativ e prize-col lecting pr oblem when we hav e tw o disjoint sets S 1 and S 2 and we pay the multiplica tiv e p enalty only when two v ertices, one in S 1 and the other one in S 2 , are not connected (by letting f or eac h vertex in S 1 the first typ e weigh t b e its actual weigh t and the second type weigh t b e zero and for each vertex in S 2 the fi r st type weig ht b e zero and th e second type weigh t b e its actual weigh t.) After hinting on the arising complications, we sh o w how we can extend our algorithms for this case as well. 3 Theorem 4. F or any ǫ, ǫ ′ > 0 , there is a bicriteria (1 + ǫ, 1 − ǫ ′ ) -appro ximation a lgo rithm for the Euclidean Asymmetric S -MPCSF p roblem, that runs in time p olynomial in n, 1 /ǫ ′ and exponential in 1 /ǫ . In addition, fo r any constant ǫ , th ere is a (1 + ǫ ) -appro ximation algorithm fo r the Euclidean Asymmetric M PCSF p roblem, that runs in p olynomial time. Indeed, the algorithms in Theorem 4 can b e extended to the case in wh ich there are a constant num b er of different t yp es of weigh ts for each vertex generalizi ng the case in which w e h a ve a constan t number of disjoint sets and we pay the multiplicativ e p enalt y when tw o ve rtices fr om t wo different sets are not connected. Notice that the case of two disjoint sets already generalizes the prize-c ol le cting Steiner tr e e problem (by considering S 1 = { r } and S 2 = V − { r } ) wh ose b est approximati on guarantee is curr ent ly 1 . 992. A t the end, we present in Section 6 why PC SF and k -forest problems can b e APX-hard in the general case (and thus for PT AS s we cannot go b eyond the multiplicativ e case). W e conclude with some op en p r oblems in Section 7. All the omitted pro ofs app ear in the app endix. 1.3 Our techn iques for the prize-collecting version Here, we summarize our tec hniques for the multiplicat ive p rize collecti ng S teiner forest algorithms; see S ection 5. In all th ose algorithms, we store in each DP state extra parameters, includin g the sum of th e wei ghts, as well as the multiplicativ e prize already collected in eac h comp onent. These parameters enable us to carry out the DP up d ate pro cedure. Interestingly , the s u m an d collected prize p arameters hav e their own p recision u nits. In the asymmetric version, a ma jor issue is that n o fixed unit is goo d for all sum parameters. Some may b e s mall, yet h av e significant effect w h en multiplied by others. T o remedy this, we use v ariable units, remin iscen t of the floating-p oint storage formats (mantissa and exp onent). T o the b est of our knowledge , Bateni and Ha jiagha yi [4] were the first to take adv antage of this idea in the con text of (p olynomial time) appr o ximation schemes. Th e basic idea is that a certain parameter in the description of DP states has a large (not p olynomial) range, how ever, as the v alue grows, we can afford to sacrifice m ore on the precision. Thus, we store tw o (p olynomial) integer numbers, say ( i, x ), where i denotes a v ariable unit, and x is the co efficient: th e actual number is then r eco vered by x · u i . The conv ersion b etw een th ese representations is not lossless, bu t the aggregate error can b e b ounded satisfactorily . In Section 5.3 we consider the pr ob lem wh ere the ob jective is a linear fu nction of p enalties paid and the cost of the forest bu ilt. T he challenging case is when th e cost of th e optimal f orest is very small compared to th e p enalties paid. In this case, we identify a set of vertices with large p enalties and argue they hav e to b e connected in the optimal s olution. Then, with a n o ve l trick we show how to ignore th em in the b eginning, and tak e them into accoun t only after the DP is carried out. 2 Preliminaries Let n = | V | b e the total number of terminals and let OPT b e the total length of the optimal solution. A bi tmap is a matrix with 0-1 entries. Two bitmaps of the s ame dimen sions are called disjoint if and only if they do not hav e v alue one at the same entry . Consider t wo partitions P = { P 1 , P 2 , . . . , P |P | } and P ′ = { P ′ 1 , P ′ 2 , . . . , P ′ |P ′ | } o ve r the same groun d s et. Then, P is said to b e a r efinement of P ′ if and only if any set of P is a subset of a s et in P ′ , namely ∀ P ∈ P , ∃ P ′ ∈ P ′ : P ⊆ P ′ . 4 (a) (b) Figure 1: (a) An example of a d issection square with depth 3, and depiction of p ortals f or a sample dissection sq u are w ith m = 8; (b ) the γ × γ grid of cells ins id e a sample dissection square with γ = 4. By standard p er tu rbation and scaling tec hniques, we can assume the following conditions h old incurring a cost increase of O ( ǫ OPT); see [2, 12] f or example. (I) The diameter of the set V is at most d ′ = n 2 ǫ − 1 OPT. (II ) All the vertic es of V and the Steiner p oints hav e co ordinates (2 i + 1 , 2 j + 1) wher e i and j are inte gers. F or simp licit y of exp osition, we ignore the ab ov e increase in cost. As we are going to obtain a PT AS, this increase will b e absorb ed in the fu ture cost increases. W e h a ve a grid consisting of ve rtical and horizonta l lines with equations x = 2 i and y = 2 j where i and j are intege rs. Let L denote the set of lines in th e grid. W e let L b e the smallest p o wer of tw o greater than or equal to 2 d ′ and p erform a diss ection on th e randomly sh if ted b oundin g b o x of size L × L ; see Figure 1(a). F or eac h d issection square R and each side S of R , designate m + 1 equally spaced p oin ts along S (includ ing the corners) as p ortals of R where m is th e smallest p ow er of 2 greater than 4 ǫ − 1 log L . So the s q u are R has 4 m p ortals. There is a notion of level asso ciated with eac h dissection squ are, lin e, or side of a square. Th e b ounding b ox has level zero, and leve l of each other dissection square is one more than the level of its parent dissection square. The lev el of a line ℓ is the minimum level of a squ are R a side of which falls on th e line ℓ . Thus, the fi rst tw o lines dividing the b ounding b o x h a ve level one. If a side S of a square R falls on a line ℓ , we d efine lev el ( S ) = lev el( ℓ ). So level( S ) ≤ lev el( R ). The th ic kness of the lines in Figure 1 denotes their level: the thick er th e line, the low er is its lev el. F or a (possib ly infin ite) set of geometric p oin ts X , let comp( X ) denote the number of connected compon ents of X ; we will use the sh orthand “comp onent” in this paper. With sligh t abuse of notation, ℓ ∈ L is used to refer to the set of p oints 3 on ℓ . In addition, we us e L to denote the union of p oints on the lines in L . Similarly , we us e R to d en ote the set of all p oints on or inside the square R . The set of p oints on (the b oun dary of ) the square R is referred to by ∂ R . T he total length of all line segments in F is denoted by length ( F ). The follo w ing theorem is mentioned in [12] in a stronger form. W e only need its fi rst half whose pro of follows fr om [2]. 3 not necessarily terminals 5 Theorem 5. [12] T h ere is a soluti on F having expected length at most (1 + 1 4 ǫ )OPT such that each dissection square R sat i sfies the following tw o properties: fo r each si de S of R , F ∩ S has a t most ρ = O ( ǫ − 1 ) non-corner comp onents 4 ( boundary comp onents property ); a n d each component of F ∩ ∂ R contains a p ortal of R ( p ortal property ). 3 Structural theorem Let R b e a dissection squ are. Divide R into a regular γ × γ grid of c el ls , where γ is a constant p o wer of tw o determined later; see Figure 1(b). W e s a y R is the owner of these cells. The level of these cells, as well as the new lines they introduce, is defin ed in accordance with the d issection. That is, we assign them lev els as if they are normal dissection squares and w e ha ve contin ued the dissection pro cedure for log γ more lev els. Ther e are several lemmas in the work of [12] to prov e the structur al pr op erty they r equire (this is the main cont ribution of that work). W e mo dify the dyn amic p rogramming defin ition such that its pr o of of correctness needs a simpler structural prop erty . T h e pr oof of this prop erty is simp ler than that in the aforementio ned pap er. Theorem 6. There is a s olut ion F having expected length a t most (1 + 1 2 ǫ )OPT such that each dissection square R satis fi es t h e lo cality property : i f the terminal s t 1 and t 2 are insid e a cell C of R and are connected to ∂ R via F , then they are connected in F ∩ R . The p roof h as ideas similar to [12, Th eorem 3.2, an d Lemm as 3.3, 3.4, 3.5 and 3.9]. W e firs t mentio n and p rov e a lemma we need in order to p rov e Th eorem 6. The lemma more or less app ears in [2 , 12]. Lemma 7. Fo r th e forest F output by Theorem 5, comp( F ∩ L ) ≤ length( F ) . W e can now prov e the main structural result. A side S of a square R is called private if it do es not lie on a side of the parent square R ′ of R . Ob serve that out of any two opp osite sides of a dissection square, exactly one is priv ate. Pro of of Theorem 6. W e start with a solution F satisfying Th eorem 5. Th e fin al solution is pro duced by iteratively find in g the smallest cell C owned by a square R that violates the lo calit y prop erty , and adding σ ( C, F ) to F , where σ ( C, F ) is defi n ed as the u nion of the priv ate sid es of C and any side of C having non-empty intersectio n with F . W e claim th e locality prop erty is realized after fin itely many such additions. If after adding σ ( C, F ) to F , the cell C still violates the lo calit y prop erty , there has to b e exactly t wo opp osite sides of the cell having non-empty intersect ion with F ; otherw ise, the σ ( C, F ) is clearly connected. Ho wev er, in case of the opp osite sides, one midd le side w ill b e a priv ate side of C an d hence included as well . Next, we argue that th e conditions of Th eorem 5 still h old. T ak e a side S of any s q u are R . I f the conditions are to b e affected for S , it has to b e du e to an addition invo lving a cell C that has a s ide S ′ such that (1) S ′ has n on-empty intersecti on with S , and (2) S ′ is add ed to F as part of σ ( C, F ). The condition will b e trivial if S ′ con tains S . Thus, we assume th at C is a sm aller square than R . So S ′ cannot b e a priv ate side of C . Ho wev er, the num b er of comp onents on S cannot increase if S ′ has already an intersect ion with F . 4 Non-corner comp onents are those not including any corners of squares. Note that eac h square can hav e at most four corner comp onents. 6 Finally we show that the additional length is not large. Let F ∗ = F ∩ L , and let G = { ( x, y ) : x = 2 i, y = 2 j } b e the set of all grid p oin ts. W e will charge the additions to the connected comp onents of F ∗ − G . Notice th at comp( F ∗ − G ) ≤ comp( F ∗ ) + 3 | F ∗ ∩ G | (1) ≤ comp( F ∗ ) + 3 · (length( F ∗ ) + comp( F ∗ )) (2) = 4 comp( F ∗ ) + 3 length ( F ) ≤ 7 length( F ) , by Lemma 7 . (3) Inequalit y (1) holds b ecause r emov al of eac h grid p oint on F ∗ increases the number of comp onents by at most three. T o obtain (2), n otice that in an y connected component of F ∗ , the distance b et wee n any tw o p oints of F ∗ ∩ G is at least 2. Hence, if there are m ore than one su c h p oints, there cannot b e more than length( F ∗ ) ones. W e charge this addition to a connected comp onent of ( ∂ R ∩ F ) − G , in such a wa y that eac h connected comp onent is c harged to at most twice: once from eac h side. F or simp licit y , we dup licate eac h connected comp onent of ( F ∩ ℓ ) − G : they corresp ond to squares from either s id e of ℓ . F or any dissection square R , let C R refer to th e connected comp onents of F ∩ R th at reach ∂ R . F urther, let K R b e the set of connected compon ents of ( F ∩ ∂ R ) − G . When σ ( C, F ) is add ed wh ere R is th e o wner of C , there are k ≥ 2 comp onents c 1 , . . . , c k ∈ C R that b ecome connected. An y element of K R connected via F ∩ R to a comp onent c ∈ C R is said to b e an interfac e of c . The addition will b e charge d to a fr e e interface of some c ∈ C R with m axim um level. T his elemen t will no longer b e free for the rest of the pro cedure. W e argue this p r ocedu r e successfully charges all th e add itions to app ropriate b order comp onents. T o this end, we sh ortly prov e the follo wing str onger claim via induction on the number of additions p erf ormed. W e call a dissection squ are R v i olate d if the localit y p rop erty d oes not hold for a cell C owned by R . Claim 8. At all times during the exec ution of this procedure, any com p onent c ∈ C R has a free interface, for each violated square R . As a result, any addition can b e charg ed to a free component. The second statement of the claim follo ws fr om the first part. T h e first part is pr o ve d as follo w s. The claim clearly h olds at the b eginnin g, since all interfaces are free, and eac h comp onent has an inte rface. Su pp ose the addition σ ( C, F ) is p erformed and let R b e th e owner of C . W e sh o w any dissection square R ′ will stay fine. Notice that the size of the squares R for which the addition is p erformed is increasing in time. Hence, any dissection square R ′ smaller than R is irrelev ant in the statemen t of the claim, since they cannot b e violated. F or R itself, eac h c i has at least one free inte rface. O n e of the interfaces is used , and thus the new comp onent formed by their union has a free interface . Supp ose for the sake of reac hing a contradict ion that a comp onent c ′ ∈ C R ′ has no free interface after the addition. T hus R ′ con tains R , and the charging was not done to a pr iv ate side of R . Recall th at prior to the addition, c ′ is connected to some comp onents of C R with at least t wo free interfaces in R . One of them s till remains free. W e charged to the interface of maximum lev el and it was in ∂ R ′ . Hence, the free interface is also in ∂ R , leading to a contradict ion. Let (the rand om v ariable) c ℓ,j denote the number of c harges to comp onents on ℓ ∈ L due to cells C owned by squares R of leve l j . Ind epen dently of the randomness P ℓ P j c ℓ,j ≤ 2 comp( F ∗ − G ) by the ab ov e discussion and Claim 8. Note the cost of adding σ ( C, F ) (c harged to a comp onent on R ) is at m ost 4 L ′ /γ where L ′ is the sid e length of R . The total increase d ue to charges to ℓ is at most P j ≥ depth( ℓ ) c ℓ,j 4 L γ 2 j where L is the side length of the b ound ing b ox. Due to the randomization 7 in the d issection, we hav e Pr [depth( ℓ ) = i ] = 2 i /L ; see [2] for ins tance. The exp ected increase in length is thus X ℓ X i 2 i L X j ≥ i c ℓ,j 4 L γ 2 j ≤ 4 γ X ℓ X j c ℓ,j 2 j X i ≤ j 2 i ≤ 8 γ X ℓ X j c ℓ,j ≤ 16 γ comp( F ∗ − G ) by Claim 8 ≤ 112 γ length( F ) by (3) . W e p ick γ to b e the smallest p ow er of two larger than 112(1 + ǫ ) · 2 ǫ − 1 to fi nish the pro of. Therefore, w ith probability 1 / 2, we hav e length ( F ) ≤ (1 + ǫ )OPT. In the entire argument, no attempt was made to optimize th e parameters. 4 The algorithm A subsolution for R is a fin ite set of lin e segmen ts F ⊂ R satisfying conditions of Theorems 5 and 6, with the extra prop erty that any terminal t in R is connected via F either to its mate or to ∂ R . A c onfigur ation χ = ( K , P ) for R has tw o p ortions: a set K of p airs κ i = ( P i , M i ) and a partition P whose ground set is K , such th at • P i is a su bset of p ortals of R ; • M i is a b itmap of size γ × γ ; • P i and P j are disjoint if i 6 = j ; • the total number of p ortals, namely P i | P i | , is at most 4( ρ + 1); and • bitmaps M i and M j are disjoint if i 6 = j . The configuration captur es suffi cien t inform ation ab out F so as to m ak e it p ossible to tak e care of the interac tion b etw een R and the outside. In particular, eac h pair ( P , M ) d escrib es a connected compon ent of F , by sp ecifying the set of p ortals on its b oundary and the set of cells connected to these p ortals. Roughly sp eaking, the p artition P tells us wh ic h comp onents κ i and κ j need to b e connected from outside R : this imp lies the existence of a pair of terminals that are in κ i and κ j , resp ectiv ely , b ut they are n ot connected in R . W e will see b elow w hy this restrictive abstraction do es not lose any crucial subs olutions. W e say a sub solution F is c omp atible with a configur ation χ = ( K , P ) if 1. for any connected comp onent κ of F th at intersects ∂ R , there exists a pair κ ′ = ( P , M ) ∈ K such that • κ sp ans P ; • eac h conn ected comp onent of κ ∩ ∂ R con tains a p ortal of P ; 8 • the bitmap M has v alue one in the p ositions corresp onding to any cell C con taining a terminal t of κ ; and 2. an y terminal pair located in d ifferent comp onents κ 1 and κ 2 of K are either connected via F ∩ R , or κ 1 and κ 2 are in the same set of P . 4.1 The dynamic programming In the dynamic p rogram, we build a table T R [ χ ], indexed b y configur ations for eac h dissection square R . Th e goal is to p opulate this table so th at T R [ χ ] is the minimum length of a subs olution for R that is compatible with χ . First of all, we show that for each R , the num b er of configurations is sm all. Consider χ = ( K , P ). There are at most λ = 4( ρ + 1) pairs in K . F or a particular κ = ( P , M ), there are P λ i =0  m +1 i  = O ( m λ +1 ) options for th e set of p ortals P . The bitmap M has 2 γ 2 p ossibilities. A crude up p er b ound of 2 λ 2 is trivial for p ossibilities of P . Thus, the total num b er w ill b e at most Φ = h O  m λ +1  · 2 γ 2 i λ · 2 λ 2 = O (p oly( m )) = O (p oly log( n )) . Theorems 5 and 6 guarantee the existence of a near-optimal solution all whose subp roblems are compatible with a configuration: The connected comp onents of F reaching ∂ R can b e decomp osed into disjoint bitmaps b ecause of T h eorem 6. Theorem 5 on the other hand ensu res each conn ected compon ent on ∂ R contains a p ortal, and the total number of such comp onents is small. The details of the DP up d ate, as well as its correctness pr oof, app ears b elow. The final solution of the p r oblem is obtained from the minimum T R [ χ ] where R is the b oundin g b o x, and P of χ do es not require any connections: i.e., all sets of the partition are singletons. Th is wo uld imply all the n ecessary connections hav e b een made inside R . T o actually construct the solution, we need to store add itional inform ation in eac h d y n amic p rogramming state indicating which configurations it was last u p dated fr om. It is then straight forward to recur s iv ely construct the solution, by taking the union of the p ertinent configurations. Here we show how th e dyn amic programming table f or Eu clidean pr ize-collecting Steiner forest is u p dated f r om the already-computed v alues. And finally we show why th e upd ate routine is sound and complete. The table T R [ χ ] is p opu lated in the order of increasing size for R . F or a base dissection squ are R , finding the v alue of T R [ χ ] is s traigh tforward. Notice that there is at most one p oint (p ossibly w ith several terminals collocated) inside R . Depend ing on w h ether the mates of those terminals are collocated w ith them or not, we may need to connect some of them to the b oundary ∂ R . There are on ly a constant number of p ortals in χ , h ence we can go ov er all the ways to connect them u p and find the smallest v alue. Note that there cannot b e any Steiner p oin t inside R . No w we get to the up date ru le. C on s ider a dissection square R and a corresp ond ing configur ation χ = ( K , P ). Let R i for i = 1 , 2 , 3 , 4, be the c hildren of R in th e dissection. T ake corresp ond ing configurations χ i = ( K i , P i ). Notice th at each cell of R consists of exactly four cells of one R i . W e can expan d a b itmap M of R i to a bitmap M ′ of d imensions 2 γ × 2 γ for R , by placing three all-zero bitmaps of dimensions γ × γ at appr opriate lo cations around M . W e do this in su ch a wa y that the p ortion corresp ondin g to M still p oints to R i inside R . Consider all the comp onents κ = ( P , M ) corresp onding to the four subsquares, expand their bitmap, and collect th em in K 1 . Merge the partitions P i to get P ′ . If there is a terminal pair ( s, t ) w h ere s is in R i and t is in a different R j , 9 Algorithm E uclideanSteinerF orest Input: Set of terminals V i n the plane, and set D of p airs of terminals Output: A for est F c onne cting p airs in D 1. Carry out the p ertur bation and scaling. 2. Let L b e smallest p owe r of tw o larger than 2 n 2 ǫ − 1 d , wher e d is the maximum distance of a pair. 3. P erform a random dissection in the b oun ding b ox of side L . 4. Place m + 1 p ortals on eac h side of a dissection square, where m is the smallest p ow er of tw o larger than 4 ǫ − 1 log L . 5. Solv e th e base cases T R [ χ ] for leaf d issection squares R : Go ov er all p ossible wa ys of connecting the p ortals an d the center p oint. 6. P opulate the table T R [ χ ] in increasing order of size for R : F or any χ = ( K , P ) corresp onding to R consisting of R 1 , . . . , R 4 : (a) Go ov er all configurations χ i = ( K i , P i ) corresp onding to R i . (b) Build K 1 from the un ion of all comp onents of K i with exp anded bitmaps. (c) Build P ′ from the un ion of P i . (d) If there is a termin al p air ( t 1 , t 2 ) where t 1 ∈ R i 1 and t 2 ∈ R i 2 for i 1 6 = i 2 , • If th ere is n o bitmap in χ i 1 (or χ i 2 ) containing the cell containing t 1 (or t 2 resp ectiv ely), the confi gu r ation is bad. • Otherwise, m erge the sets corresp ondin g to the appropr iate comp onents in P ′ . (e) Build K 2 by merging comp onents having the same p ortals, and make appr opriate c hanges to P ′ . (f ) Build K 3 by removing p ortals not on ∂ R . (g) If any comp onent with empty p ortal set has u n satisfied connectivit y requ irement in P ′ , the cur rent configurations are not consisten t. (h) Build K 4 by eliminating comp onents with empty p ortal set. (i) If any b itmap con tradicts the locality prop erty , these configurations are not con- sistent . (j) I f the configurations are consistent, up date T R [ χ ] with . min ( T R [ χ ] + 4 X i =1 T R i [ χ i ] ) . 7. Find the fi nal solution among T R [ χ ] wh ere R is the b oun ding b ox and χ has no unsat- isfied requirement. 8. Construct the solution F by r ecursively following the v alues f rom T R [ χ ]. Figure 2: The algorithm for Euclide an Steiner for est p roblem. 10 there sh ould b e a comp onent corresp onding to eac h of these in R i and R j , resp ectiv ely . Otherw ise, these configur ations do n ot corresp ond to any (v alid) sub s olution. Merge th e sets corresp onding to these comp onents in P ′ : i.e., they hav e to b e connected. Next mer ge any tw o comp onents of K 1 if they s hare a p ortal, and build K 2 . F ur ther, mak e appropriate change s in P ′ . Build K 3 by remo ving from K 2 all p ortals not on ∂ R . Some of these comp onents reach ∂ R and some do not, namely those with an empty p ortal set P . If th er e is any comp onent w ith emp ty p ortal set that is not one partition s et, we deem the configurations χ i as inc onsistent : in this case, some comp on ents that are required to b e connected together d o not reac h the b oun dary . Otherwise, remov e all the pairs in K 3 with empty p ortal set to obtain K 4 . No w, if there is a cell of R whose four constituent cells reac h the b oundary as more than one connected comp onent, the configur ations are not consisten t either: this contradicts the prop erty of Theorem 6. Finally , reduce the dimensions of th e bitmaps to γ × γ such that a cell of the new bitmap acquires v alue on e if and only if there is a one in one of the p ositions corresp onding to the constituent cells in the original bitmap. No w, χ = ( K , P ) is s aid to b e consistent with the four confi gurations χ 1 , . . . , χ 4 if an d only if P contains all the requirements of P ′ , i.e., P ′ is a r efi nement of P , and in add ition, th er e exists a κ = ( P, M ) ∈ K for any κ ′ = ( P ′ , M ′ ) ∈ K ′ such th at P ⊆ P ′ and M = M ′ . In case these configur ations are consistent , T R [ χ ] will take the minimum of its current v alue and P i T R i [ χ i ]. Y ou can refer to Figur e 2 for a summary . 4.2 Pro of of correct ness Correctness follo ws fr om induction on the size of the squ are R that all dynamic programming states hav e their intended v alue. In particular, we know that there is a near-optimal solution all whose subsolutions are compatible w ith one configur ation. He nce, th ese will b e compu ted correctly and giv e th e fin al solution. More sp ecifically the following claim holds for all DP states. Lemma 9. A dynamic programming state T R [ χ ] ends up hav i ng the minimum va l ue corresponding to a solution F of R , such that for any dissection square R ′ which is a descendant of R in the dissection tree, th e subsoluti on F ∩ R ′ of R ′ is compatible with a configuration χ ′ fo r R ′ . No w, we are at th e p osition to p rov e the main Theorem regarding the Euclide an Steiner for est problem. Pro of of Theorem 1. By Lemm a 9, the prop osed d y n amic programming is sou n d and complete. There are Φ = O (p oly( n )) DP states. T o solve eac h non -b ase state, we go o ver at most Φ 4 c hild states and then p erform a p olynomial consistency c hec k. Each base case s tate is computed in constan t time. Hence, the total algorithm runs in time O (p oly( n )). 4.3 Highligh ts of the new ideas Here, we p oint out the d ifferences b etw een our work and the p revious work of [12]. Borrad aile et al. use closed p aths to identify the connected zones of the dissection squ are. These paths consist of ve rtical and horizonta l lines and all the br eak-points are the corners of the cells. As part of their structural prop erty , they p r o ve that they can guarantee a solution in which these zones can b e identified via p aths whose total length is at most a constant η times the p erimeter of the squ are R . Then eac h p ath is repr esent ed by a c hain of { 1 , 2 , 3 } of length at most O ( η γ ): the three v alues are used to denote moving one unit forward, or tur ning to the left or right. This results in a storage of 3 O ( ηγ ) which is a constant p arameter. Instead, we use a bitmap of size γ × γ to addr ess this issue. 11 Eac h zone is repr esent ed by a bitmap th at h as an entry one in the cells of th e zone. T he b oun d that we obtain, 2 γ 2 , ma y b e slightly worse than th e previous work, howe ve r, a simp ler stru ctural prop erty , namely the lo calit y pr operty , suffices as the pro of of correctness. Borradaile et al. in con trast need a b ound on the total length of the zone b oun daries, as noted ab o ve . In addition to the s implification made due to this change, b oth to the p ro of and the treatment of the dynamic programming, we simplify the p roof fu r ther. Borradaile et al. c harge the additions of σ ( C, F ) to three different structures, and th e argument is describ ed and analyzed separately for eac h . W e manage to p erform a universal treatment and charging all the additions to the simplest of the three structures in th eir work. But this can b e d one only after showing F ∗ − G has a limited num b er of comp onents. The pr oof is simp le yet elegant—a weak er claim is p rov ed in [12], b ut ev en the statement of the claim is h ard to r ead. 5 Multiplicativ e prizes W e first tac kle the S -multiplic ative prize-c ol le cting Steiner for est p roblem. Then, we will tak e a look at its asymmetric generalization. Finally , we show how the multiplic ative prize-c ol le cting Steiner for est p roblem can b e reduced to S -MPCSF. 5.1 Collecting a fixed prize Supp ose we are giv en S , the amount of prize we s h ould collect. Let OPT b e the minimum cost of a forest F that collects a prize of at least S , and su pp ose Q ⊆ D is the set of terminal p airs connected v ia F . W e sh ow how to find a forest with cost at m ost (1 + ǫ )O P T th at collects a prize of at least (1 − ǫ ′ ) S . By the structur al prop erty , we know that there is a solution F ′ connecting the same set of terminal pairs Q wh ose cost is at most (1 + ǫ )O P T, yet it s atisfies the conditions of Theorems 5 and 6. Roun d all the vertex weig hts down to th e next int eger multiple of θ = ǫ ′ √ S / 2 n . In a connected comp onent of F ′ of total weigh t A i that lost a weigh t a i due to round ing, the lost prize is A 2 i − ( A i − a i ) 2 ≤ 2 a i A i ≤ 2 a i √ S , b ecause the total weigh t of the comp onent is at most √ S . Thus, F ′ collect s at least S − 2 nθ √ S ≤ (1 − ǫ ′ ) S from the round ed weigh ts. Eac h dyn amic programming state consists of a d iss ection square R , a set of comp onents K , and a new p arameter Π which denotes the total prize collected inside R by conn ecting the terminal pairs. Each element of K —corresp ondin g to a connected comp onent in the sub solution—now has the form κ = ( P , Σ) wh ere P denotes th e p ortals of κ , and Σ is the total s um of the weigh ts in κ . The DP is carried out in a fashion similar to that of [2]. The v alues of Σ and Π are easy to determine f or the base cases. It is n ot difficult to up date them, either. When ev er tw o comp onents κ 1 = ( P 1 , Σ 1 ) and κ 2 = ( P 2 , Σ 2 ) m erge in the DP , the sum Σ for the new comp onent is simp ly Σ 1 + Σ 2 . Besides, the merge increases th e Π v alue of the DP state by 2Σ 1 Σ 2 . Pro of of Theorem 2. The soundness and completeness is simple and is along the same lines as the pro of of Th eorem 1. Carr y in g out the ab ov e op eration assumes th e v alues of Σ and Π could b e stored accurately . Ho wev er, as they describ e the d ynamic programming states, th eir size sh ould b e su ffi cien tly small or else th e algorithm will not run in p olynomial time. Here do es th e roun ding help us. All v alues of Σ are stored as multiples of θ and the v alues of Π are stored as multiples of θ 2 . Notice that as we roun d the vertex weigh ts at the b eginnin g, throughout the algorithm the v alues of Σ and Π will b e multiples of their resp ectiv e un its. Hence, n o extra pr ecision err or will o ccur and we find the aforementio ned solution. If at any time dur ing th e execution of the algorithm, 12 the v alue of Σ goes ab ov e √ S , we trun cate it to √ S . Similarly , the v alue of Π is not allo we d to surpass S . This d oes not eliminate any solution, b ecause at the p oint of truncation, th e subs olution has already gathered sufficient p rize. Hence, the range of Σ is fr om zero u p to √ S , and this giv es √ S /θ = 2 n/ǫ ′ different v alues. Similarly f or Π, there are at most S/θ 2 = 4 n 2 /ǫ ′ 2 options. There are at m ost Φ 1 = h O  m λ +1  · 2 γ 2 · 2 n/ǫ ′ i λ · 2 λ 2 · 4 n 2 /ǫ ′ 2 = O  p oly  n, 1 ǫ ′   DP states f or each square R . T h e ru nning time is p olynomial in Φ 1 and the claim follows. T o start the algorithm, we n eed to guarante e the instance s atisfies the conditions at the b egin- ning of Section 2. See App end ix A.1 for d etails of how this is ac h iev ed. 5.2 The asymmetric prizes The b asic id ea is to store tw o parameters Σ s and Σ t for each comp onent of K . These parameters store the total weigh t of the first and second t yp e in the comp onent, namely P i φ s i and P i φ t i , resp ectiv ely . The d ifficulty is that to collect a p r ize of A = A s A t in a comp onent, only on e of the parameters A s or A t needs to b e large. In particular, we cannot do a round ing with a pr ecision lik e ǫ ′ √ A/n . It may even happ en that A s is large in one comp onent, whereas we hav e a large A t in another. In fact, we cann ot store the v alues of th e Σ s or Σ t as multiples of a fixed unit. T o get around the problem, Σ s is stored as a pair ( v , x ), wh ere v is a vertex of the graph and x is an inte ger. T ogether they show that Σ s is x · ǫ 1 φ s ( v ) /n 2 ; the v alue of ǫ 1 will b e chosen later, and v is supp osed to b e the vertex of largest type-one weigh t p resent in the comp onent. A similar provision is made for Σ t . Finally , the v alue of Π is stored as a multiple of ǫ 2 A/n ; we will shortly pick the v alue of ǫ 2 . Wheneve r Σ s 1 = ( v 1 , x 1 ) and Σ s 2 = ( v 1 , x 1 ) are added to giv e Σ s = ( v , x ), we do th e calculatio n as follo w s: let v b e the vertex v 1 or v 2 that has the larger φ s v alue, and then x =  x 1 φ s ( v 1 ) /n 2 + x 2 φ s ( v 2 ) /n 2 ǫ 1 φ s ( v ) /n 2  . Pro of of Theorem 4 . The precision error for Σ s = ( v , x ) is at m ost n · ǫ 1 φ s ( v ) /n 2 = ǫ 1 φ s ( v ) /n , b ecause there is an accumulati on of at most n round ing errors eac h of which h as b een less than ǫ 1 φ s ( v ) /n 2 . Notice that if Σ s is stored in terms of the vertex v , it has to include v and thus its t yp e one weigh t is at least φ s ( v ). Hence, the precision error is at most a ǫ 1 /n multiplicativ e factor. Therefore, when we do a multiplicat ion of Σ s Σ t to get an addition to Π, the error is at most a multiplica tiv e 2 ǫ 1 /n : (1 − ǫ 1 /n ) A s (1 − ǫ 1 /n ) A t ≥ (1 − 2 ǫ 1 /n ) A s A t . Next a rou n ding err or may happ en to s tore the v alue in terms of ǫ 2 A/n . Each Π on the other hand is made up of at most n addition terms, so the total error is at most n (2 ǫ 1 /n + ǫ 2 /n ) A . W e p ic k ǫ 1 = ǫ 2 = ǫ ′ / 3 to conclude that the total error is b ound ed by ǫ ′ A . All the discussion applies to Σ t as well. Due to trun cation and roundin g, there are at most n/ǫ 2 options for Π. And each Σ s (or Σ t ) has at most n 2 /ǫ 1 p ossibilities. Thus, the total num b er of DP states for eac h dissection squ are is Φ 2 = p oly( n, 1 /ǫ ′ ). Therefore, we obtain a bicriteria approximati on to the asymmetric v ariant of the p roblem. 13 5.3 The prize-collecting v ersion: trade-off b et ween p enalt y and forest cost In the p rize-colle cting v ariant, we pay for th e cost of the forest, and for the p rizes not collected. If the total weigh t is ∆, the prize not collecte d is ∆ 2 minus the colle cted prize. One d ifficulty here is to determine the correct r ange f or the collected pr ize so that we can u se the algorithm of Section 5.1. The tr ivial range is zero to ∆ 2 . Ho we ver, the round ing precision we pick for the p en alties should also tak e into account the cost of th e f orest. If the cost of the intended solution is much smaller than ∆ 2 , we cann ot simply go with roundin g errors like ǫ ∆ /n . Otherwise, the error caused due to rounding the p enalties will b e too large compared to the s olution v alue. The trick is to fin d an estimate of the solution v alue, an d then consider tw o cases d ep en ding on how the cost compares to the total p enalty . Using a 3-approximation algorithm, we obtain a solution of v alue ω . W e are guarantee d that OPT ≥ ω / 3. If ∆ 2 ≤ ω / 3, the optimum solution is to collect no prize at all. Otherwise, assume ∆ 2 > ω / 3. T o b eat the solution of v alue ω , we should collect a pr ize of at least ∆ 2 − ω . W e fir st consider the simpler case when ω / ∆ 2 > 1 /n 2 : F or an ǫ ′ > 0 wh ose precise v alue will b e fix ed b elow, we use the algorithm of Section 5.1 to fin d a bicriteria (1 + ǫ/ 2 , 1 − ǫ ′ )-approxima te solution for collecting a pr ize S ; this is done for any S which is a multiple of ǫ ′ ∆ 2 in range [(1 − ǫ ′ )∆ 2 − ω , ∆ 2 ]. W e select the b est one after addin g the u ncollected pr ize to eac h of these solutions. Supp ose th e optimal solution OPT collects a prize S ′ . Let OP T f = OP T − (∆ 2 − S ′ ) b e the length of the forest. Round S ′ down to the next multiple of ǫ ′ ∆ 2 , say S . F ed with p rize v alue S , the algorithm finds a solution that collects a prize of at least (1 − ǫ ′ ) S with f orest cost at most (1 + ǫ/ 2)OPT f . Claim 10. T h e total cost of this solution is at most (1 + ǫ )O P T i f ǫ ′ = min ( ǫ 3 , 1 ) 6 n 2 . Pro of. The total cost of this solution is  1 + ǫ 2  OPT f +  ∆ 2 − (1 − ǫ ′ ) S  ≤  1 + ǫ 2  OPT f +  ∆ 2 − (1 − ǫ ′ )(1 − ǫ ′ ) S ′  ≤ OPT + ǫ 2 OPT f + (2 ǫ ′ + ǫ ′ 2 ) S ′ = OPT + ǫ 2 OPT + (2 ǫ ′ + ǫ ′ 2 ) S ′ OPT OPT ≤ OPT + ǫ 2 OPT + (2 ǫ ′ + ǫ ′ 2 ) ∆ 2 OPT OPT ≤ OPT + ǫ 2 OPT + (2 ǫ ′ + ǫ ′ 2 )3 n 2 OPT (4) ≤ OPT + ǫ 2 OPT + ǫ 2 OPT (5) = (1 + ǫ )OPT , where (4) follows from ∆ 2 OPT ≤ n 2 ω ω / 3 = 3 n 2 , and (5) u ses the defin ition of ǫ ′ . The other case, i.e., ω / ∆ 2 ≤ 1 /n 2 , is more chall enging. Notice that in order to carry out the same pro cedure in this case, ǫ ′ ma y n ot b e b oun d ed by 1 / p oly( n ) an d thus the run ning time may not b e p olynomial. The solution, how ever, has to collect almost all the prize. Thus, one of the connected comp onents includes almost all the ve rtex weig hts. W e set aside a subset B of ve rtices of large weigh t. The ve rtices of B hav e to b e connected in the solution, or else the paid p en alt y 14 will b e to o large. T h en, dynamic programming pro ceeds by ignoring the effect of these vertices and only keeping tabs on how many vertic es fr om B exist in each comp onent. A t the en d , we only tak e into account the solutions that gather al l the vertices of B in one compon ent and compute the actual cost of those solutions and pick the b est one. In the following, we provide the details of our method and p rov e its correctness. Let B b e th e set of all vertices w hose weigh t is larger than nω / ∆. Lemma 11. All the vertices of B are connected in the optimal soluti on. Pro of. There are at most n comp onents, s o there is a comp onent, say C , w hose total weigh t is not less than ∆ /n . W e claim all the vertices of B are insid e this comp onent. The p en alty paid by the optimal solution is at most ω ≤ ∆ 2 /n . If there is any vertex of B outside C , the p enalty of the solution is more than ∆ /n · nω / ∆ = ω , yieldin g a contradictio n. Next, w e r ou n d up all the weigh ts to the next multiple of θ = ǫ ′ ω / ∆ for vertice s not in B . Define O PT ′ as the optimal solution of the resulting instance. Let OP T f b e th e length of the forest in OPT, and defi ne OPT ′ f similarly . Let OPT π and OPT ′ π denote the p enalty paid by O PT and OPT ′ , resp ectiv ely . Ass u me that ǫ ′ ≤ 1. Lemma 12. OPT ′ π ≤ OPT π + 12 nǫ ′ OPT . Pro of. W e recompute th e p enalties paid by OPT usin g the roun ded weigh ts. Th e pair ( s , t ) not connected in OPT is either of the tw o kinds : (1) one of s and t is in B ; or (2) none of th em is in B . The total round ing err or for the p enalties of the fi rst typ e is b ounded by n ∆ θ . There are at most n 2 pairs of th e second t yp e. Since the weig hts of th ese terminals are at most n ω / ∆, the er r or is not more than n 2 [2( nω / ∆) θ + θ 2 ]. Hence, the total error is at most n 2 [2( nω / ∆) θ + θ 2 ] + n ∆ θ ≤ n 2  ( ǫ ′ 2 + 2 nǫ ′ ) ω 2 ∆ 2  + n ǫ ′ ω ≤ n 2  3 nǫ ′ ω 2 ∆ 2  + n ǫ ′ ω b ecause ǫ ′ ≤ 1 =  3 n 2 ω ∆ 2 + 1  nǫ ′ ω ≤ 4 nǫ ′ ω b ecause ω ∆ 2 ≤ 1 n 2 , which is no more than 12 nǫ ′ OPT as d esired. Supp ose we use a dynamic p rogramming appr oac h similar to th e previous sub sections to fi nd the approximately min imum forest length for any sp ecified collect ed prize amount; in particular, we obtain a b icriteria (1 + ǫ/ 2 , 1 − ǫ ′ )-approxima te solution. Dur ing th is pro cess, we ignore the we ight s associated with vertices in B . Consider a DP state χ = ( K , Π) corresp onding to a diss ection squ are R . Each comp onent κ ∈ K lo oks lik e ( P , Σ , µ ): the n ew piece of information, µ , is an intege r num b er denoting the number of vertice s of B insid e κ . Extending the p revious algorithm to p opu late the new DP table is simple. Finally , we look at all the configu r ations χ for the b oundin g b ox su c h that the µ v alue of one comp onent is exactly |B | w hereas it is zero for all other comp onent s. This guarant ees that all elements of B are inside the f ormer comp onent and hen ce we can add u p the p enalties invo lving those vertices. Let K = { κ 1 , κ 2 , . . . , κ q } w here κ i = ( P i , Σ i ), and let κ 1 b e the 15 compon ent containing B . The additional cost d ue to vertice s of B is X v ∈B φ ( v ) ! · q X i =2 Σ i ! . Finally , we r ep ort th e b est solution corresp onding to these configurations. Pro of of T heorem 3. L et u s fi rst s ee that the algorithm describ ed r uns in p olynomial time. It is sufficient to b oun d the number of configurations. The new p iece of information has at most n p ossibilities. F urther, Σ ≤ n 2 ǫ ′ θ is alwa y s a multiple of θ . Similarly , Π will not exceed n 4 ǫ ′ 2 θ 2 and is alw a ys a multiple of θ 2 . W e p ick ǫ ′ = 1 24 n . By Lemmas 11 and 12, the round ing do es n ot increase the p enalties p aid by the optimal solution by more than ǫ / 2OPT. W e then utilize the algorithm describ ed for S -MPCSF to find a solution of cost at m ost (1 + ǫ/ 2)OPT f + OPT π + ǫ/ 2OPT ≤ (1 + ǫ )OPT . Finally , c hanging the weigh ts back to the original v alues clearly do es not increase the cost. 6 Evidence for Hardness So far PT ASs for geometric problems in E uclidean p lane including our s and those of Arora [2] and Mitchell [29] can be easily generalized for Euclidean d -dimensional space, for any constant d > 2. How ever we can pr o ve the f ollo win g th eorem on the hardn ess of th e problem for Euclidean d -dimensional space. Theorem 13. If noto rious densest k -subgraph is hard to appro xima te within a factor O ( n 1 d ) for some constant d , then for any d ′ > 2 d + 1 , the k -fo rest problem in Euclidean d ′ -dimensional space is hard to appro ximate wi t hin a factor O ( n 1 2 d − 1 d ′ − 1 ) . Pro of. Ha j iagha yi and J ain [22] sh ow that if densest k -subgraph is hard to approximate within a factor O ( n 1 d ), then th e k -forest problem on stars is hard to approximate within a factor O ( n 1 2 d ). On the other hand , Gupta [20] shows that a tree metric of s ize n can b e embedd ed into Euclidean d ′ -dimensional space with d istortion in O ( n 1 d ′ − 1 ). Thus for any d ′ > 2 d + 1, we cannot obtain an approximati on factor o ( n 1 2 d − 1 d ′ − 1 ) for k -forest in Eu clidean d ′ -dimensional s pace, since otherwise by solving the prob lem in Euclidean d ′ -dimensional space, fi nding an Eulerian tour and shortcutting it, an d fi nally emb edding it back int o the star, we can obtain a b etter approximatio n than O ( n 1 2 d ), a con tradiction. Note as mentioned ab ov e th at, desp ite extensive stud y , the curr ent b est app ro ximation factor for n otorious d en sest k -sub graph is O ( n 1 / 3 − ǫ ) [16] and thus we do n ot exp ect to hav e any PT AS for k -forest in 8-dimens ional Eu clidean space. Unlike the general cases of these problems, as far as PT ASs f or th e case of Euclidean spaces are concerned, it seems k -for est and prize-c ol le cting Steiner for est problems are essentially equiv alent. Indeed in L emm a 15, we pr ov e that any PT AS for k -for est r esults in a PT AS for prize-c ol le cting Steiner for est , and we b elieve that any DP algorithm giving a PT AS for PCSF compu tes along its wa y the optimal solution to different k -forest instances. Thus based on the evid en ces ab ov e, we do b eliev e Euclide an k -for est and E u clide an prize- c ol le cting Steiner for est hav e no PT ASs in their general forms. 16 7 Conclusion Besides presen ting a simpler and correct analysis of th e PT AS for the Euclide an Steiner for est pr oblem , we show ed how the app roac h can b e generalized to solve multiplicativ e prize-collect ing problems. Generalizing ou r r esu lts to p lanar graphs, esp ecially obtaining a PT AS f or Steiner forest, h as b een a long-standing op en pr oblem in this field. The qu estion was settled very recently by Bateni, Ha j iagha yi and Marx [6 ]. While Borradaile, Klein and Keny on-Mathieu [11] ga ve a PT AS for Steiner tr e e on planar graphs, a m ain ingred ient of their algorithm is s olving Steiner tr e e on graphs of b ounded-treewidth. Ho we ver in a sh arp contrast, Gassner [18] sho wed recently that Stei ne r for est is NP-hard even on graphs of treewidth at most 3. Bateni et al. [6] gives a PT AS for the problem on graphs of b ounded treewidth , and uses it to obtain a PT AS for planar and b oun ded-genus graphs. Last but n ot least, obtaining any improv ement ov er the app roxima tion factor 2.54 in [22] for multiplica tiv e prize-collect ing S teiner forest in general graphs seems very interesting. References [1] A. A rcher, M. Ba teni, M. Hajiagha yi, and H. Ka rloff , Impr ove d appr oximation al- gorithms for prize-c ol le cting steiner tr e e and TSP , in Pro ceedings of the 50th Annual IEE E Symp osium on F ound ations of Compu ter Science (F OCS), 2009. [2] S. Arora , Polynomial time appr oximat ion schemes for E u clide an tr aveling salesman and other ge ometric pr oblems , Journ al of the ACM, 45 (1998), pp . 753–78 2. [3] E. Balas , The prize c ol le cting tr aveling salesman pr oblem , Netw ork s , 19 (1989), pp. 621–63 6. [4] M. Ba teni and M. 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R obins and A. Zeliko v sky , Tighter b ounds for gr aph Steiner tr e e appr oximatio n , SI AM Journal on Discrete Mathematics, 19 (2005) , pp. 122–134. [32] F. S. Salma n, J. Che riy an , R. Ra v i, and S. Su bramanian , A ppr oximating the single- sink link-instal lation pr oblem in network design , SIAM Jour nal on Op timizatio n, 11 (2000), pp. 595–610. [33] Y. S harma, C. S w amy, and D. P. Williamson , A ppr oximation algorithms for prize c ol- le cting for est pr oblems with submo dular p enalty functions , in Pro ceedings of the eigh teenth annual ACM-SIAM symp osium on Discrete algorithms (S ODA ), 2007, pp. 1275–12 84. A Deferred pro ofs and further discussion Pro of of Lemma 7. T he p roof of Theorem 5 (although not rep r od uced here) do es not incr ease comp( F ∩ L ). Hence, it su ffices to p rov e the result for the forest F sp ecified at the b eginning of Section 2. Obs erve that by (I I), F ∩ L consists merely of singleton p oints, b ecause n o Steiner p oint lies on a line ℓ ∈ L . F urth er notice that the ℓ 1 -length of F is at most √ 2 length ( F ). Let F x b e the total absolute distance F tra ve ls in the x dir ection. Since x -coordinate d ifference of any tw o consecutiv e b r e ak-p oints of F is a multiple of 2 and the inte rsection with ve rtical lines of L occurs at co ordinates of the form (2 i, y ), the total number of intersect ions with vertical lines is exactly F x / 2. W e can similarly argue f or the intersecti ons with horizontal lines, and fi nally conclude that comp( F ∩ L ) ≤ √ 2 2 length( F ). Pro of of Lemma 9. This is clearly tr ue for the base cases of the DP since w e go ov er all the p ossibilities. Next, take any configuration χ = ( K , P ) corresp onding to a non-leaf dissection square R , and s u pp ose there is a subsolution F with resp ect to R compatible with χ , su ch that any sub s olution F ′ formed by restricting F to a dissection squ are R ′ which is a descendant of R is compatible with some configur ation χ ′ of R ′ . Let F i b e th e subsolutions r estricted to the su bsquares 19 R i . Each of th em is thus compatible with an appropriate χ i = ( K i , P i ). By ind u ctiv e hyp othesis, the dynamic p rogramming states T R i [ χ i ] hav e b een correctly compu ted. Each connected compon ent of F not connected to ∂ R has to h a ve all its terminal p airs satisfied. This is taken care of by c hec king the partition P ′ : the terminals in comp onents that d o not adv ance in the dyn amic table to T R [ χ ] hav e their demand s satisfied internally . In addition, the lo calit y prop erty for F ensures the configur ations will b e consistent, and hence we p erform an u p date of T R [ χ ] from T R i [ χ i ]. Th is finishes th e completeness pro of. V erifying that the up d ate rule is soun d is trivial. If the four configur ations χ i up date χ , then there exists a subs olution F formed b y the union of the corresp ondin g subsolutions F i , that is compatible with χ . A.1 The preliminary conditions for mu ltiplicativ e prizes In Section 2, we said that standard p erturbation and s caling techniques allo w u s to assum e with a cost increase of at most O ( ǫ OPT) th at the b oun ding b ox of the ins tance has side length at m ost n 2 ǫ − 1 OPT, while restricting all vertice s and Steiner p oints to p oints of the form (2 i + 1 , 2 j + 1) for inte gers i and j . The claim is based on the f ollowing tw o premises: 1. If d is the maximum d istance of a pair in D , then OPT ≥ d . 2. If u and v are farther than n 2 d , they cannot b e connected in the op timum solution. Using this, the ins tance can b e br oken up into disjoint s u binstances and then the p erturb ation can b e carried out. Ho wev er, the first p remise is f alse in the case of multiplicati ve prizes since not all the pairs n eed to b e connected. Next we show h o w similar conditions can b e guaranteed in this case. The v alue of O PT can b e g u esse d using binary search. T o b egin the search, we can get crud e b ounds of ω /n ≤ OPT ≤ ω , usin g simple app roxima tion algorithms for th e general cases of PC SF and k -forest. 5 Knowing OPT, we bu ild a graph G ′ on the vertices: th ere is an edge b etw een u and v if and only if their distance is at most OPT. The diameter of each connected comp onent is at most n OPT. W e consider each of them separately , since t wo vertic es in different comp onents cannot b e connected in the optimal solution. The side length of th e b oundin g b ox is at most n OPT. Scale the in s tance by 8 ǫ − 1 and let OPT ′ = 8 ǫ − 1 OPT denote the new optimal v alue. Build a grid in the boun ding b y lines with equations x = 2 i and y = 2 j for integers i, j . Mov e each vertex and Steiner p oint to the closest p oint of the form (2 i + 1 , 2 j + 1). Notice that there are at most n Steiner p oin ts. Assuming OPT > 0, th e c hange in the solution v alue du e to the p erturbation is at most 2 n · 4 = 8 n ≤ ǫ OPT ′ . Hence, we can assume that • the sid e length of the b ounding b o x is at most nǫ − 1 OPT ′ , and • the vertices and Steiner p oints are at co ord inates (2 i + 1 , 2 j + 1) for intege rs i, j . A.2 k -MST as a sp ecial case of S -MPCSF Here we show that (even the symmetric) S -MPCSF is a generalization of the ro oted k -MST pr oblem (for which the b est approximati on guarante e is 2). Su pp ose w e are given an instance I of the ro oted 5 The b est known approximatio n algorithms known for these problems are 2 . 54 and min { √ k, √ n } , resp ectively . 20 k -MST problem. I t consists of a graph G ( V , E ), edge lengths c e , a ro ot vertex r and a number k . Supp ose r is not to b e counte d among the k v ertices. W e bu ild the new in stance I ′ of the S -MPCSF problem as follows. The graph G ′ is the same as G . The weigh ts of all vertice s are on e, except for r whose weigh t is n 2 . Then, the goal w ill b e to fin d the chea p est forest that gathers a pr ize of at least S = ( n 2 + k ) 2 = n 4 + 2 n 2 k + k 2 . Theorem 14. The instan ce I of the ro oted k -MST problem is equi valent t o the instance I ′ of the S -MPCSF pro blem. Pro of. As we noted in S ubsection 1.2, in case of p olynomially b ound ed intege r weigh ts, we can mak e su r e the r etur ned solution collects a p r ize of at least S (without any appr oximation factor). This can b e achiev ed by picking ǫ ′ < 1 /S . Obviously , any tree connecting k ve rtices to the ro ot is translated to a forest that collects a prize of at least S . Let each vertex not sp anned by the tree b e a singleton comp onent in the forest. Finally , we claim that any solution of v alue S or higher translates to a solution of v alue at least k for the original instance. The resu lting tree is jus t th e comp onent of the forest conta ining the root vertex. Sup p ose for the sake of r eaching a contradiction that the comp onent spans k ′ < k non-ro ot vertices. The total prize collected is at most ( n 2 + k ′ ) 2 + ( n − k ′ − 1) 2 < n 4 + 2 n 2 k ′ + k ′ 2 + n 2 = S + 2 n 2 ( k ′ − k ) + k ′ 2 + n 2 − k 2 < S + 2 n 2 ( k ′ − k + 1) ≤ S, yielding a contradictio n, and pr o ving the s u pp osition is false. A.3 PCSF vs. k -forest Lemma 15. An α -appro ximati on algorithm for the k -forest problem gives an α (1 + ǫ ) -appro ximati on algo rithm for the prize-collecting Steiner fore st problem, for any constant ǫ > 0 . Pro of. W e show h o w to approximate a PCSF instance I by inv oking seve ral (p olynomially m any) instances I ′ of the k -forest p roblem. Obtain an estimate ω for I , such th at ω 3 ≤ OPT ≤ ω using a general-case 3-approximation algorithm. Let π i b e the p enalty of the pair i in I . Without loss of generalit y , we can assume that π i ≤ 2 ω for any pair i . Let θ = ǫω / 3 n . Place p i = ⌊ π i θ ⌋ copies of the pair i in I ′ . Find an α -approximat e solution to the resulting k -forest in stance for every v alue of 0 ≤ k ≤ n ′ , where n ′ is the num b er of pairs in I ′ . C ompute the PCSF v alue for each of these solutions and rep ort the b est one. W e show that at least one of these candidate solutions is go od . Let OPT f and OPT π b e the length of the forest and the p aid p enalt y of the optimal solution, r esp ectiv ely . Supp ose OPT connects a su bset of terminal p airs Q . Then, OP T π = P i 6∈ Q π i . F o cus on the candidate solution with k = P i ∈ Q ⌊ π i /θ ⌋ . T he length of the corresp ond in g k -forest instance is at most OPT f , b ecause a p ossible solution is that of connecting th e copies of Q . T o compute the PCSF v alue, we add the p enalt y of p airs in Q that are not connected u sing this tree. W e can assume either all or n o copies of eac h p air is connected. The number of p airs n ot conn ected is at most n ′ − k , and their p enalties 21 sum to no more than X i not connec ted π i ≤ X i not connec ted ( p i + 1) θ ≤ X i not connec ted p i θ + nθ ≤ ( n ′ − k ) θ + nθ ≤ OPT π + n θ = OPT π + ǫω / 3 ≤ OPT π + ǫ O PT . Thus, the PCSF v alue of the b est candidate solution is at most α OPT f + O PT π + ǫ O PT ≤ α (1 + ǫ )OPT. It r emains to show the instances I ′ hav e p olynomial size. Since π ≤ 2 ω , each p air i w ill hav e p i ≤ 6 nǫ − 1 copies. Hence, I ′ has p olynomial size and we can u se the approximati on algorithm f or the k -forest. 22