On the Problem of Wireless Scheduling with Linear Power Levels

On the Problem of Wireless Sc heduling with Linear P o w er Lev els Tigran Tonoy an ⋆ TCS Sensor Lab, Centre Univers itaire d’Informatique, route de Drize 7, 1227 Carouge, Genev a, Switzerland Abstract. In this paper w e consider the prob lem of comm unication scheduling in wireles s netw orks with respect to the S INR(Signal to Interference plus Noise R atio) constraint in metric spaces. The nod es a re assigned line ar p owers , i.e . for ea ch sender nod e the p ow er is constant t imes the path loss b etw een the sender and corresp onding receiver. This is the minimal p ow er for a successful transmission. W e present a constant factor deterministic app ro ximation algorithm, which w orks for at least Eu clidean fading metrics . S imultaneo usly w e obtain the approximate v alue of th e optimal schedule length with error at most a constan t factor. T o giv e an in sight into t h e complexity of the p roblem, we show that in some metric spaces the problem is NP-hard and cannot be approximated within a factor less than 1.5. 1 In t ro duction The pro blem of scheduling is the following: given some set of tr ansmission reques ts or links (sender-receiver no de pa irs in the netw ork), the goal is to find a schedule, so that all the transmiss ions b etw e en those no des can b e done success fully in the minim um time. The ma in factor a ffecting the successful data transmissions in wir eless net works is the signa l interference of the co ncurrently tra nsmitting nodes, whic h in genera l makes it imp ossible to do all the needed transmissions co nc ur rently: there can b e a receiver no de, which cannot deco de the data intended to it b ecaus e of the “no ise” made by other tra nsmitions. So one nee ds to split the set of requests into subgro ups, in eac h of whic h all sender no des can transmit concurrently . Then all the data transmission can b e done in a time pr op ortiona l to the num b er of different groups in the schedule. The goal is to minimize the num b er of s uch subsets . The solutio n of the problem depends crucially on the mo del of interference which is a dopted for the given case. There are several models considered in th e literature, such as pr oto c ol mo del and physic al mo del . W e consider the physic al mo del , which is sho wn to b e more realis tic than (traditional) proto co l mo del. It assumes that the influence of a transmitting node on other no des decreas es propo rtionally to a cons tant p ow er of the distance from that node (if there a r e no obstacles). Based o n this model, the SIN R (S ignal to Interfer enc e plus Noise R atio) constra int is cons idered fo r reflecting the possibility o r imp ossibility of a ccepting the sig nal of some sender by the corr esp onding receiver. The so lution of the s cheduling problem dep ends also on the p ow er levels of the no des in the ne tw ork: each no de can trans mit the data with a sp ecific p ow e r , so the more is the p ow er of a no de, the stronger is the signal r eceived by the intended re ceiver (als o the more is the “noise” made by that no de to other transmissions ). Our r e sults a r e for line ar p ower as signments, which a long with uniform p ow er assig nmen ts are the most p opula r p ower schemes us ed in the literature. In case of the unifor m pow er assignment a ll the no des us e the sa me p ow er level, so this assig nment is very simple to implement, but can le a d to long schedules (hence an incr eased delay in the netw ork). In case of linear power assignment the p ower of the sender of a link is consta nt times the p ath loss be tw een the sender a nd corr esp onding receiver, so this is the minim um possible p ower for a successful transmission. O n the other hand, the optimum schedule length of the linear p ow er can also b e to o long co mpared to some other p ow er assig nments. R elate d Work and Our R esults The algorithmic study of the pr oblem of sc heduling for arbitra ry ne tw orks in SINR mo del g ained c o nsiderable attention in last years b ecause of the attempts to sho w that this mo del describ es the physical reality more precis e than mo re tr aditional gr aph-b ase d mo dels , suc h as the pr oto c ol mo del [19]. ⋆ Researc h partially founded by FRONTS 215270 2 There are sev eral v ariants of sc heduling pr oblem co nsidered in the litera tur e. In [18 ], [2], c onne ct ivity problem is co nsidered from the scheduling p ersp ective, where it is needed to find and sc hedule (in to minimal nu mber of slots) a set of links which form a co nnec ted structure. Sche duling with p ower c ontr ol , where for obtaining small schedules it is allowed a s well to control the power levels of the transmitters, is cons idered in [20], [5], [10], [11], [22], [16 ]. In [1 6] a log n -approximation algo rithm is designed (where n is the n umber of links), which uses p ow er as s ignments of non- lo c al nature. In [10 ], [11 ], [2 2] it is shown that the me an p ower assignment is r elatively efficie nt from the p o int of view of sc heduling, when compar e d to other lo ca l p ow er assignments, and appr oximation guara nt ees are pr ov e n fo r this power assig nment . In fact in [5], g eneralizing the construction fr o m [18], it is shown that for each local (or oblivious , as they call it) power assignment , there are net work instances, for whic h no non-trivial schedules ca n b e obtained using this p ow er a ssignment. On the o ther hand, in [10] it is shown that if the lengths o f links differ not more than a co nstant factor, then the uniform p ower assignment is a constant factor approximation for this problem. The problem of sc heduling with fixed power levels is consider for several p ower assignments, such as uniform, line ar and me an p ow er assignments. There are O (log n ) approximation algo rithms desig ned for the mean p ow e r assignment in [22]. The case o f uniform pow er as signment is co nsidered in [3], [8], [9], [1], [14]. T o the b es t of our kno wledge, the b est approximation ratio obtained for this case is O (log n ), although constant factor approximation algor ithms are desig ne d for a rela ted problem of c ap acity maximizatio n for a large family of power as signments [13] (in [14 ] it is claimed th at their algorithm a pproximates the optimal schedule leng th within a co nstant facto r , but a flow was found in the pro of ). In [17] O (log 2 n )-approximation randomized distributed algorithms are des igned for a r ange of p ower assig nment s, which is improv ed to O (log n ) fa c tor in [12]. The sp ecific case of linear p ow e r a ssignments is co nsidered e.g. in [4], [6]. In [6] a randomized a lgorithm is pro po sed, which finds a schedule of length O ( I + log 2 n ), wher e I is a lower bound on the o ptimal schedule length. The pap er is an ex tended version of [21], where we propos e a constant facto r a lgorithm for scheduling w.r.t. fixed linear p ow er as s ignment, and s how that the optimal schedule leng th is Θ ( I ), where I is as defined in [6]. W e also show that the pro ble m ca nnot b e approximated within a factor less than 1 .5 (with assumption P 6 = N P ). Whith the same metho ds as in [6], our a lgorithm for linear p ow ers also leads to a randomized algorithm fo r Cro s s-Lay er Optimization problem, which improv es the algo rithms from [6] by a facto r log n . 2 F orma l Definition of the Problem Throughout this work we assume the wireles s net work nodes to b e sta tica lly lo cated (i.e. the netw ork is not mobile) in a metric measure space X with a dista nce function d and a measure µ . The ball in X with center p and radius r > 0 is the set B ( p, r ) = { q ∈ X | d ( p, q ) < r } , and the r ing with center p , width w > 0 and outer radius r > w is the set R ( p, r , w ) = { q ∈ X | r − w ≤ d ( p, q ) < r } . F or a ring R = R ( p, r , w ) w e denote b ( R ) = B ( p, r ) a nd B ( R ) = B ( p, r − w ). W e a ssume that the measure µ satisfies the follo wing condition: for any tw o balls A and B with radii a and b respectively , µ ( A ) µ ( B ) ≤ K  a b  m (1) holds for some constants K ≥ 1 and m ≥ 1 , which are sp ecific to the metric space. W e are given a set of links L = { 1 , 2 , . . . , n } , where each link v r e presents a communication r equest from a sender no de s v to a receiver no de r v . The asymmetric distanc e from a link v to a link w is d vw = d ( s v , r w ). The length of the link v is d vv = d ( s v , r v ). Each transmitter s v is assig ned a p ow er le vel P v , which does not change. W e assume that the strength o f the sig nal decrea ses with the distance fr om the transmitter, i.e. the 3 received sig nal strength from the sender of w at the receiver of v is P wv = P w d α wv , where α > 0 denotes the p ath-loss exp onent . F or interference w e ado pt the SINR mo del, where the transmission corresp onding to a link v is succes sful if and only if the following condition holds: P vv ≥ β   X w ∈ S \ v P wv + N   , (2) where N ≥ 0 denotes the ambien t no ise, β > 1 denotes the minimum SINR requir ed for message to be successfully received, a nd S is the set of concurrently scheduled links . W e say that S is fe asible if (2) is satisfied f or all v ∈ S . The linear p ower ass ignment a ssigns ea ch sender s v a p ower level P v = c l d α vv , where c l . The uniform p ow er a ssignment assigns to eac h sender no de the same p ow er level P . A pa rtition of the set L into feasible subse ts (or slots ) is c alled a sche dule . The num b er of subsets in a schedule is called the length of the sc hedule. The problem we are in ter ested in is to find a schedule of a minim um length, ass uming that the linear power assignment is used. 3 Auxiliary F acts Here is a set of lemmas, which we will use in subsequent sections. The follwing is a known b ound for Riemann zeta function: ζ ( s ) = ∞ X i =1 1 i s ≤ s s − 1 , if s > 1 , (3) which can b e prov en by noticing that ∞ X i =1 1 i s ≤ Z ∞ 1 1 x s dx + 1. A pro o f of the f ollowing lemma can b e found, for exa mple, in [1 5], page 28. Lemma 1. F or r e al numb ers a 1 , a 2 , . . . , a m ( a i ≥ 0 , i = 1 , 2 , . . . , m ) , and r , s (0 < r < s ) , m X i =1 a s i ! 1 s < m X i =1 a r i ! 1 r holds, unless al l a i but one ar e zer o. F or the next le mma , consider a ny g iven real n umber s a ≥ 1 and c > 0, and the function f ( t ) = ( a + c ) t − a t . Note that f ( t ) is a monotonically increa sing function on [1 , ∞ ], as f ′ ( t ) > 0 for t ≥ 1. So f ( t ) ≤ f ( ⌈ t ⌉ ) for t ≥ 1. F or a n integer k ≥ 1 we hav e ( a + c ) k − a k = c k − 1 X i =0 ( a + c ) i a k − 1 − i ≤ k c ( a + c ) k − 1 , so we hav e the following lemma: Lemma 2. F or r e al numb ers a ≥ 1 , c > 0 , t ≥ 1 ( a + c ) t − a t ≤ ⌈ t ⌉ c ( a + c ) ⌈ t ⌉− 1 . 4 4 The sc heduling algorithm In this sectio n we presen t a scheduling algor ithm, whic h is very simple and a pproximates t he o ptimal schedule length within a co nstant factor. It is assumed that the linear p ow er assig nmen t is used for the p ow er levels. As in other sc heduling algorithms, instead of using the SINR formula in the form of (2), w e use the inv ers e of it, which has the useful prop er ty of b eing additiv e, and is eas ie r to dea l with. Definition 1. T he affe ctanc e of a link v , c ause d by a set S of links, is the fol lowing sum of r elative inter- fer enc es of t he links fr om S on v , a S ( v ) = X w ∈ S \ v  d ww d wv  α . With the affectance defined, SINR co nstraint for a set o f links S and a link v can b e written as a S ( v ) ≤ 1 β − N c l . (4) F or simplicity of writing we deno te the r ig ht side by 1 /β . 4.1 F orm ul ation of the algorithm The algor ithm (pseudo co de is presented as Algo rithm 4.1 ) is a gr e e dy algo rithm , which s orts a ll the links in descending or der of the length, and starting fro m the first one, adds each link to the first slot, in whic h already s cheduled links influence this one no more than a predefined constan t. As w e will see afterwards, this sp ecial ordering is needed only for feasibility of the res ulting sc hedule, whereas th e pro o f of approximation factor do es not dep end on this or der. The pre cise v alue of the constant c > 3 used in the algor ithm will b e defined afterwards. Algorithm 4.1 Scheduling w.r.t. linear p ow er assignment. 1. In put: the links 1 , 2 , . . . , n 2. sort th e links in descending order of their lengths: l 1 , l 2 , . . . , l n 3. S i ← ∅ , i = 1 , 2 , . . . 4. for t ← 1 to n do 4.1 find th e smallest i , such that a S i ( l t ) ≤ 1 c α 4.2 schdule l t with S i : S i := S i ∪ l t 5. outp ut: ( S 1 , S 2 , . . . ) 4.2 Correctness of the alg orithm Consider the set of links S assigned to the sa me slot by the algo rithm, and v ∈ S . Let S − denote the subset of S , which co ntains the links shorter than v . It is eno ugh to show that a S − ( v ) is small for eac h slot S and v ∈ S . T o show this we will use a standard area argument. W e start with a simple lemma, which shows that if t wo links are scheduled in the s ame s lot, then they should b e spatially separ ated. Le t the links w and v b e assigned to the same slo t b y the algorithm, and d = max { d vv , d ww } . Lemma 3. F or any two links w and v , which ar e as ab ove, the fol lowing holds: d vw ≥ ( c − 2) d , d wv ≥ ( c − 2) d and d ( s v , s w ) ≥ ( c − 3) d. 5 F easibility of the schedule is shown Lemma 4. Ther e exists a c onstant c 0 , dep ending only on m , K and α , such that for the link v and the set of li nks S − as ab ove, a S − ( v ) ≤ c 0 ( c − 3 ) α , holds, if α > m m + 1 − ⌈ m ⌉ . Pr o of. F or simplicity , thro ughout this pro of we deno te q = c − 2. Consider the partition of the metric space int o conce nt ric rings R i = R ( r v , ( i + 1) q d vv , q d vv ) for i = 1 , 2 , . . . , a nd the ball B ( r v , q d vv ). F ro m Lemma 3 and definition of S − it follows that there ar e no sender s from S − inside B ( r v , q d vv ). Now for so me i > 0 consider the links from S − with senders inside R i , a nd denote that set b y S − i . F or each link w deno te ρ w = ( q − 1) d ww 2 . Then it follows from the last ineq uality of Lemma 3 that for eac h suc h link w the ball B ( s w , ρ w ) do esn’t intersect the cor resp onding ball of any other link . F ur ther, a ll s uch balls are contained in the ring R ′ i = R ( r v , ( i + 1) qd vv + ρ v , q d vv + 2 ρ v ). So from the countable additivity of µ it f ollows that X w ∈ S − i µ ( B ( s w , ρ w )) ≤ µ ( R ′ i ) = µ ( B ( R ′ i )) − µ ( b ( R ′ i )) o r, as K ≥ 1 , X w ∈ S − i µ ( B ( s w , ρ w )) µ ( B ( R ′ i )) ≤ 1 − µ ( b ( R ′ i )) µ ( B ( R ′ i )) ≤ K − µ ( b ( R ′ i )) µ ( B ( R ′ i )) . (5) F rom (1) we hav e the fo llowing inequa lities for ea ch link w : µ ( b ( R ′ i )) µ ( B ( R ′ i )) ≤ K  iq d vv − ρ v ( i + 1) q d vv + ρ v  m and µ ( B ( s w , ρ w )) µ ( B ( R ′ i )) ≥ 1 K  ρ w ( i + 1) q d vv + ρ v  m , which combined with (5) leads to the following: X w ∈ S − i ρ m w ≤ K 2 ((( i + 1 ) q d vv + ρ v ) m − ( iqd vv − ρ v ) m ) ≤ ≤ K 2 ( q d vv ) m (( i + 3 / 2) m − ( i − 1 / 2) m ) ≤ 3 m − 1 2 m − 2 ⌈ m ⌉ K 2 ( q d vv ) m i ⌈ m ⌉− 1 (6) where we used Lemma 2 and the fact, that ρ v < q d vv / 2. Dividing b oth sides of (6) by  q − 1 2  m and replacing q − 1 by q / 2 in deno minator, we get X w ∈ S − i d m ww ≤ 2 · 3 m ⌈ m ⌉ K 2 d m vv i ⌈ m ⌉− 1 . (7) On the other hand, from the triang le inequality and the definition o f ring R i , the following holds: d wv ≥ d ( s w , s v ) − d ( s v , r v ) ≥ ( q − 1) d vv i , so a S − i ( v ) ≤ P w ∈ S − i d α ww (( q − 1) d vv i ) α (8) 6 Using Le mma 1, from (7) and (8) we get an upp er b ound o n the a ffectance o f the senders from R i : a S − i ( v ) <  P w ∈ S − i d m ww  α/m (( q − 1) d vv i ) α ≤  2 · 3 m ⌈ m ⌉ K 2 d m vv i ⌈ m ⌉− 1  α/m (( q − 1) d vv i ) α ≤ 3 α  2 ⌈ m ⌉ K 2  α/m ( q − 1) α i α ( m +1 −⌈ m ⌉ m ) , i = 1 , 2 , . . . By s umming ov er i , and using (3), we co mplete the pro of of t he lemma (as we have α > m m + 1 − ⌈ m ⌉ ): a S − ( v ) ≤ 3 α  2 ⌈ m ⌉ K 2  α/m ( q − 1) α ∞ X i =1 1 i α ( m +1 −⌈ m ⌉ m ) ≤ 3 α  2 ⌈ m ⌉ K 2  α/m ( q − 1) α · α ( m + 1 − ⌈ m ⌉ ) α ( m + 1 − ⌈ m ⌉ ) − m , so we hav e c 0 = 3 α  2 ⌈ m ⌉ K 2  α/m α ( m + 1 − ⌈ m ⌉ ) α ( m + 1 − ⌈ m ⌉ ) − m ⊓ ⊔ Having Lemma 4, the proof of the following theorem is easy . Theorem 1. If c ≥ α p β ( c 0 + 1) + 3 and α > m m + 1 − ⌈ m ⌉ , then the outpu t of the algorithm is a fe asible sche dule. 4.3 The appro ximation ratio In this section we s how that the algo r ithm outputs a schedule, which is longer than the o ptima l one no more than by a c onstant factor. The following definition is taken from [6]. Definition 2. L et S b e a set of tr ansmission r e quest s and p a n o de i n the network, then we define I p ( S ) = X w ∈ S min  1 ,  d ww d ( s w , p )  α  , and I ( S ) = max p I p ( S ) . When S is the set of a ll links (which w e denoted by L ), we use the notation I ( L ) = I . I is a measure o f int erference, whic h in [6] is sho wn to b e a low er bound(with a consta nt factor ) for optimal schedule length in case of linear p ow er assig nment s. Theorem 2. [6] If T is the minimu m sche dule length, then T = Ω ( I ) . Using Theo rem 2 it is ea sy to pr ov e the approximation ra tio. Theorem 3. a) If c > 1 and the output of Algorithm 4 .1 is a fe asible sche dule, then it is a c onstant factor appr oximation for sche duling with line ar p owers, b) the optimal sche dule length in c ase of line ar p owers is Θ ( I ) . Pr o of. Suppose A 1 , A 2 , . . . , A t is the o utput o f Algor ithm 4.1 . Let v b e a link fro m A t . By definit ion of the algorithm we hav e a A i ( v ) > 1 c α , if i < t. Since we assume c > 1, we hav e als o I r v ( A i ) > 1 c α , so I r v ( L ) = t − 1 X i =1 I r v ( A i ) > ( t − 1) /c α . On the o ther hand we hav e I r v ( L ) ≤ I ( L ), so t < c α I ( L ) + 1, which together with Theorem (2) completes the pro of. ⊓ ⊔ 7 5 On the complexit y of sc heduling with linear p o w ers W e define the pr o blem E Q SCHEDULING, which is a simplified case of the problem o f scheduling (again w.r.t. linear pow er ass ignment), when all links in the netw ork ha ve “almos t the same” length (i.e. the lengths differ only by a constant factor). EQSCHEDULING: giv en a set of links L = { l 1 , l 2 , . . . , l n } in a netw ork, which have lengths differing no t more than a co nstant factor, and a na tur al num b er K > 0 , the question is if there is a partition of tha t set int o not more than K SINR-feasible subsets or slots . T o sho w that the pr oblem is NP- ha rd, we reduce to it the NP- complete pr oblem P AR TITION(see [7]), which is defined as follows. P AR TITION: g iven a finite set A of p ositive integers, the ques tio n is if there is a subset A ′ ⊆ A , for which X a ∈ A ′ a = X a ∈ A \ A ′ a holds, i.e. it is exactly the half o f A by s um. When it is not ambiguous, we will ide ntify an instance of P AR TITION with the corresp onding set of int egers A . Here we use no t the g eneral pr oblem P AR TITIO N, but some sp ecific case, which is equiv alent to the general one. W e need the following lemma. F or a finite set of integers A let S ( A ) deno te the s um P a ∈ A a . Lemma 5. F or e ach instanc e A of P AR TITION ther e is another instanc e B , which is p olynomial ly e quiva- lent t o A , and satisfies the following pr op ert ies: A ⊂ B , and | B | = 3 | A | for a ∈ A, a S ( B ) ≤ 1 2 | A | 3 (9) for a ∈ B \ A, a S ( B ) ≤ 1 2 | A | (10) Pr o of. Let m deno te a maximal element in A . Then we cons tr uct B by adding 2 | A | new e le ment s with the same v alue | A | 2 m to A . A s each element in B \ A is not less tha n S ( A ), then it’s easy to c heck, that eac h partition o f A c o rresp onds to a partition of B , and vice versa, so tw o instances of the problem ar e equiv alent. The other tw o prop erties ar e str aightforw ard. ⊓ ⊔ Theorem 4. Ther e ar e metric sp ac es, wher e EQS CHEDULING is NP-har d. Pr o of. W e give a reduction from P AR TITIO N. F or a given instance A of P AR TITION let’s construc t the instance B , as describ ed in Lemma 5 . W e will co nstruct an insta nc e o f EQSCHEDULING with a s et of links L in a net work, so that the answ er of B is “yes” if a nd only if it is possible to schedule L in t wo subsets. Let B = a 1 , a 2 , . . . , a n , and n = | B | = 3 | A | . The net work has 2 n + 4 no des s i , r i , i = 0 , 1 , . . . , n, n + 1. The set of links is L = { l 0 , l 1 , . . . , l n , l n +1 } , where ea ch link l i represents the s ender-rece iver pair ( s i , r i ). The distances ar e defined a s follows: we use d ij for denoting the dis tance d ( s i , r j ). The links l 0 and l n +1 hav e lenght d 00 = d n +1 ,n +1 = 1 α √ 3 . All other links have length 1. W e se t d ( r 0 , r n +1 ) = 0, s o that th e links l 0 and l n +1 cannot be sc heduled together in the same group. F urther, d i 0 = d i,n +1 = α s β S ( B ) 2 a i , d 0 i = d n +1 ,i = d ii + d i 0 = α s β S ( B ) 2 a i + 1 , d ij = d ( s i , r j ) = 1 + d ( s i , r 0 ) + d ( r 0 , s j ) = 1 + α s β S ( B ) 2 a i + α s β S ( B ) 2 a j 8 for i, j = 1 , 2 , . . . , n, i 6 = j . The other distances can be arbitr ary , satisfying the a xioms of metric spa ces. First we show that the set of links S = { l 1 , l 2 , . . . , l n } is SINR-feasible. T o do so we select any link , sa y the link l 1 , and show, that the co nstraint (2) is satisfied for S and l 1 . T aking into account the definition of the distances, the left part of (2) b ecomes a S ( l 1 ) = 1 3   α s β S ( B ) 2 a 1 + 1   α + n X i =2 1   1 + α s β S ( B ) 2 a i + α s β S ( B ) 2 a 1   α ≤ ≤ 1 3 β S ( B ) 2 a 1 + 3 + n X i =2 1 1 + β S ( B ) 2 a i + β S ( B ) 2 a 1 = 1 β     1 3 S ( B ) 2 a 1 + 3 β + n X i =2 1 1 β + S ( B ) 2 a i + S ( B ) 2 a 1     , where we used the Lemma 1 with s = α a nd r = 1, as we ass ume α > 1. F o r e v aluating the last express ion, we consider tw o case s : 1) if 1 ∈ A , then accor ding to (9) we hav e a 1 S ( B ) < 1 2 | A | 3 , so a S ( l 1 ) ≤ 1 β     1 3 | A | 3 + 3 /β + X i ∈ B \ A 1 1 /β + S ( B ) 2 a i + | A | 3 + X i ∈ A \{ 1 } 1 1 /β + S ( B ) 2 a i + | A | 3     ≤ ≤ 1 β  1 3 | A | 3 + 3 /β + 2 | A | | A | + | A | 3 + 1 /β + | A | − 1 2 | A | 3 + 1 /β  . As it is easy to see, the rig ht s ide is less than 1 /β for instances with | A | larg e enough. 2) if 1 / ∈ A , then acco rding to (10) we have a 1 S ( B ) < 1 2 | A | , so a S ( l 1 ) ≤ 1 β     1 3 | A | + 3 / β + X i ∈ B \ ( A ∪{ 1 } ) 1 1 /β + S ( B ) 2 a i + | A | + X i ∈ A 1 1 /β + S ( B ) 2 a i + | A |     ≤ ≤ 1 β  1 3 | A | + 3 /β + 2 | A | − 1 2 | A | + 1 /β + | A | | A | 3 + | A | + 1 /β  = = 1 β  1 − | A | + 3 /β | A | + 3 /β 2 + 2 /β 6 | A | 2 + 9 /β | A | + 3 /β 2 + | A | | A | 3 + | A | + 1 /β  . The expr e s sion in parentheses is less than 1 when | A | is larg e enoug h, so again a S ( l 1 ) ≤ 1 / β holds . The affectance on the link l 0 by the s et S is equa l to a S ( l 0 ) = n X i =1  d ii d i 0  α = n X i =1       1 α s β S ( B ) 2 a i       α = 2 / β , which is the same as the affectance on the link l n +1 by S . So it follows, that L can b e scheduled into t wo sets if a nd o nly if the set S can b e partitioned in to tw o subse ts , so that each one affects the link l 0 (and l n +1 ) exactly b y β . As it’s not hard to chec k, this is the same as to so lve the P AR TITION problem instance B . The reduction is p olynomial. ⊓ ⊔ 9 Corollary 1. Ther e ar e metric sp ac es, wher e E QSCHEDULING c annot b e appr oximate d within a c onstant factor less than 3 2 unless P = N P . Pr o of. Let P 6 = N P . F ro m the pr o of of Theorem 4 we see, that P AR TITION ca n b e polynomia lly reduced to EQSCHEDULING with K = 2 in some metric spaces. Suppose there is a p olynomial algorithm, which ap- proximates EQSCHEDULING within a factor γ < 3 2 . Then let’s co nsider an a rbitrar y instance A (sufficiently large) of P AR TITION. There is an instance L of EQSCHEDULING with K = 2 , so that the answ er for A is ’yes’ if and only if the linkset of L ca n b e scheduled in no mor e than 2 groups. Applying the approximation algorithm to L , we g et the o ptimal sc hedule length with erro r at most a factor γ . If the resulting sc hedule has complexity not less than 3 , then the optimal schedule length is at leas t 3 γ > 2, so the the answer of A is ’no’. If the le ng th of the r esulting schedule is les s than 3, then t he optimal schedule le ngth is no mo re than 2, s o the answer o f A is ’yes’. T his shows that the γ -appr oximation algo rithm for E QSCHEDULING could be used to p olynomially so lve P AR TITION, which is a contradiction to the a ssumption tha t P 6 = N P . ⊓ ⊔ References 1. M. A ndrews, M. Dinitz. 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