On some sufficient conditions for distributed Quality-of-Service support in wireless networks

On some sufficient conditions f or distributed Q uality-of-Se rvice support in wir eless networks Ashwin Ganesan (F ormerly with ECE dep t, University of W isconsin at Madison, USA) Curr ent Ad dr ess: 53 Deo nar House Deonar V illage Road, Mumbai-88, In dia Email: ashwin.ganesan @r ediffmail.com Abstract —Giv en a wireless network where some pairs of communication links interfere with ea ch other , we study sufficient conditions for determining whether a giv en set of minimum bandwidth Quality of Service (QoS) requirements can be satisfied. W e are esp ecially interested in algorithms which hav e l ow communication over head and low p rocessing complexity . The interference in the n etwork is modeled using a conflict graph whose vertices are the communication links in the n etwork. T wo links ar e adjacent in thi s graph if and only if t hey interfere with each oth er due to being i n the same vicinit y and hence cannot be simultan eously active. The problem of scheduling the transmission of the vario us links is then essentially a fractional, weighted vertex coloring problem, fo r which u pper b ounds on the fractional ch romatic number are sought u sing only localized information. W e present some distributed algorithms for th is problem, and discuss their worst-case perf ormance. These algorithms are seen to be withi n a bound ed factor away from op timal for some well known classes of networks and interference models. Key words -Quality of ser vice (QoS); distributed algorithms; conflict graph; w ireless networks; interference models; frac- tional, weighted vertex coloring. I . I N T RO D U C T I O N In rec ent year s there has b een an increasing in terest in using data networks to support a wide variety of applica - tions, each req uiring a different Quality of Service (QoS). For exam ple, real- time ap plications such as voice, video and industrial contro l are tim e-sensitiv e and require that the delay be small, while for oth er data applica tions the sender may r equire that a co nstant, minimum bit-rate service be provided. In the simplest and lowest lev el of service, such as th e one provided in the In ternet Protocol service model, the network makes a best-ef fort to deliv er data from the source to destination , but it m akes n o g uarantee s of any kin d, so it is possible that packets can get dropp ed, delayed, or deli vered out of order . Howe ver , this basic lev el of ser vice is in sufficient for many data application s such as vide oconfe rencing that a lso have a minim um bandwidth requirem ent. W e co nsider in th is work ap plications requ iring a minimum b andwidth quality-of- service. Consider a wireless com munication network where n odes (which repre sent wir eless devices such as laptops, phon oes, routers, sensors, e tc) wish to co mmun icate with each othe r using a sh ared w ireless medium. Any given pair of no des may m ake a request f or a dedicated point-to-po int lin k between th em tha t suppo rts their required bit-rate Quality of Service. The objecti ve of the the admission co ntr ol mechanism is to d ecide whether the d esired service can be provided, given the av ailable resou rces, with out d isrupting the serv ice guar anteed to previously admitted de mands. This mechanism ne eds to take into accou nt the fact that nodes in the same v icinity contend fo r the sha red wir eless medium and hence can cause interferenc e if simultaneou sly active. Also, for reason s such as low commun ication overhead and scalability , it is d esired that this decision b e made in some decentralized fashion . Th us, each n ode may ha ve access only to in formatio n pertainin g to its local neighborh ood and not a bout the entire comm unication network. These two req uirements - that the decision take into account interfer ence due to neighbo ring nodes and th at it be ma de in a dece ntralized m anner - are cr ucial aspects of th e pa rticular problem we stud y . If a decision to admit a demand is made, the schedu ling pro blem is to schedule the transmissions of the various nodes so as to provide the service level that was guaran teed. The focus of most of o ur work here is o n the admission contro l p roblem and not th e sche duling pro blem. For an introd uction to the flow ad mission con trol problem , see [1]. More formally , the wireless network model and desired QoS are specified as f ollows. Let V b e a set of no des an d L ⊆ V × V be a set of comm unication links. Each link ( i, j ) ∈ L makes a demand (to transmit information from i to j ) at a rate of f ( u, v ) b/s. The total band width of the shared wireless mediu m available for the co mmun ication network G = ( V , L ) is C b/s. The main p roblem stu died her e is to determine wheth er a set of demand s ( f ( ℓ ) : ℓ ∈ L ) ca n b e satisfied. Of course, if all the links can be simultaneously activ e, the set of demand s ( f ( ℓ ) : ℓ ∈ L ) can be satisfied as long a s each individual d emand is at m ost C . Howe ver , d ue to inte rference effects nodes in the same vicinity con tend for the shared wir eless m edium and h ence cannot be active at the same time. For example , in I EEE 80 2.11 MAC protoco l- based networks, any nodes adjacent to n ode i or to n ode j are required to be idle while the comm unication ( i, j ) takes place. The interference i n the network is modeled using a con flict graph . Gi ven a network grap h G = ( V , L ) , define its conflict graph to be G C = ( L , L ′ ) , wh ere two link s ℓ 1 and ℓ 2 are adjacent in the con flict graph if an d only if they canno t be simultaneo usly activ e. The co nflict grap h G C = ( L, L ′ ) specifies which pairs of link s interfer e with each oth er . Th is interferen ce model h as b een stu died r ecently by a number of auth ors; for example, see Jain et al [1 1], Hamdao ui an d Ramanathan [10], and Gupta, Musacchio and W alra nd [6]. A special case o f this mod el, where two lin ks are con sidered to be in terfering if an d o nly if they ar e incid ent (in th e ne twork graph) to a co mmon no de, has been studied earlier by Hajek [7], Hajek and Sasaki [8], and Kodialam and Nadagopal [12]. In the ad mission control problem studied in th is work, the vertices of the conflict g raph co rrespon d to links in the com munication network. The quality-o f-service metric is specified in terms of the ban dwidth desired by each link. This gives rise to one d emand value fo r each vertex of the conflict grap h, and this value could possibly be non- integral. The admission control p roblem is then to determine wh ether these deman ds can be satisfied using a spe cified amou nt of resour ce (to tal av ailable bandwidth). This prob lem is different from the classical weighted verte x co loring problem in some ways. First, f ractional solu tions to the coloring problem are also admissible, as indicated by the linear pro- grammin g formulation giv en b elow . Second, our e mphasis is on decision s that c an b e ma de in a d ecentralized man ner, i. e. using on ly localized inf ormation . Finally , the conflict gra ph sometimes has a dditional structure derived f rom the structure of the links in the ne twork or the interfer ence mo del. The admission control p roblem studied here is essentially th at of obtaining , u sing only localized informatio n, an up per bound on the resou rce required to satisfy a demand pattern . The scheduling prob lem, which we d o not study here, con cerns how these resources ar e actually allocated or managed. This paper is organize d as follows. The admission control problem studied here is form ulated pr ecisely in Sectio n I-A. Decentralized solutio ns to this pr oblem are then studied: Section II is on the row constraints, Section III is o n the degree and mixed conditions, and Section IV is o n the clique constraints. These results provide sufficient condition s and distributed algo rithms fo r a dmission con trol. The m ain new results here con cern the worst-case perform ance of these algorithm s. Finally , in Sec tion V the results obtained thus far are applied to some specific examples suc h as networks with primary in terferen ce c onstraints. A. Model and pr o blem formulation W e first state the flow co ntrol pro blem form ally . Th en, in order to av oid repeating trivialities th rough out the pa per, we will pr esent an equiv alent refo rmulation of the pro blem that ignores many o f the constants and variables an d in volves just th e essential details. W e will work o nly with this reform ulation in the r est of this paper . Let G = ( V , L ) be a ne twork graph, where L ⊆ V × V . Each lin k ℓ ∈ L has a maximum tran smission capacity of C ℓ ≤ C b/s, an d there is a deman d to use th at link at some rate f ( ℓ ) b/s; the to tal av ailable band width o f the shared wireless m edium is C b /s. T he ma in pr oblem we study is to d etermine, using only localized infor mation, whether th e set of demand s ( f ( ℓ ) : ℓ ∈ L ) can be satisfied. Setting τ ( ℓ ) = f ( ℓ ) /C ℓ , we g et an eq uiv alent r eformu lation wher e each link m akes a demand to b e active for a certain fractio n of e very unit of time. More precisely , an ind epend ent set o f a gr aph G C = ( L, L ′ ) is a subset I ⊆ L o f eleme nts that a re pairwise nonad jacent. If the set of link s that ar e simultaneou sly acti ve is an inde penden t set, then these lin ks cause no interferen ce with eac h o ther and can have (a part of) their dema nds satisfied d uring th e same time slot. Let I ( G C ) d enote the set of all ind epend ent sets of G C . A schedu le is a map t : I ( G c ) → R ≥ 0 . Th e schedule assigns to each ind ependen t set I j a time duration t j = t ( I j ) , which specifies the fr action of time th at the links in I j are active. A schedule t is said to satisfy a set of dema nds ( f ( ℓ ) : ℓ ∈ L ) if, fo r each ℓ ∈ L , P I j : ℓ ∈ I j t j C l ≥ f ( ℓ ) , and if the duration of th e schedule P I j t j is at m ost 1. A schedule is said to b e op timal if it satisfies the deman d of all the links and has min imum duration . Reformulation. Suppose we are given a n etwork graph G = ( V , L ) and a con flict graph that specifies wh ich links interfere with each o ther . Let τ ( ℓ ) d enote the amo unt of time when lin k ℓ demands to be acti ve. A link d emand vector ( τ ( ℓ ) : ℓ ∈ L ) is said to be feasible within time duration [0 , T ] if there exists a schedu le o f duratio n at most T th at satisfies the dem ands. W e will o ften assume, fo r simplicity of exposition, that T = 1 . Note that a sch edule is a map t : I ( G c ) → R ≥ 0 that assigns to each indepen dent set I j of the conflict graph a time duratio n t ( I j ) . A link ℓ is then activ e f or total d uration P j : ℓ ∈ I j t ( I j ) . The flow admission problem is to d etermine whether th ere exists a sched ule of duration at most T th at satisfies the link demand vector τ . The sch eduling problem is to realize such a schedule. W e are interested in solutions that can be impleme nted u sing only localized info rmation and with low pro cessing cost. Notatio n. It will be conv enient to use the following notatio n. Let G = ( V , E ) b e a simple, undirected grap h. For v ∈ V , Γ( v ) d enotes the ne ighbor s of v . α ( G ) de notes t he maximum number of vertices o f G that are pairwise nonadjacen t. For V ′ ⊆ V , G [ V ′ ] denotes the ind uced sub graph who se vertex- set is V ′ and whose ed ge-set is those edges of G that h ave both endp oints in V ′ . For any τ : V → R and any W ⊆ V , define τ ( W ) := P v ∈ W τ ( v ) . B. Prior work and our con tributions The take off point for ou r w ork is the prior work of [6] and [10]. T heir work proves th at certain distributed algor ithms provide suf ficient conditions for a dmission control. They call these con ditions the row co nstraints [6] (o r rate cond ition [10]), the degree con dition [10], mixed cond ition [10], and scaled clique con straints [6]. The main r esults of this paper are along the following lines: the exact worst-case perfo rmance of these d istributed admission control mechanisms is chara cterized and it is thereby shown that these m echanisms can b e arbitr arily far away from optimal; we then show that for some well known c lasses of networks and interfer ence models, these distributed algo rithms are actually within a bou nded factor away from optimal. I I . R OW C O N S T R A I N T S W e now present a suf ficient cond ition fo r flow admission control that can be impleme nted d istributedly . Given a co n- flict graph G C = ( L, L ′ ) , link demand vector ( τ ( ℓ ) : ℓ ∈ L ) and T , a sufficient condition f or feasibility is given by th e following result (see [ 10, Thm. 1], [6, Thm. 1]): Proposition 1. If τ ( ℓ ) + τ (Γ( ℓ )) ≤ T fo r each ℓ ∈ L , th en the dema nd vector ( τ ( ℓ ) : ℓ ∈ L ) is feasible within d uration T . These co nstraints are called the row con straints in [6] and the r ate cond ition in [1 0]. Let A = [ a ij ] be the 0-1 valued n × n adjacency matrix of G C where a ij = 1 iff i = j o r ℓ i and ℓ j are in terfering . Let 1 den ote the vector whose every entry is 1. Then the sufficient co ndition above is equiv alent to the condition Aτ ≤ T 1 o n th e rows of A , hence th e nam e r o w constraints . The proof of Pro position 1 gives a very efficient algorithm for che cking feasibility . It pr ovides both a distributed admis- sion contr ol m echanism as well as a distributed sched uling algorithm : wh en a link ℓ i that is curren tly inacti ve makes a de mand to be active f or du ration τ ( ℓ i ) , the admission control mechanism can be implemented efficiently by just checking the condition ab ove fo r ℓ i and its neigh bors. The informa tion requ ired by a link to ch eck this co ndition is just its de mand and the deman d of its neighbo rs. Further more, the distributed schedu ling algo rithm that meets the demand for lin k ℓ i needs to know only th e tim e in tervals already assigned to the n eighbo rs of ℓ i in order to de termine th e time interval fo r ℓ i . A. Row constraint poly tope an d in duced sta r n umber Giv en a conflict g raph, let P I denote its indep endent set polytop e. This poly tope is de fined as the conve x hull of the characteristic vectors of the in depend ent sets of the g raph. Note that P I is exactly equ al to the set o f all link dema nd vectors which are feasible within one u nit of time . For the giv en conflict graph, let P row denote th e set of all lin k demand vectors that satisfy the row con straints for T = 1 ; that is, P row := { τ ≥ 0 : τ ( ℓ i ) + τ (Γ( ℓ i )) ≤ 1 ∀ i } . Since the r ow co nstraints ar e suf ficient, P row ⊆ P I . Also, note that β P row = { τ : τ ( ℓ i ) + τ (Γ( ℓ i )) ≤ β } . Define th e scaling facto r β row := inf { β ≥ 1 : P I ⊆ β row P row } . Equiv alently , β row = sup τ ∈ P I max i { τ ( ℓ i ) + τ (Γ( ℓ i )) } . So P row ⊆ P I ⊆ β row P row , and β row is the sm allest scaling factor wh ich conv erts the sufficient co ndition into a necessary o ne. It has b een pointed out ( cf. [6 ]) th at the row con straints can b e a rbitrarily far aw ay f rom optimal. For example, suppose the network consists of lin ks ℓ 1 , . . . , ℓ d +1 , where ℓ 1 interferes with each of ℓ 2 , . . . , ℓ d +1 and there is no in ter- ference between the rem aining links. Th en the con flict graph is a star grap h. The link dem and vector ( ε, 1 − ε, . . . , 1 − ε ) is f easible within one unit o f time , but the row constrain t for ℓ 1 has value τ ( ℓ 1 ) + τ (Γ( ℓ 1 )) = ε + (1 − ε ) d which can be made arbitrarily close to d as ε approach es 0. This sho ws that β row ≥ d . W e prove next that the o pposite inequality also ho lds, i.e. th e row constraints can be a factor s away from the optimal s ched ule time for some demand vector only if the co nflict graph co ntains a star on ⌈ s + 1 ⌉ vertices as an in duced subgraph. Definition 2. The indu ced star nu mber of a graph H is defined by σ ( H ) := max v ∈ V ( H ) α ( H [Γ( v )]) . Hence, th e in duced star n umber of a graph is the numb er of leaf vertices in th e max imum sized star o f th e gr aph. This number determin es e xactly how close the ro w constraints are to optimal in the worst case: Theorem 3. Let G C be a conflict graph. The exact worst- case performance of the r ow con straints is g iven by β row = σ ( G C ) . Pr o of: See [ 4]. It f ollows that the row constraints, which are sufficient condition s, are a lso necessary if f P row = P I , which is the case iff σ ( G C ) = 1 , w hich is th e case iff each compo nent of G C is a complete gr aph. Thus, the row constrain ts above are also a necessary con dition if and only if the conflict grap h is the d isjoint u nion of c omplete graphs. While the induced star numbe r of a g raph ca n b e arb itrarily large, for special classes of networks stud ied in the literatu re th is quantity is b ounded by a fixed con stant. Th is happ ens to be in the case for unit disk graph s and for networks with primary interferen ce c onstraints. B. A str engthenin g of the r ow constraints W e sho wed above that the perfor mance of the row con- straints is determined b y the in duced star number σ ( G C ) . W e now show that a sligh t imp rovement to σ ( G C ) − 1 can be obtaine d. For simplicity of exposition, we shall a ssume in this sectio n that G C is conn ected; if this is not the ca se we can work with e ach c onnected co mponen t separately and the results her e still app ly . Recall that th e row c onstraint correspon ding to link ℓ i is that the sum total of the demand τ ( ℓ i ) and the dem ands of all its interfering ne ighbor s τ (Γ( ℓ i )) n ot exceed the av ailable resource T . It is easy to see that all the links in the network, except for any on e de signated link, say ℓ 1 , can ignor e the demand of up to one of its in terfering neig hbor s. Proposition 4. Given a ne twork an d its confl ict g raph G C , pick any d esignated link ℓ 1 ∈ L . A sufficient condition for τ to be feasible within duration T is that τ ( ℓ i ) + τ (Γ( ℓ i )) ≤ T , i = 1 τ ( ℓ i ) + { τ (Γ( ℓ i )) − min ℓ j ∈ Γ( ℓ i ) τ ( ℓ j ) } ≤ T , i = 2 , . . . , m. Pr o of: See [ 4]. Note that this su fficient c ondition is equ iv alent to the row constraints wh en G C is complete. The fo llowing stronger result can b e ob tained wh en th e graph is not an odd cycle. Proposition 5. Sup pose G C is n ot comp lete. Then the set of constraints τ ( ℓ i ) + { τ (Γ( ℓ i )) − min ℓ j ∈ Γ( ℓ i ) τ ( ℓ j ) } ≤ T , i = 1 , . . . , m is a suf ficient condition for τ to be feasible within du ration T if G C is n ot an odd cycle. Furthermor e, the smallest scaling factor th at co n verts this sufficient co ndition in to a n ecessary one is eq ual to exactly σ ( G C ) or σ ( G C ) − 1 , dep ending on the structur e of G C . Pr o of: See [ 4]. I I I . D E G R E E A N D M I X E D C O N D I T I O N S It was shown that the row constra ints pr ovided a simple, distributed sufficient condition f or feasibility of a giv en demand vector . I n this co ndition, there was exactly on e constraint associated with eac h link , n amely , the sum total of th e demand o f the link and demand s of its neighb ors not exceed th e av ailable re source. W e n ow d escribe an even simpler cond ition. W e call this the d egree condition since it requires k nowing, for each link, the d emand o f th at link and just the nu mber ( not actual dem ands) of links inter fering with it. Suppose link ℓ i interferes with exactly d ( ℓ i ) othe r links, i.e. in the conflict graph ℓ i has degree d ( ℓ i ) . Then , the following result p rovides another su fficient condition fo r admission control [1 0]: Proposition 6. A given dem and vector ( τ ( ℓ i ) : ℓ i ∈ L ) is feasible within duration T if τ ( ℓ i )( d ( ℓ i ) + 1) ≤ T for each ℓ i ∈ L . The p erform ance of th e degree c ondition is de termined, not surprisingly , by the maximum d egree of a vertex in the conflict graph. Mo re p recisely , d efine P degree := { τ ≥ 0 : τ ( ℓ i )( d ( ℓ i ) + 1) ≤ 1 , ∀ i } . Then P degree ⊆ P I by Proposition 6. Define β degree := inf { β ≥ 1 : P I ⊆ β P degree } . Let ∆( G C ) den ote the maximu m degree of a vertex in G C . Lemma 7. F or any confl ict graph G C , the exact worst- case performance of the degr ee condition is given b y β degree ( G C ) = ∆( G C ) + 1 . Pr o of: See [ 4]. This implies that th e sufficient degree con dition is also necessary (and hen ce o ptimal) iff G C is the empty graph, i.e. iff n o two links interfere with each other . I t is po ssible to combine the row c onstraints and degree con straints to get a sufficient condition which is strictly stronger, as shown in [10]: Proposition 8. A link deman d vector τ is fea sible within duration T if min { τ ( ℓ i ) + τ (Γ( ℓ i )) , τ ( ℓ i )( d ( ℓ i ) + 1) } ≤ T , ∀ ℓ i ∈ L. In general, P row , P degree ( P row ∪ P degree ) P mixed P I , where P mixed := { τ : min { τ ( ℓ i ) + τ (Γ( ℓ i )) , τ ( ℓ i )( d ( ℓ i ) + 1) } ≤ 1 , ∀ ℓ i ∈ L } . Let β mixed denote th e smallest scaling facto r that converts the sufficient mixed co ndition into a ne cessary one; henc e, giv en th e co nflict graph G C = ( L, L ′ ) a nd its indep endent set po lytope P I , we have th at P mixed ⊆ P I ⊆ β mixed P mixed and β mixed = sup τ ∈ P I max ℓ i ∈ L min { τ ( ℓ i )+ τ (Γ( ℓ i )) , τ ( ℓ i )( d ( ℓ i )+1) } . Theorem 9. The worst-case p erformance of the mixed condition is bo unded a s 1 + σ ( G C ) 2 ≤ β mixed ≤ σ ( G C ) , wher e σ ( G C ) d enotes th e indu ced star n umber o f G C . Mor eover , the lower and upp er bounds ar e tight; the star graphs r ealize the lower bo und, an d there exist g raph sequences for which β mixed appr oaches the upp er bou nd arbitrarily closely . Pr o of: See [ 4]. One general class of grap hs that inc ludes the star graph s, the even and o dd cycles, the com plete graphs and bip artite graphs ar e tho se that satisfy th e fo llowing property : for each vertex ℓ ∈ L in the graph G C = ( L, L ′ ) , the n eighbo rs o f ℓ induce a disjoint unio n of co mplete grap hs. For this gener al class of graph s there is a simp le expression for the exact value of β mixed : Theorem 10. Suppo se G C = ( L, L ′ ) satisfies σ ( G C [Γ( ℓ )]) ≤ 1 , ∀ ℓ ∈ L . Let d ( ℓ ) de note the numb er of neighbo rs of ℓ and le t η ℓ denote the number of connected compon ents induce d by the neig hbors of ℓ . Then β mixed = max ℓ ∈ L η ℓ (1 + d ( ℓ )) η ℓ + d ( ℓ ) . I V . C L I Q U E C O N S T R A I N T S A necessary condition for a given link demand vector to be feasible can be obtained as f ollows. Suppose th ere exists a sched ule of d uration 1 satisfy ing dem and τ . T hen if K is a clique in the co nflict graph, the time intervals assigned to the distinct links in K m ust be d isjoint, hence τ ( K ) ≤ 1 . Thus, a n ecessary cond ition for τ to be feasible within duratio n T is th at τ ( K ) ≤ T for every maxim al clique K in the co nflict graph. These con straints ar e called cliqu e co nstraints [6 ]. As before, we can associate a p olytop e with this nec essary condition ; define P clique := { τ : τ ( K ) ≤ 1 , ∀ K } ⊇ P I , where K runs over all the cliq ues (or eq uiv alently , over just all the m aximal cliqu es) o f th e conflict g raph. Using the notio n o f the imper fection ratio of graphs, bound s o n the sub optimality of clique co nstraints were obtained [6] fo r th e case of un it disk graph s. More p recisely , giv en a conflict graph G C and demand vector τ , let T ∗ ( τ ) denote th e min imum du ration o f a schedu le satisfying τ (the optimal value of this linear prog ram is also the smallest β such that τ ∈ β P I , and T ∗ ( 1 ) is often referr ed to as th e fractional chr omatic nu mber of G C ). L et T clique ( τ ) denote the ma ximum value of τ ( K ) over all cliques K in the conflict graph; so T clique ( τ ) ≤ T ∗ ( τ ) . The imperfectio n ratio o f a gr aph G C is d efined as imp( G C ) := sup τ 6 =0 { T ∗ ( τ ) / T clique ( τ ) } . This quantity h as been stu died in [5] ; it is finite and is achieved fo r any given grap h. I n th e definition above, for a given deman d vector τ , the n umerato r specifies the exact amount of resour ce required to satisfy the dem and, as determ ined by an optimal, cen tralized algorithm . T he denomin ator specifies a lower b ound on the resour ce re- quired to satisfy the demand, as d etermined by a particular distributed algorithm (the cliqu e co nstraints). Their ratio is the factor by which the distributed algorithm is away fro m optimal fo r th e given demand vector . The im perfection ra tio, which maximizes this r atio over all d emand p atterns, is the n the worst-case perf orman ce of the distributed algor ithm. The following gene ral result is implicit in [6] (where the authors f ocus on un it disk gra phs) and in [5]: Proposition 11. The la r gest scaling factor which converts the n ecessary cliqu e co nstraints into a sufficient c ondition is 1 / imp( G C ) ; i.e. the worst-case performan ce of the clique constraints is given by sup { β ≤ 1 : β P clique ⊆ P I } = 1 imp( G C ) ; and 1 imp( G C ) P clique ⊆ P I ⊆ P clique . V . E X A M P L E S In th is section w e apply th e results obtained so far to some special classes of network s and interference models. In Section V -A, we examine a mode l of interfer ence called pri- mary interfere nce c onstraints, which ha s b een well-studied in the literatu re for the centr alized setting; we examine the distributed version o f th e problem here . A. Primary interfer ence model Giv en a network G = ( V , L ) , su ppose two lin ks are considered to be interf ering if f they share one o r more endvertices in commo n. W e refer to this kind of interfe rence as prim ary interfere nce. This interf erence mod el arises, for example, fro m the assumption th at each node ca n co m- municate to only one other node at any given time. T his interferen ce mode l is perha ps the m ost well studied in the literature; f or examp le, see [7], [8], [ 12]. Th e co nflict gra ph for such a network is called a line graph . It can be s hown th at that if the con flict graph G C is a line grap h then σ ( G C ) ≤ 2 . It follows that fo r such networks the row co nstraints will b e at most a factor 2 away f rom o ptimal. More specifically , for this inter ference mo del, the row constraints o n the conflict g raph can b e r eformu lated o n the network g raph G = ( V , L ) as follows. Suppose link ℓ i is in cident between nodes u i and v i . Given link d emand vector τ , let τ ( u ) deno te th e sum of the demand s of all links inc ident in G to node u . Then the row c onstraint τ ( ℓ i ) + τ (Γ( ℓ i )) ≤ T in the conflict g raph G C = ( L, L ′ ) is equiv alent to the co nstraint τ ( u i ) + τ ( v i ) − τ ( ℓ i ) ≤ T in the network grap h. This e quiv alence yields th e f ollowing sufficient conditio n: Corollary 12. Let G = ( V , L ) b e a network graph, a nd suppose two links inte rfer e if an d o nly if th ey ar e inciden t to a common node. Then ( τ ( ℓ ) : ℓ ∈ L ) is feasible within duration T if for ea ch ℓ = { u , v } , τ ( u ) + τ ( v ) − τ ( ℓ ) ≤ T . This sufficient c ondition is a facto r of at most 2 awa y fr om optimal. Another distrib uted algorithm th at can be used i n networks having p rimary interference constraints is g iv en by th e clique constraints. T ri vially , a n ecessary cond ition for τ to be feasible within duration T is that τ ( K ) ≤ T fo r all cliq ues K in the con flict g raph. By Pr oposition 11, the p erfor mance of the cliq ue co nstraints is determ ined by the imper fection ratio of the con flict grap h. It is known th at th e im perfection ratio of a lin e graph is at most 1.2 5 [5, Prop. 3. 8]. T his means that a suf ficient cond ition for τ to be feasible wtihin duration T is that 1 . 25 τ ( K ) ≤ T for all cliques K in the conflict gra ph. Since G C is a line grap h, each cliqu e K in G C correspo nds eith er to a set of links K ⊆ L that are all incident to a co mmon node in the network grap h G = ( V , L ) or to a set of thre e link s that for m a trian gle in th e network graph. For v ∈ V , let τ ( v ) d enote the sum of the demand s of all link s in cident to v in the network g raph. Theorem 13. Let G = ( V , L ) b e a network graph, and suppose two links inte rfer e if an d o nly if th ey ar e inciden t to a common node. Then ( τ ( ℓ ) : ℓ ∈ L ) is feasible within duration T if τ ( v ) ≤ 0 . 8 T , ∀ v ∈ V and τ ( uv ) + τ ( v w ) + τ ( uw ) ≤ 0 . 8 T , ∀ u, v , w ∈ V . This sufficien t con dition is a factor o f at most 1. 25 away fr o m optimal. An imp ortant aspe ct of this r esult is that, though the number of maxima l clique s in a g eneral gr aph can g row exponentially with th e size of the grap h, the num ber of maximal clique s in a line grap h g rows o nly poly nomially in the size of the graph . Thu s, unlike for general g raphs, for lin e graphs the clique con straints provid e an efficient d istributed algorithm for ch ecking feasibility of a giv en deman d vecto r . Remark : A result d ue to Shann on on the edge-colorin g of multigrap hs [13] implies that: for a g iv en network graph G = ( V , L ) , a sufficient con dition for ( τ ( ℓ ) : ℓ ∈ L ) to be feasible with in duration T is that τ ( v ) ≤ 2 T / 3 , ∀ v ∈ V . Theorem 13 improves this bo und fro m a factor of 2 /3 to 0.8. This improvement is possible because, unlike in the classical edge-co loring problem , fractional co loring solution s are a lso admissible in our framework. Furthermo re, the sufficient condition in Theor em 13 is less loc alized, in that each no de in the network graph needs to know not only the sum total of the dem ands of all links incid ent to it, but also the demands of all link s b etween its neig hbors. V I . C O N C L U D I N G R E M A R K S W e introduced the notion of the induc ed star num ber of a grap h and show that it determines the exact worse-case perfor mance of the row constra ints. Th e pe rforman ce of two other sufficient con ditions - nam ely the d egree condition and mixed con dition - was also studied. Finally , the results obtained thus far are ap plied to some specific classes o f networks an d interference mod els. These results imply that for some spec ial classes o f net- works an d interf erence mod els, th ere exist simple, efficient, distributed ad mission con trol mechan isms an d sched uling mechanisms wh ose p erform ance is within a bou nded factor away from that o f an op timal, centr alized m echanism. A detailed repo rt con taining proof s of the r esults presented here is av ailable from the au thor [4]. A C K N O W L E D G E M E N T S Thanks are du e to pro fessor Parmesh Raman athan for suggesting this direc tion o f scaling th e sufficient c ondition s. R E F E R E N C E S [1] D. Bert sekas and R. G. Gallager . Data Networks . Prentice Hall, 1992. [2] R. L. Brooks. On colouring the nodes of a network. Pr oc. Cambridge Phil. 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