The Primal Simplex Approach o he QoS Rouing Problem Ying Xiao, Krihnaiyan Thulairaman Univeriy of Oklahoma, Norman, OK {ying_xiao, hulai}@ou.edu Guoliang Xue Arizona Sae Univeriy, Tempe, AZ xue@au.edu Abrac Qualiy-of-Service (QoS) rouing problem require he deerminaion of a minimum co pah from a ource node o a deinaion node in a daa nework uch ha he delay of he pah i bounded by (> ). Thi problem alo known a he conrained hore pah (CSP) problem i NP-hard. So, heuriic and approximaion algorihm have been preened in he lieraure. Among he heuriic, he LARAC algorihm, baed on he dual of he LP relaxaion or he Lagrangian relaxaion of he CSP problem i very efficien. In hi paper we udy he primal implex approach o he LP relaxaion of he CSP problem and preen an approximaion algorihm o hi problem. Several iue relaing o efficien implemenaion of our approach are dicued. Experimenal reul comparing he performance of he new algorihm wih ha of he LARAC algorihm are preened.. Inroducion Recenly here ha been coniderable inere in he deign of communicaion proocol ha deliver cerain performance guaranee ha are uually referred o a Qualiy of Service (QoS) guaranee. A problem of grea inere in hi conex i he QoS rouing problem ha require he deerminaion of a minimum co pah from a ource node o a deinaion node in a daa nework ha aifie a pecified upper bound on he delay of he pah. Thi problem i alo known a he conrained hore pah (CSP) problem. The CSP problem i known o be NP-hard []. So, in he lieraure, heuriic approache and approximaion algorihm have been propoed. Heuriic, in general, do no provide performance guaranee on he qualiy of he oluion produced, hough hey are uually fa in pracice. On he oher hand, -approximaion algorihm deliver oluion wihin arbirarily pecified preciion requiremen. Reference [2] [4] and he reference herein conain mo of he curren lieraure on approximaion algorihm for he CSP problem. A regard heuriic, he LHWHM algorihm [5] i a imple heuriic which i very fa (requiring only one or wo invocaion of Dijkra hore pah algorihm) and produce oluion which are uually found o be of accepable qualiy in pracice. Reference [6] alo dicue furher enhancemen of he LHWHM algorihm. There are heuriic ha are baed on ound heoreical foundaion. Thee algorihm are baed on oluion o he dual of he linear programming relaxaion of he CSP problem. The fir uch algorihm wa repored in [7] by Handler and Zang. Thi i baed on a geomeric approach (wha i alo called he hull approach by Mehlhorn and Ziegelmann [8]). More recenly, in an independen work, Jüner e al. [9] developed he LARAC algorihm which alo olve he Lagrangian relaxaion of he CSP problem. In conra o he geomeric mehod, hey ued an algebraic approach. In anoher independen work Blokh and Guin [] defined a general cla of combinaorial opimizaion problem of which he CSP problem i a pecial cae, and propoed an approach o hi problem. In a recen work, Xiao e al. [] drew aenion o he fac ha he algorihm in [7]- [] are equivalen. In view of hi equivalence, we hall refer o hee algorihm imply a he LARAC algorihm. Mehlhorn and Ziegelmann [8] have developed everal inighful reul. In an unpublihed work [2], Jüner eablihed he rong polynomialiy of he LARAC algorihm. Ziegelmann [3] provide a fairly complee li of reference o he lieraure on he CSP problem. Anoher reference where one could find a urvey of QoS rouing reearch i [4]. A recen work on QoS rouing i [5] where he auhor udy and preen approximaion algorihm for minimum co dijoin pah elecion under delay conrain. In [6] he auhor udy he dijoin pah elecion problem uing he primal implex mehod. Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
In hi paper we preen a novel approach o he QoS rouing problem, making a deparure from currenly available approache. We udy he problem uing he primal implex mehod of linear programming and exploiing cerain rucural properie of nework. The re of he paper i organized a follow. In ecion 2, we define he CSP problem and preen i ineger linear programming (ILP) formulaion a well a i linear programming (LP) relaxaion. Thi formulaion i he ame a he LP formulaion of he minimum co flow problem excep for an addiional conrain due o he delay requiremen. Thi addiional conrain give rie o everal queion ha need o be inveigaed o achieve an efficien implemenaion of he primal implex mehod. Thi lead u o he definiion of an equivalen problem on a ranformed nework, called he TCSP problem. Secion 3 9 dicu he revied implex mehod and i applicaion on RELAX-TCSP, he relaxed form of he TCSP problem. The complee algorihm called NBS algorihm and i peudo-polynomial ime complexiy are preened in ecion. Thi ecion alo how how o exrac an approximae oluion o he original CSP problem from he opimum oluion o he RELAX-TCSP problem and derive bound on he qualiy of hi oluion wih repec o he opimum oluion. In ecion, experimenal reul comparing he NBS algorihm wih he LARAC algorihm are preened. Secion 2 conclude wih a ummary of he main conribuion. To conerve pace proof of reul are omied. 2. The CSP problem and an equivalen ranformed problem (TCSP) In hi ecion, we fir define he conrained hore pah problem. To achieve an efficien implemenaion of our algorihm, we hall define an equivalen problem on a ranformed nework. Then we preen he ineger linear programming formulaion of hi ranformed problem called TCSP problem. Relaxing he inegraliy conrain of he TCSP problem lead o he RELAX-TCSP problem. Definiion. Conrained Shore Pah (CSP) Problem: Conider a direced nework G(V, E) where V i he e of node and E i he e of link of he nework. Each link (u, E i aociaed wih wo ineger weigh c uv > (co) and d uv > (delay). For any pah p (or cycle wih a given orienaion) define he co c(p) and delay d(p) of p a c( p) c c, d ( p) d d, uv uv uv uv ( u, p ( u, p ( u, P ( u, P where p + (p - ) i he e of forward (backward) link on p a we ravere p from he ar node o he end node. A pah i called a direced pah if here are no backward link in he pah. Given wo node, and ineger >, a direced - pah p i aid o be feaible if d(p). The CSP problem i o find an - pah p* = arg min{c(p) p i a feaible - pah}. We ue he erm link and arc inerchangeably. Someime we may ue c(u,, d(u, and e uv o repreen he link co c uv, link delay d uv and he link e = (u,, repecively. Wihou lo of generaliy we aume ha for every node i, here i a direced pah from i o he deinaion node. Tranformaion of he CSP problem: To olve he CSP problem efficienly, we ranform i o an equivalen one on a ranformed nework defined a follow: The graph of he ranformed nework i he ame a ha of he original problem, ha i, G(V, E). For (u, E, d' uv and c' uv in he ranformed problem are given by d' uv = 2 d uv and c' uv = c uv. The new upper bound ' in he ranformed problem i given by ' = 2 +. Theorem. An - pah p* i a feaible oluion (an opimal oluion) o he CSP problem iff i i a feaible oluion (an opimal oluion) o he TCSP problem. In he re of he paper, we only conider he TCSP problem. Even hough many of our reul hold for he CSP problem oo, Lemma 4 and 7 and Theorem 4-5 hold rue only for he TCSP problem. Alo we ue (being odd) and d uv (being even) o denoe he delay bound and link delay in he ranformed problem, repecively. Noice ha he ranformaion doe no change he co of any pah in he nework. The TCSP problem can be formulaed a he following ILP problem. TCSP: Min ( u, E c uv x uv () Subjec o, if u ; xuv xvu, if u ; { v ( u, E} { v ( v, u) E}, oherwie. (2) d u v E uv xuv w (, ) (3) (u, E, x uv = or (4) Noe: The variable w in (3) i he lack variable correponding o he delay conrain. Relaxing he inegraliy conrain in (4) reul in he RELAX-TCSP problem given below. TCSP: Min Subjec o ( u, E c uv x uv (5) Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
, if u ; xuv xvu, if u ; (6) { v ( u, E} { v ( v, u) E}, oherwie. d uv xuv w (7) ( u, E (u, E, x uv (8) Le he link be labeled a e, e 2, e m and he node be labeled a, 2., n. We hall denoe he delay of edge e i a d i and he co of e i a c i. The incidence marix of G ha m column, one for each link and n row, one for each node. The rank of hi marix i (n - ), and removing any row of hi marix will reul in a marix of rank (n - ). We denoe hi reuling marix a H. We alo aume ha he row removed from he incidence marix correpond o node n. Alo we aume ha he column of H correponding o link e k will be denoed by he vecor h k. For e k = (i, j), we have h k = (h,k,, h i,k, h j,k, h n-,k ) wih all componen being excep for h i, k = and h j, k = -. Le H A (, 2..., m) a a a a m, (9) D where D = (- d,- d 2 - d m ), () hi ai i m d,, and () i a m. (2) Alo, le x be he column vecor of he m flow variable x uv and he lack variable w, and c be he row vecor of he co (c, c 2, c m, ). Noe ha he co of he lack variable i. The above LP formulaion of he RELAX- TCSP problem can now be wrien in marix form a follow. RELAX-TCSP: Min: c x Subjec o Ax = b, (3) x for (u, E, where b = (b, b n-, -) wih b =, b = - and b i = for i,. The re of he paper i concerned wih he implex mehod baed oluion of RELAX-TCSP. 3. Baic oluion of RELAX-TCSP Simplex mehod of linear programming ar wih a baic oluion and proceed by conrucing one baic oluion from anoher. A baic oluion coni of wo e of variable, baic and non-baic. For he RELAX- TCSP problem under conideraion, all he non-baic variable in a baic oluion will have zero value. Given a baic oluion, we hall denoe by G b he ubgraph of G correponding o he baic variable (excep he lack variable if i i in he baic oluion) in hi oluion. The ubgraph G b will be called he ubgraph of he baic oluion. The non-ingular ubmarix of A defined by he baic variable i called a bai marix or imply, a bai. In hi ecion we preen cerain imporan properie of he baic oluion of he RELAX-TCSP problem. - Branching Poin flow Fig.. Srucure of baic oluion graph - - Lemma [7]: Le G (V, E) be a direced nework wih a lea one cycle W (no neceary direced). Aigning an arbirary orienaion o W, le U = (u, u 2, u 3 u m ), where, if e j W and he orienaion of e j agree wih he orienaion of W; u j, if e j W and he orienaion of e j diagree wih he orienaion of W;, oherwie. Then, H U =. We hall denoe by d(w) he igned algebraic um of he delay of he link in a cycle W a we ravere around he cycle along he given orienaion. Lemma 2. The ubgraph G b of a baic oluion conain a mo one cycle (See Fig. ). Lemma 3. If here i a cycle W in G b, hen d(w). Lemma 4. The flow vecor x in (3) aifie x and a link aume poiive flow (> ) iff i i on ome direced - pah in G b (See Fig. ). If G b conain no cycle or he cycle i link dijoin wih he - pah in G b, hen he link flow are ineger ( or ). Lemma 5.The bai marix conain he la row of A a) Tree baic oluion b) Baic oluion wih a cycle < < Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
defined in (9) and he (n - ) row of H. 4.2 Solve he yem Y B = c B 4. Revied implex mehod on he RELAX- TCSP problem In hi ecion, we fir briefly preen he differen ep in he revied implex mehod of linear programming which i decribed in deail in [8]. We hen derive formula required o idenify he enering and he leaving variable ha are needed o generae a new baic oluion from a given baic oluion. 4. Revied implex mehod Conider an arbirary linear programming (LP) problem in he andard form. Min c x Subjec o A x = b, x. Here A i an n (m + ) marix wih rank (A) = n, x = (x, x m + ), c = (c c m + ), b = (b, b 2, b n ). Each feaible baic oluion x* i pariioned ino wo e, one e coniing of he n baic variable and he oher e coniing of he remaining m + n non-baic variable. Thi pariion induce a pariion of A ino B and A N, a pariion of x ino x B and x N and a pariion of c ino c B and c N, correponding o he e of baic variable and he e of non-baic variable, repecively. The bai marix B i noningular. Revied Simplex Mehod: Sep : Solve he yem Y B = c B, Y = (y, y 2 y n ). Sep 2: Chooe an enering column. I may be any column a i of A N uch ha Y a i i greaer han he correponding componen of c N. The curren oluion i opimal if here i no uch column. Sep 3: Solve he yem B V = a i, V = (v, v 2, v n ). Sep 4: Find he large uch ha x* B V. If here i no uch, hen he problem i unbounded; oherwie, a lea one componen of x* B V i equal o and he correponding variable leave he bai. Sep 5: Se he value of he enering variable a and replace he value x* B of he baic variable by x* B V. Replace he leaving column of B by he enering column and in he bai heading, replace he leaving variable by he enering variable. Then go o ep. In he following we olve he yem of equaion in ep and 3 and derive explici formula for Y and V. Le Y = (y, y n, ). Here y, y n, are called poenial (or dual variable) and Y i called he poenial vecor. Each y i, i =, 2, n - i he poenial aociaed wih node i (or he row i) and i he poenial aociaed wih he la row (delay conrain row) of A. Now conider Y B = c B (4) Thi yem of equaion ha n equaion in n variable. We ge he following from (4). For each link e k = (i, j) in G b, (y, y n-,) h k = c ij. Tha i, y i y j d ij = c ij, if i n or j n; y i d in = c in, if j = n and -y j d nj = c nj,if i = n (5) From he above, we can ee ha we can e he poenial of he node n a any conan, for example, in all compuaion ha follow. Definiion 2: ) For link e k = (i, j), c(e k, ) = d ij + c ij i called he acive co of link (i, j). 2) r(i, j) = y j y i + d ij + c ij i called he reduced co of link (i, j). 3) The reduced co of w i given by r(w) =. 4) The reduced co of a pah p i defined a r ( p ) r ( i, j ) r ( i, j ) ( i, j ) p ( i, j ) p I can be een from (5) ha for any link (i, j) in G b r(i, j) = y j y i + d ij + c ij =. (6) From (6) we alo have ha for any pah p from i o j and any cycle W in G b r(p) = y j y i + d(p) + c(p) =, and (7) r(w) = d(w) + c(w) =. Lemma 6: If G b conain a cycle W, hen = - c(w)/d(w); Oherwie, =. I can be een from ep 2 of he revied implex mehod ha a non-baic variable i eligible o ener he bai if i reduced co i negaive. Noe ha he lack variable i eligible o ener he bai if <. Once we have compued he value of a in Lemma 6, he oher poenial y i can be calculaed uing equaion (7) and elecing he pah in G b from node n o node i. 4.3 Solve he yem B V = a k We how how o olve he yem of equaion B V = a k. Conider hree cae: Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
Cae : G b conain only n - link, i.e., here i no cycle in G b and he lack variable w i a baic variable, and ome link e k = (i, j) i he enering variable. Cae 2: The baic variable are aociaed wih n link and he enering variable i e k = (i, j). Cae 3: The baic variable are aociaed wih n link and he enering variable i he lack variable w. Soluion in all he hree cae are ummarized in he following heorem. Theorem 2. a) If G b conain no cycle and he enering variable i an in-arc e k = (i, j), hen he vecor V defined below i he deired oluion o B V = a k, where W' i he new cycle formed by adding he in-arc e k and he orienaion of W' i choen o agree wih he direcion of e k. The vecor V = (v v n ) i defined a :, if i n and he link correponding o he v i i h column of B i in W' and i orienaion agree wih he cycle orienaion;, if i n and he link correponding o he i h column of B i in W' and i orienaion diagree wih he cycle orienaion; d( W' ), if i n, oherwie b) If G b conain a cycle W and link e k = (i, j) ener he bai, hen V = -V' p + (d(w')/d(w)) V i he oluion of B V = a k, where d(w') and d(w) are he delay of cycle W' and W, repecively and V' p and V are defined by he cycle W' and W, repecively. c) If G b conain a cycle W and he enering variable i he lack variable w, hen V = ( / d(w)) V i he oluion o B V = a k, where V i defined by cycle W. 5. Iniializaion To conruc an iniial baic feaible oluion we fir deermine a panning ree conaining a feaible - pah. Thi can be done by applying Dijkra algorihm o compue he hore pah ree wih repec o he delay from all node o he deinaion node. If he reuling - pah in he ree i infeaible, hen no feaible pah exi and he algorihm erminae. Wihou lo of generaliy we aume ha he - pah i feaible. Clearly in he baic oluion correponding o he panning ree eleced a above, he flow in all he link in he - pah in he panning ree will be equal o one, and flow in all oher link will be zero. Since he delay of every link in he TCSP problem i even and he upper bound on pah delay i odd we can ee ha he lack variable w ha nonzero value and hu i i in he iniial baic feaible oluion. 6. Pivo rule and rucure of baic oluion for he TCSP problem In hi ecion we udy he rucure of he ubgraph of baic oluion generaed by he implex mehod. The ubgraph G b of he iniial baic feaible oluion ha (n - ) link and he n h variable in hi baic oluion i he lack variable w, and w > in he oluion. A hi iniial ep, =. Define d(g ) = x b u v G uv d uv. (, ) b By (7), d(g b ) = w. Now one of he following wo poibiliie occur in he following pivo.. The implex mehod conruc a new panning ree oluion wih he lack variable w remaining nonzero in he new oluion. 2. The implex mehod conruc a G b ha conain one cycle W (formed by adding he in-arc). If he lack variable w become fir when we puh he flow along he cycle W, hen w become nonbaic wih repec o hi oluion. If he cycle W doe no hare link wih he - pah in G b, hen he flow on all he link in W will be zero and hu all he link flow will be eiher or. Thi would hen imply ha he delay of he - pah (being even) will no be equal o he upper bound (being odd), making he value of w nonzero. Thi i a conradicion. So he only cycle W in G b mu have ome common link wih he - pah in G b. The cycle W canno be a direced cycle. If i were a direced cycle, hen he reduced co of he enering link will be equal o he um of he co of he link in W. Thi um i a poiive number conradicing he requiremen ha he reduced co of he enering link mu be negaive (ep 2 of he revied implex mehod). Alo, he flow value on all he link in W mu be nonzero, for oherwie all he link flow will be eiher or making w nonzero. Summarizing, when he fir ime a G b wih a cycle i encounered i will be necearily of he form hown in Fig.. (b). Flow on he link in he cycle will be or. The implex mehod will elec he value of > in uch a way uch ha d(g b ) =. Though he cycle in he G b encounered he fir ime afer iniializaion will no be a direced cycle, in a ubequen ep a G b wih a direced cycle may be creaed. To achieve an efficien implemenaion of he implex mehod, we would like o avoid generaing any G b conaining a direced cycle. Thi can be achieved by he pivo rule P given nex. Pivo Rule P: The lack variable w aume he highe prioriy o be eleced a enering variable. Theorem 3. If he pivo rule P i followed and he implex mehod on he TCSP i iniialized a in ecion 5, hen no baic oluion ubgraph G b conaining a direced cycle will be creaed. Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
7. Ani-cycling raegy A baic oluion in which one or more baic variable aume zero value i called degenerae. Simplex pivo ha do no aler he baic oluion are called degenerae. Furhermore, a baic oluion generaed a one pivo and reappearing a anoher will lead o cycling. Since degenerae pivo do no reul in any improvemen of he oluion, hey are alo a caue of inefficiency. We preen wo raegie o handle degeneracy. The fir one o be preened in hi ecion i he ani-cycling raegy which i he modificaion and exenion of Cunningham ani-cycling raegy [8]. The econd raegy o be preened in he following ecion i deigned o avoid performing degenerae pivo. In our dicuion we hall aume ha he node ha been eleced a he roo node. Definiion 3. Given a feaible baic oluion ubgraph G b, we ay ha he link (u, G b i oriened oward (rep. away from) he roo if any of he pah in G b from he roo o u (rep. pae hrough v (rep. u). A feaible baic oluion G b wih correponding flow vecor x i aid o be rongly feaible if every link (u, of G b wih x uv = i oriened oward he roo. If he ou-arc (u, i no a link of he cycle in he bai oluion, hen G b - (u, conain exacly wo componen G b (u) and G b ( uch ha u G b (u) and v G b (. If he roo i in G b (, link (u, i oriened oward he roo; oherwie i i oriened away from he roo. See Fig. for an example of a rongly feaible G b. Acually, by Lemma 4, all link in G b are oriened oward if he bai i rongly feaible. Lemma 7. For any degenerae pivo, he ou-arc i no on he cycle of he curren G b. Theorem 4. If he ubgraph G b of feaible baic oluion generaed by he implex mehod are rongly feaible hen he implex mehod doe no cycle. 8. Avoiding degenerae pivo In hi ecion, we develop a raegy which avoid performing degenerae pivo. Enhanced Pivo Rule P2: If here i a choice for elecing he enering variable, hen elec an enering variable in he following order of preference: a) The lack variable if i i eligible o ener. b) Eligible link whoe ail node are on ome direced - pah in he curren G b. A we dicued in ecion 6, rule a) above guaranee ha every G b i of one of he wo form in Fig.. Boh hee ubgraph of baic oluion are rongly feaible. Conider nex rule b). Suppoe we can find an in-arc e = (u, according o rule b). Le W' denoe he new cycle in G b + e wih i orienaion defined a he direcion of e. I can be een ha he flow on all link in W' whoe direcion diagree wih ha of W' are of nonzero and hu we can puh poiive amoun of flow along he cycle unil he flow on ome link of he - pah (whoe direcion diagree wih he orienaion of W') reach zero. By removing one uch link wih zero flow, we obain a new G b. In fac, we can elec he ou-arc in uch a way ha he reuling G b i alo rongly feaible (ee nex ecion). Thi pivo will no lead o degeneracy. On he oher hand, if no uch link i eligible o ener he bai (noe: in hi cae i nonnegaive), hen we have no opion bu o perform a degenerae pivo. To avoid performing degenerae pivo we proceed a follow. Le P be he e of node on he - pah in he curren baic ubgraph G b. Aign co o link in he nework a follow: Link co c(u, wih u P and v P i e a c(e uv, ) + y v ; Oherwie c(u, i e a c(e uv, ). Now condene all he node in P ino a ingle node, ay, R. and revere he direcion of all he link. Le he reuling nework be called N'. Noe ha none of he link wih boh i end in P will be in N'. Now ue Dijkra algorihm on N' and obain he hore pah ree wih node R a he ar node. The link of G correponding o he link of he hore pah ree of N' and he link wih heir boh end node in P will be a new G' b (Noice ha hi operaion preerve he rongly feaibiliy of G b and will no change he value of ). Le he hore diance value of he node u compued by he algorihm be denoed a d(u). Then he poenial of he node wih repec o G' b will be:. For each node u P, y u = d(u), and 2. For all oher node (all he node in P) he poenial are he ame a in he previou G b. Now, (u,, u P, y u = d(u) d( + c(e uv, ) = y v + c(e uv, ), which implie ha for all uch link, r(u, = y v y u + d uv + c uv and hu link whoe ail are no in P are no eligible for choice a in-arc. Since he above operaion doe no affec he value of, w i no eligible eiher. Thu we can only conider arc whoe ail are in P (par (b) of enhanced rule P2). If we ill canno find an in-arc according o enhanced rule P2 afer he above operaion, i implie ha we have go he opimal baic oluion becaue no enering variable i available. Thi procedure i denoed a Modified-Dijkra and i implemenaion i much impler han he decripion above. 9. Finding a leaving arc (ou-arc) Suppoe he curren feaible baic oluion G b i rongly feaible and link e = (u, i he in-arc. If G b conain a cycle W, hen he flow can be decompoed ino exacly wo - pah. We define he branching poin a he fir node on W a we ravere he pah from node Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
o (ee Fig.. (b)). In hi ecion, e and e' alway denoe he in-arc and ou-arc, repecively. Claim : If he curren baic oluion G b i rongly feaible and i no opimal, hen one of he arc e' inciden o he branching node or he ail node of he in-arc e i eligible for choice a ou-arc and G b + e e' i ill rongly feaible. We prove he claim by enumeraing all poible cae and deermining he leaving variable in each cae uing Theorem 2 and ep 4 of he revied implex mehod. Le he cycle creaed by adding he in-arc be denoed by W' wih i orienaion defined a ha of he in-arc. When we chooe he leaving variable, we alway chooe i uch ha he rong feaibiliy of he baic oluion i preerved. Cae : Slack variable w i in he baic oluion (he curren G b i a ree, = and w > ). Thi correpond o Theorem 2 (a). According o ep 4 of he revied implex mehod, we need o conider only he enrie of V ha are or d(w') if d(w') >. Wihou lo of generaliy, aume d(w') >. Thee enrie correpond o he link of W' ha lie on he - pah of he curren G b or he lack variable w. The correponding enrie in he curren baic oluion x* B are for he link and i curren value for w. The minimum value of ha aifie he conrain x* B V will be deermined by he inequaliie and w d(w'). Thu he minimum value of will be min{, w / d(w')}. Since w = d(g b ) i odd and d(w') i even, w / d(w'). So, if w < d(w'), w will leave he bai. Oherwie, he link in W' ha lie on he - pah in he curren G b are eligible o leave he bai. We hall elec he unique link e' on he - pah in G b ha i inciden o he ail node of he in-arc. Thi guaranee ha he new G b, denoed a G' b i rongly feaible. Noice ha if w leave he bai, w = in G' b. Thi mean ha d(g' b ) =. In hi cae, G' b conain wo - pah p and p 2 wih flow and, repecively (ee Fig. ). The value of can be calculaed from he equaion d(p ) + ( ) d(p 2 ) =. Thi line of argumen will be ued in deermining he leaving arc in Cae 2 dicued below. Cae 2: The baic oluion coni of n link, i.e., here i a cycle W wih branching poin a in he baic oluion. The lack variable w i eligible o ener he bai if <. Then according o par a) of pivo rule P2, we le w ener he bai and hall elec one of he wo link in he curren G b ha are inciden on he branching poin a o leave he bai. The choice can be made according o Cae 3 in ecion 4.3. Suppoe >. An in-arc will creae a new link W'. Thi correpond o Cae 2 in ecion 4.3. We need o conider four ub-cae ha capure all poibiliie. Wihou lo of generaliy, we aume ha he orienaion of W i clockwie and he orienaion of W' agree wih he direcion of he in-arc. Cae 2.: Poible ou-arc: (, 2), (3, 5) and (3, 4). Here, (x 2, x 35, x 34 ) = (,, - ) and hu he ou-arc correpond o he fir zero componen in he following formula a increae from. (,, - ) (, d(w') / d(w), - d(w') / d(w)) = (, d(w') / d(w), - + d(w') / d(w)). Cae 2.2: Poible ou-arc: (, 2), (2, 7) and (2, 3). in-arc 7 6 W' W 2 3 4 Then he ou-arc correpond o he fir componen reaching in he following formula a increae. (x 2, x 27, x 23 ) - (, + d(w') / d(w), - d(w') / d(w)) = (,, ) (, + d(w') / d(w), - d(w') / d(w)). Cae 2.3: Poible ou-arc: (2, 3), (2, 9) and (4, 5). The ou-arc correpond o he fir zero componen in he following formula when increae. (x 23, x 29, x 45 ) (- d(w') / d(w), d(w') / d(w), - d(w') / d(w)). Cae 2.4: Poible ou-arc: (2, 3), (, 6) and (, 2) in-arc W' W 2 3 4 W 2 W 6 5 9-3 4 6-8 in-arc W' 5 5 in-arc 2 3 7 Ou-arc correpond o he fir zero componen in he following formula a increae. (x 23, x 6, x 2 ) ( d(w') / d(w), d(w') / d(w), d(w') / d(w)).. The Nework Baed Simplex algorihm (NBS): complexiy and performance We fir preen he complee algorihm (given below) for he Nework Baed Simplex (NBS) algorihm for he RELAX-TCSP problem. Then we how ha our 6 5 W' 4 Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
algorihm i of peudo polynomial ime complexiy. We alo how ha we can exrac from he opimum oluion o he RELAX-TCSP problem a feaible oluion o he TCSP (and hence he original CSP) problem and derive bound on how much he co of hi oluion deviae from he co of he opimum oluion o he original CSP problem. Procedure NBS Tranform he original nework a in ecion 2. Find an iniial feaible baic oluion a in ecion 5. loop { if ( < ) hen Le lack variable w be he enering variable (rule (a) of Pivo rule P2) ele if an in-arc aifying rule (b) of Pivo Rule P2 i available hen Chooe one of hem a he enering variable. ele { invoke procedure Modified-Dijkra on he acive co o updae he poenial. if an in-arc aifying rule (b) of Pivo Rule P2 i available hen Chooe one of hem a he enering variable. ele op /*ha reached he opimal condiion*/ } } Deermine a leaving variable a in ecion 9. Updae he flow and he poenial a in ecion 4... Complexiy analyi We nex preen a complexiy analyi of he NBS algorihm and eablih i peudo-polynomial ime complexiy. Fac : If here i no cycle in he baic oluion ubgraph, hen for each link e uv, he aociaed flow x uv i eiher or. If here i a cycle W in G b, x ij i eiher or no le han / d(w). Fac 2: If e uv i he in-arc and W' and W are he newly creaed cycle and he old cycle (if i exi), repecively, we have < y u y v d uv - c uv = d(w') + c(w') = c( W '), ; c( W ') d( W ) d( W ') c( W ) / d( W ),. Fac 3: Le be he maximal flow ha can be puhed on he new cycle W'. Suppoe ha e uv and x uv are a link and i flow in he baic oluion, repecively. Then i only conrained by x uv ( + d(w') / d(w) ), and and hence max min{, /( d(w) + d(w') )}=/( d(w) + d(w') ). Fac 4: Le T and T' be wo conecuive feaible baic oluion in he implex mehod and c(t) denoe he co of he flow aociaed wih he baic oluion T. If c(t') < c(t) and D i he maximal link delay, hen c(t') c(t) = y u y v - d uv - c uv /(2n 2 D 2 ). Theorem 5. NBS algorihm erminae wihin 2n 3 D 2 C pivo, where n i he number of node and D (rep. C) i he maximal link delay (rep. co)..2. An approximae oluion o he TCSP / CSP problem and performance bound If he opimal baic oluion ubgraph for he RELAX- TCSP problem conain no cycle, hen clearly he - pah in hi ubgraph i alo he opimum oluion o he original CSP problem. On he oher hand, if he opimal baic oluion graph conain a cycle, hen he opimum flow can be decompoed ino flow along wo direced - pah p and p 2 wih poiive flow along each pah. Lemma 8. If c(p 2 ) c(p ), hen c(p 2 ) c(p*) c(p ) and d(p 2 ) d(p ), where p* i he opimal pah of he original CSP problem and i he delay bound. I follow from he above lemma ha he pah p i a feaible oluion o he CSP problem. We may ue hi a an approximae oluion o he original CSP problem. We nex evaluae he qualiy of hi approximae oluion. Theorem 7. Le p and p 2 be he wo pah derived from he opimal oluion o he RELAX-TCSP problem wih c(p ) c(p 2 ), hen c( p) c( p2 ) ( ), and c( p*) c( p ) d( p2 ) d( p ) ( ), where i he flow on pah p a erminaion and i he delay bound. Uing a pecial example below, we can how ha no conan facor approximaion oluion baed on relaxaion approach (including our approach and LARAC) i poible. However, numerical imulaion how ha he approximae oluion i very cloe o he opimum. For cloing he gap beween he opimum value and he approximae value ee [9]. Fig. 2. Couner example, 4, - 2 4, - 4, Le OPT, OPT S and denoe he opimal co, he co of he pah reurned by relaxaion mehod and he delay upper bound. In Fig.2, he olid link correpond o he baic variable in he opimal bai. Thu OPT S = - 4. Co, Delay Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
Obviouly, OPT = 4. So OPT S OPT / OPT = ( - 8) / 4, where can be pecified arbirarily. algorihm wih a rongly polynomial algorihm baed on parameric earch [9], denoed a PARA. Co 5 4 3 2 (a) Qualiy of Soluion on Regular graph (Ou-Degree = 6) LHWHM NBS OPT 2 3 4 5 (b) Compuaional Time on Regular Graph (number of node = 2 ) Co 5 45 4 35 3 25 2 5 5 (a) Qualiy of Soluion on Waxman' Random Graph( =.6, =.9) 5 5 2 25 3 35 4 OPT LHWHM NBS LRARC (b) Compuaional Time Time(m) 8 7 6 5 4 3 2 NBS-TIME LARAC-TIME PARA-TIME 6 32 48 64 8 96 2 28 Time(m) 7 6 5 4 3 2 NBS-TIME LHWHM-TIME LARAC-TIME PARA-TIME Ou-Degree Time(m) Time(m) 35 3 25 2 5 5 (c) Compuaional Time on Regular Graph (Ou-Degree = 6) NBS-TIME LHWHM-TIME LARAC-TIM E PARA-TIME 2 3 4 5 7 6 5 4 3 2 (d) Compuaional Time on Regular Graph (Ou-Degree = 36) NBS-TIME LHWHM-TIME LARAC-TIM E PARA-TIME 2 3 4 Fig. 3. Simulaion on regular graph. Simulaion and comparaive evaluaion We compare NBS algorihm wih he LARAC and LHWHM algorihm. We alo compare he NBS Time(m) 3 25 2 5 5 2 3 4 Fig. 4. Waxman random graph Compuaional Time on Power-Law Ou-Degree graph NBS-TIME LHWHM -TIM E LARAC-TIME PARA-TIME 2 3 4 Fig.5. Power-Law Ou-Degree graph. We ue hree clae of nework opologie: regular graph H k, n (ee [7]), Power-Law Ou-Degree graph [2] and Waxman random graph [2]. For a nework G(V, E), he node are labeled a,, n = V. n / 2 and n are choen a he ource and arge node. For he Power-Law Ou-Degree graph and Waxman random graph, he hop number of feaible - pah i uually very mall even when he nework i very large. Thi will bia he reul in favor of he LHWHM algorihm. So, for Waxman random graph, a link joining node u and v i added if u v < V / 5 beide oher rule generaing Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE
he random graph. We keep he original verion of Power- Law Ou-Degree graph a in [2]. Even hough hi kind of graph favor he LHWHM algorihm, he comparion of he performance of he LARAC and NBS algorihm i ill an indicaor of he meri of NBS. The link co and delay are randomly independenly generaed even ineger in he range from o 2. The delay bound i.2 ime he delay of he minimum delay - pah in G. The reul are hown in Fig. 3-5. Experimen how ha NBS algorihm may find beer oluion han he LARAC algorihm by elecing he be feaible pah encounered during he execuion inead of he opimum pah o he RELAX-TCSP problem. We alo find ha for pare graph, NBS ake more ime han he LARAC algorihm. However, when he nework i dene (larger ou-degree), NBS bea LARAC (ee Fig. 3. (b)). 2. Summary In hi paper, we udied he QoS rouing problem (or equivalenly he CSP problem) from he primal perpecive in conra o mo of he currenly available approache ha udied he problem from a dual perpecive. Specifically we applied he revied implex mehod on he primal form of he RELAX-TCSP problem. Several raegie are employed o achieve efficien implemenaion of he revied implex mehod. We how ha our algorihm i of peudo-polynomial ime complexiy. We have alo hown how o exrac an approximae oluion o he original CSP problem from he opimum oluion o he RELAX-TCSP problem and derive bound on he qualiy of hi oluion wih repec o he opimum oluion. Exenive imulaion reul are preened o demonrae ha our approach compare favorably wih he LARAC and i faer on dene graph. Acknowledgmen: The work of K. Thulairaman and Guoliang Xue have been uppored by NSF ITR gran ANI-32435 and ANI-32635. REFERENCES [] Z. Wang and J. Crowcrof, Qualiy-of-Service rouing for upporing mulimedia applicaion, IEEE JSAC, vol.4, no.7, pp.228-234, Sep. 996. [2] R. Hain, Approximaion cheme for he rericed hore pah problem, Mah. of Oper. Re., 7(), 992, pp.36-42. [3] C. Phillip, The nework inhibiion problem, Proceeding of he 25 h Annual ACM Sympoium on he Theory of Compuing, May, 993 [4] D. Lorenz and D. Raz, A imple efficien approximaion cheme for he rericed hore pah problem, Oper. Re. Leer, vol. 28, pp. 23-29. [5] G. Luo, K. Huang, J. Wang, C. Hobb and E. Muner, Muli-QoS conrain baed rouing for IP and ATM nework, in Proceeding of IEEE Workhop on QoS Suppor for Real-Time Inerne Applicaion, Vancouver Canada, June, 999 [6] R. Ravindran, K. Thulairaman, A. Da, K. Huang, G. Luo and G. Xue, Qualiy of ervice rouing: heuriic and approximaion cheme wih a comparaive evaluaion, ISCAS, 22. [7] G. Handler and I. Zang, A dual algorihm for he conrained hore pah problem, Nework, 293-3, 98 [8] K. Mehlhorn and M. Ziegelmann, Reource conrained hore pah, Proc. 8h European Sympoium on Algorihm (ESA2) [9] A. Jüner, B. Szviaovzki, I. Méc and Z. Rajkó, Lagrange relaxaion baed mehod for he QoS rouing problem, IEEE INFOCOM-2, pp. 859-868. [] D. Blokh and G. Guin, An approximaion algorihm for combinaorial opimizaion problem wih wo parameer, Auralaian Journal of Combinaoric, vol. 4, 996, pp.57-64. [] Y. Xiao, K. Thulairaman and G. Xue, Equivalence, unificaion and generaliy of wo approache o he conrained hore pah problem wih exenion, Alleron Conference on Conrol, Communicaion and Compuing, Univeriy of Illinoi, Urbana-Champaign, Oc. - 3, 23 [2] A. Jüner, On reource conrained opimizaion problem, ubmied for publicaion. [3] M. Ziegelmann, Conrained hore pah and relaed problem, PhD hei, Max-Planck-Iniu für Informaik, 2. [4] S. Chen and K. Nahred, An overview of Qualiy-of- Service rouing for he nex generaion high-peed nework: problem and oluion, IEEE Nework Magazine, 2(6), Nov. / Dec, 998. [5] A. Orda and A. Sprinon, Efficien algorihm for compuing dijoin QoS pah, IEEE INFOCOM, 24. [6] Y. Xiao, K. Thulairaman and G. Xue, Dijoin QoS pah elecion for proecion again failure: a nework programming baed approach, Alleron Conference on Conrol, Communicaion and Compuing, Univeriy of Illinoi, Urbana-Champaign, Oc. 24. [7] K. Thulairaman and M. N. Swamy, Graph: Theory and algorihm, Wiley Inercience, New York, 992. [8] V. Chval, Linear programming, W. H. Freeman and Company, New York (983) [9] Y. Xiao, K. Thulairaman, G. Xue and A. Jüner, The conrained hore pah problem: algorihmic approache and an algebraic udy wih generalizaion, ubmied for publicaion. [2] C. R. Palmer and J. G. Seffan, Generaing nework opologie ha obey power law, IEEE GLOBECOM, 2. [2] B. M. Waxman, Rouing of mulipoin connecion, IEEE J. Seleced Area in Comm. 6(9), Dec. 988 Proceeding of he Fir Inernaional Conference on Qualiy of Service in Heerogeneou Wired/Wirele Nework (QSHINE 4) -7695-2233-5/4 $2. 24 IEEE