ROUTING is a fundamental problem in communication

Transcription

1 QoS Rouing in Communicaion Neworks: Approximaion Algorihms Based on he Primal Simplex Mehod of Linear Programming Ying Xiao, Memer, IEEE, Krishnaiyan Thulasiraman, Fellow, IEEE, and Guoliang Xue, Senior Memer, IEEE Asrac Given a direced nework wih wo ineger weighs, cos and delay, associaed wih each link, Qualiy-of-Service (QoS) rouing requires he deerminaion of a minimum cos pah from one node o anoher node such ha he delay of he pah is ounded y a specified ineger value. This prolem also known as he consrained shores pah prolem (CSP) admis an Ineger Linear Programming (ILP) formulaion. Due o he inegraliy consrains, he prolem is NP-hard. So, approximaion algorihms have een presened in he lieraure. Among hese, he LARAC algorihm, ased on he dual of he LP relaxaion of he CSP prolem is very efficien. In conras o mos of he currenly availale approaches we sudy his prolem from a primal perspecive. Several issues relaing o efficien implemenaions of our approach are discussed. We presen wo algorihms of pseudo-polynomial-ime complexiy. One of hese allows degenerae pivos and uses an ani-cycling sraegy and he oher called he NBS algorihm is ased on a novel sraegy which avoids degenerae pivos. Experimenal resuls comparing he NBS algorihm, he LARAC algorihm, and general purpose LP solvers are presened. In all cases he NBS algorihm compares favoraly wih ohers and eas hem on dense neworks. Index Terms Consrained shores pah, linear programming, simplex mehod, graph algorihms, communicaion neworks, rouing proocols, QoS rouing. I. INTRODUCTION ROUTING is a fundamenal prolem in communicaion neworks. In radiional daa neworks, rouing is achieved y es effor rouing. Bes effor rouing is primarily concerned wih providing conneciviy. FIFO provides eseffor service. Here, flows are no differeniaed and are serviced on a firs-come, firs-served asis. In es effor rouing he rouing proocol usually characerizes he nework wih a single meric such as hop-coun or delay and uses a shores pah algorihm for pah compuaion. Whereas he es-effor rouing paradigm is adequae o serve he needs for radiional applicaions such as FTP (File Transfer Proocol) i is quie inadequae in providing he sringen Qualiy of Service (QoS) guaranees demanded y popular mulimedia applicaions such as real ime digial video or audio ransmission. To suppor a road range of QoS requiremens, rouing proocols need o consider more complex models ha incorporae muliple The work of K. Thulasiraman has een suppored y NSF ITR gran ANI The work of G. Xue has een suppored y NSF ITR gran ANI Ying Xiao and Krishnaiyan Thulasiraman are wih Universiy of Oklahoma, Norman, OK 739, USA ( eagle7827@gmail.com, hulasi@ou.edu). Guoliang Xue is wih Arizonia Sae Universiy, Tempe, AZ 85287, USA. ( xue@asu.edu) merics such as cos, delay, delay variaion, loss proailiy, and andwidh. This has riggered effors owards proposals for QoS ased frameworks such as DiffServe and InServ, QoS rouing proocols ha accommodae muliple QoS requiremens such as Q-OSPF and PNNI, and QoS rouing algorihms (See [], [9], [], [3]). Despie hese effors, here is no sandardized QoS rouing proocol for he Inerne. To he es of our knowledge he only sandardized QoS rouing proocol is ATM PNNI []. Two aciviies are involved in rouing: i) Capuring he nework sae informaion and disseminaing he informaion hroughou he nework. This requires deecion of significan changes, opology updaes, disriued roadcasing (flooding) of he informaion o each node in he nework ec. (ii) Rouing algorihms ha compue he pahs ha saisfy cerain performance guaranees. In his paper we are concerned wih he laer, namely, QoS rouing algorihms. QoS measures can e classified ino wo ypes of merics, non-addiive (also called oleneck, e.g., andwidh) and addiive consrains. Each measure is modeled y associaing a weigh wih each link. For a non-addiive measure QoS weigh of a pah is he minimum weigh along he pah. In he case of addiive measures such as cos, delay, reliailiy and delay-jier he QoS weigh of a pah is he sum of he QoS weighs of he links on he pah. Non-addiive measures can e handled easily y simply removing from he nework he links ha do no saisfy he required QoS measure. In his paper we are concerned wih finding pahs ha saisfy addiive QoS merics. In paricular, we are ineresed in he QoS rouing prolem ha requires he deerminaion of a minimum cos pah from a source node o a desinaion node in a nework ha saisfies a specified upper ound on he delay of he pah. This prolem is also known as he Consrained Shores Pah (CSP) prolem. The CSP prolem is NP-hard [33]. Thus, here has een a good deal of effors in developing efficien approximaion algorihms and heurisics. Heurisics, in general, do no provide performance guaranees on he qualiy of he soluion produced, hough hey are usually fas in pracice. On he oher hand, ɛ-approximaion algorihms deliver soluions wihin arirarily specified precision requiremen u are usually very slow in pracice. References [2], [2], [29] and he references herein conain mos of he curren lieraure on approximaion algorihms for he CSP prolem. As regards heurisics, he LHWHM algorihm [22] is a simple heurisic which is very fas (requiring only one or wo invocaions of Dijksra s shores pah algorihm) and produces

2 2 soluions which are usually found o e of accepale qualiy in pracice. Reference [3] also discusses furher enhancemens of he LHWHM algorihm. There are heurisics ha are ased on sound heoreical foundaion. These algorihms are ased on soluions o he dual of he linear programming relaxaion of he CSP prolem. The firs such algorihm was repored in [] y Handler and Zang. This is ased on a geomeric approach (wha is also called he hull approach [23]). More recenly, in an independen work, Jüner e al. [6] developed he LARAC algorihm which also solves he dual of he CSP prolem using Lagrangian relaxaion mehod. In conras o he geomeric mehod, hey used an algeraic approach. In [4] Xue developed an algorihm ha is similar o he LARAC algorihm. In [5] Blokh and Guin defined a general class of cominaorial opimizaion prolems of which he CSP prolem is a special case and proposed an approach o his prolem. In [35], [39], Xiao e al. drew aenion o he fac ha he algorihms in [] and [6] are equivalen. In view of his equivalence, we shall refer o hese algorihms simply as he LARAC algorihm. In [23], Mehlhorn and Ziegelmann have provided several insighs on he QoS rouing prolem. In [5] Jüner esalished he srong polynomialiy of he LARAC algorihm. Ziegelmann [43] provides a fairly complee lis of references o he lieraure on he CSP prolem. Anoher prolem, Muli-Consrained Pah (MCP) prolem has also een a opic of exensive sudy. In his prolem, each link is associaed wih l > addiive weighs. The MCP prolem is o find an s- pah ha saisfies all he l consrains. Key resuls and algorihms on he MCP prolem may e found in [7], [3], [4], [7] [2], [24] [26], [4], [42]. Recen works on he QoS rouing prolem may e found in [3], [6], [27], [37], [38]. In his paper, we presen a novel approach o he QoS rouing prolem, making a deparure from currenly availale approaches. We sudy he prolem using he primal simplex mehod of linear programming and exploiing cerain srucural properies of neworks. This is an exended and deailed version of our work in [36] and includes proofs of all resuls and more exensive experimenal resuls. The res of he paper is organized as follows. In Secion II, we define he CSP prolem and presen is Ineger Linear Programming (ILP) formulaion as well as is Linear Programming (LP) relaxaion. This formulaion is he same as he LP formulaion of he minimum cos flow prolem [2] excep for an addiional consrain due o he delay requiremen. This addiional consrain gives rise o several quesions ha need o e invesigaed o achieve an efficien implemenaion of he primal simplex mehod. This leads us o he definiion in Secion III of an equivalen prolem on a ransformed nework, called he TCSP prolem. Secion IV deals wih he srucure of he asic soluions of he RELAX-TCSP prolem, he relaxed form of he TCSP prolem. Secion V discusses he revised simplex mehod of linear programming, is applicaion on RELAX-TCSP, and several sraegies o achieve an efficien implemenaion. This resuls in an algorihm ha allows degenerae pivos and uses an ani-cycling sraegy developed in Secion V-F. Anoher algorihm called NBS algorihm presened in Secion VI avoids degenerae pivos compleely. Boh hese algorihms are of pseudo polynomial ime complexiy. In Secion VI-C.2, we show how o exrac an approximae soluion o he original CSP prolem from he opimum soluion o he RELAX- TCSP prolem and derive ounds on he qualiy of his soluion wih respec o he opimum soluion. In Secion VII, experimenal resuls comparing he NBS algorihm wih he LARAC algorihm [6], he LHWHM algorihm [22], and he general purpose LP solvers are presened. Secion VIII concludes wih a summary of he main conriuions. To conserve space proofs of a few resuls are omied. II. THE CSP PROBLEM: LP FORMULATION AND THE LARAC ALGORITHM In his secion, we firs define he Consrained Shores Pah (CSP) prolem and presen an ILP formulaion. Due o inegraliy consrains in he ILP formulaion he prolem is NP-hard. Relaxing he inegraliy consrains resuls in RELAX-CSP. We hen presen he LARAC algorihm of [6] which solves he dual of RELAX-CSP. Definiion : Consider a direced nework G(V, E) where V is he se of nodes and E is he se of links of he nework. Each link (u, v) E is associaed wih wo ineger weighs c uv > (represening cos, he expense imposed y using or insalling he link) and d uv > (ransmission delay along he link). For any pah p (or cycle wih a given orienaion) define he cos c(p) and delay d(p) of p as c(p) = c uv c uv, (u,v) p + (u,v) p d(p) = d uv d uv, (u,v) p + (u,v) p where p + (p ) is he se of forward (ackward) links on p as we raverse p from he sar node o he end node of p. Noice ha our assumpion ha link weighs are inegers does no involve any loss of generaliy ecause, in digial sysems, all numers are represened discreely and can e scaled and rounded o inegers. In order o simplify our presenaion, we assume all he values o e inegers. We also assume ha only links impose coss and delays. If he nodes impose coss and delays, we can use he node spliing echnique o ransform node coss and delays ino link coss and delays (See Chaper 2.4 of [2]). We use he erms link and arc inerchangealy. Wihou loss of generaliy, we assume ha for every node i, here is a direced pah from i o he desinaion node. In he res of he paper, m = E and n = V. A pah is called a direced pah (cycle) if here are no ackward links in he pah (cycle). Given wo nodes s, and an ineger >, a direced s- pah p is said o e feasile if d(p). In he res of he paper, a direced s- pah will e referred o simply as an s- pah. CSP(Consrained Shores Pah) prolem: Find an s- pah p op = arg min{c(p) p is a feasile s- pah}. This is illusraed wih he example in Fig..

3 3 Fig.. s 6, 64 3, 2, 3, 3 2 4, 3, 5, 5 3, Min-cos pah (no feasile), 3 = 7 4 5, 5 Min-cos feasile pah 4 3, An example of CSP prolem. u 3 5 3, 52 c uv, d uv 5, 2, 2, 5, v Cos Delay The CSP prolem can e formulaed as an ineger linear programming prolem as elow. CSP: Minimize sujec o {v (u,v) E} (u,v) E (u,v) E x uv c uv x uv () {v (v,u) E}, for u = s x vu =, for u =, oherwise (2) d uv x uv w = (3) (u, v) E, x uv = or. (4) In (3), w is he slack variale for he delay consrain. The main difficuly wih he CSP prolem lies wih he inegraliy condiion ha requires ha he variales x uv e or. Removing or relaxing his requiremen from he aove ineger linear program leads o RELAX-CSP, he relaxed CSP prolem. RELAX-CSP: Minimize sujec o {v (u,v) E} (u,v) E (u,v) E x uv c uv x uv (5) {v (v,u) E}, for u = s x vu =, for u =, oherwise (6) d uv x uv w = (7) (u, v) E, x uv. (8) We will show laer ha y using a ransformaion and applying cerain pivo rules we can enforce x uv (he discussion afer Theorem 3, Secion V-E). Dual Based Approach: LARAC Algorihm [6] The dual of he CSP prolem involves s- pahs and a variale. For each link (u, v), le he aggregaed cos c e defined as c uv + d uv. For a given, le c (p) = c(p) + d(p) denoe he aggregaed cos of he pah p. Finally define L() as: L() = min{c (p) p is an s- pah}. (9) Noe ha in he aove, min{c (p) p is an s- pah} can e easily oained y applying Dijksra s algorihm using aggregaed link coss c uv + d uv. Le he s- pah which has minimum aggregaed cos wih respec o a given e denoed as p. Then L() = c (p ) and he dual of he RELAX- CSP can e presened as follows. DUAL-RELAX-CSP: Deermine max{l() }. The value of ha achieves he maximum L() in DUAL- RELAX-CSP will e denoed y. Noe ha L, he opimum value of DUAL-RELAX-CSP is a lower ound on he opimum cos of he pah ha solves he corresponding CSP prolem [6]. From he opimum soluion o he RELAX-CSP prolem we can exrac an approximae soluion o he original CSP prolem. The key issue in solving DUAL-RELAX-CSP is how o search for he opimal. The LARAC algorihm of [6] presened in Algorihm is one such efficien search procedure. In his algorihm Dijksra(s,, c), Dijksra(s,, d), and Dijksra(s,, c ) denoe, respecively, Dijksra s shores pah algorihm using link coss, link delays, and aggregaed link coss wih respec o he muliplier. Algorihm LARAC(s,, ) algorihm {Compue he minimum cos s- pah} p c Dijksra(s,, c) if (d(p c ) ) hen reurn p c {Compue he minimum delay s- pah} p d Dijksra(s,, d) if (d(p d ) > ) hen reurn no soluion loop (c(p c ) c(p d ))/(d(p d ) d(p c )) {Compue minimum c cos s- pah} r Dijksra(s,, c ) if (c (r) = c (p c )) hen reurn p d else if (d(r) ) hen p d r else p c r end loop In [39] we have sudied several aspecs of he dual ased approach such as opimaliy condiions and oher approaches such as parameric search and inary search. III. A TRANSFORMED PROBLEM AND BASIC CONCEPTS In conras o mos oher approaches in he lieraure, we sudy he CSP prolem using he primal simplex algorihm. In order o achieve an efficien implemenaion of he approach we ransform he prolem o an equivalen one on a ransformed nework defined elow.

4 4 ) The graph of he ransformed nework is he same as ha of he original prolem, i.e., G(V, E), 2) For (u, v) E, d uv and c uv in he ransformed prolem are given y d uv = 2d uv and c uv = c uv, and 3) The new upper ound in he ransformed prolem is given y = 2 +. The ransformed prolem will e referred o as he TCSP prolem. Theorem : An s- pah p is a feasile soluion (resp. an opimal soluion) o he CSP prolem iff i is a feasile soluion (resp. an opimal soluion) o he TCSP prolem. In view of he aove resul, we consider in he res of he paper only he relaxed form of he TCSP prolem, namely, RELAX-TCSP (same as RELAX-CSP excep ha he nework is he ransformed one as defined aove). Also we use (eing odd) and d uv (eing even) o denoe he delay ound and link delay in he ransformed prolem, respecively. Noice ha he ransformaion does no change he cos of any pah in he nework. In he res of he secion we shall define cerain erminology leading o a marix represenaion of RELAX-TCSP. Le he links e laeled as e, e 2... e m and he nodes e laeled as, 2..., n. We shall denoe he delay of edge e i as d i and he cos of e i as c i. The incidence marix of G has m columns, one for each link and n rows, one for each node [8], [32]. The rank of his marix is (n ), and removing any row of his marix will resul in a marix of rank (n ). We denoe his resuling marix as H. We also assume ha he row removed from he incidence marix corresponds o node n. Also we assume ha he column of H corresponding o link e k will e denoed y 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 is componens eing excep for h i,k = and h j,k =. Le H A = = (a D, a 2..., a m, a m+ ), () D = ( d, d 2..., d m ), () hi a i =, i m, and (2) d i a m+ =. (3) Also, le x e he column vecor of he m flow variales x uv and he slack variale w, and c e he row vecor of he coss (c..., c m, ). Noe ha he cos of he slack variale is. The LP formulaion of he RELAX-TCSP prolem can now e wrien in marix form as follows. RELAX-TCSP: Minimize cx sujec o Ax =. (4) In (4), x and = (..., n, ) wih s =, =, and i = for i s,. The res of he paper deals wih he primal simplex ased soluion of RELAX-TCSP. IV. SIMPLEX METHOD: BASIC SOLUTIONS OF RELAX-TCSP Simplex mehod of linear programming sars wih a asic soluion and proceeds y consrucing one asic soluion from anoher. A asic soluion consiss of wo ses of variales, asic and non-asic. For he RELAX-TCSP prolem under consideraion, all he non-asic variales in a asic soluion will have zero values. Given a asic soluion, we shall denoe y G he sugraph of G corresponding o he asic variales (excep he slack variale if i is in he asic soluion) in his soluion. Noe ha here is no link associaed wih he slack variale. The sugraph G will e called he sugraph of he asic soluion or simply he asis graph. The non-singular sumarix of A defined y he asic variales is called a asis marix or simply, a asis. In his secion we presen cerain imporan properies of he asic soluions of he RELAX- TCSP prolem. Lemma : Le G(V, E) e a direced nework wih a leas one cycle W (no necessarily direced). Assigning an arirary orienaion o W, le U(W ) = (u, u 2, u 3..., u m ), where u j =, for e j W and he orienaion of e j agrees wih he orienaion of W, for e j W and he orienaion of e j disagrees wih he orienaion of W, oherwise. Then, HU(W ) = [32]. We shall denoe y d(w ) he signed algeraic sum of he delays of he links in a cycle W as we raverse around he cycle along he given orienaion. Lemma 2: The sugraph G of a asic soluion conains a mos one cycle. Lemma 3: If here is a cycle W in G, hen d(w ). Proof: Le En,n B = D,n e a asis marix (sumarix of A), where H n,n is a (n ) n sumarix of H and D,n is he vecor of n componens (corresponding o he asic variales) of he las row of A. Then E n,n U(W ) = y Lemma. On he oher hand, D,n U(W ) = d(w ). Since rank (B) = n, we have BU(W ) = ( En,n U(W ) d(w ) ) ( = d(w ) ). Thus he lemma follows. Lemma 4: If he asis sugraph G conains no cycle ha is no a direced cycle, here are exacly wo s- pahs in G. Thus i follows from he aove lemma ha he ransformaion we inroduced guaranees ha he srucure of he asis sugraph will e one of he hree forms shown in Fig. 2 (a spanning ree or a spanning ree plus an exra link). In a laer secion we shall inroduce a pivo rule which will ensure ha he asis sugraph will no conain any direced cycle, herey eliminaing he srucure in Fig. 2.(c).

5 5 s s (a) Branching Poin () flow < < Revised Simplex Mehod [8] ) Sep : Solve he sysem Y B = c B, where Y = (y, y 2..., y n ). 2) Sep 2: Choose an enering column. I may e any column a i of A N such ha Y a i is greaer han he corresponding componen of c N. The curren soluion is opimal if here is no such column. 3) Sep 3: Solve he sysem BV = a i, where V = (v, v 2..., v n ). 4) Sep 4: Find he larges such ha x B V. If here is no such, hen he prolem is unounded; oherwise, a leas one componen of x B V is equal o and he corresponding variale leaves he asis. 5) Sep 5: Se he value of he enering variale as and replace he values x B of he asic variales y x B V. Replace he leaving column of B y he enering column and in he asis heading, replace he leaving variale y he enering variale. Then go o Sep. < < s (c) Fig. 2. Srucure of asis graph: (a) ree asic soluion, () asic soluion wih a cycle(no direced), and (c) asic soluion wih a direced cycle. V. REVISED SIMPLEX METHOD ON THE RELAX-TCSP PROBLEM In his secion, we firs riefly presen he differen seps in he revised simplex mehod of linear programming ha is descried in deail in [8]. We hen derive formulas required o idenify he enering and he leaving variales. A. Revised simplex mehod Consider an arirary linear programming (LP) prolem in he sandard form. Minimize cx Sujec o Ax =, x. Here A is an n (m + ) marix wih rank (A) = n, x = (x..., x m+ ), c = (c..., c m+ ), and = (..., n ). Each feasile asic soluion x is pariioned ino wo ses, one se consising of he n asic variales and he oher se consising of he remaining m + n non-asic variales. This pariion induces 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, corresponding o he se of asic variales and he se of non-asic variales, respecively. The asis marix B is nonsingular. B. Iniializaion To consruc an iniial asic feasile soluion we firs deermine a spanning ree conaining a feasile s- pah. This can e done y applying Dijksra s algorihm o compue he shores pah ree wih respec o he delay from all nodes o he desinaion node. If he resuling s- pah in he ree is infeasile, hen no feasile pah exiss and he algorihm erminaes. Wihou loss of generaliy we assume ha he s- pah is feasile. Clearly in he asic soluion corresponding o he spanning ree seleced as aove, he flows in all he links in he s- pah in he spanning ree will e equal o one, and flows in all oher links will e zero. Since he delay of every link in he TCSP prolem is even and he upper ound on pah delay is odd, he slack variale w > and so i is in he iniial asic feasile soluion. In Secions V-C and V-D we solve he sysems of equaions in Seps and 3 and derive explici formulas for Y and V. These resuls are from [37] and are repeaed here for he sake of compleeness. If a link flow variale is chosen as he enering variale hen he corresponding link is called he inarc. Ou-arcs are similarly defined. C. Solving he Sysem Y B = c B. Le Y = (y..., y n, γ). Here y..., y n, γ are called poenials (or dual variales) and Y is called he poenial vecor. Each y i, i =, 2..., n is he poenial associaed wih node i (or he row i) and γ is he poenial associaed wih he las row (delay consrain row) of A. Now consider Y B = c B (5) This sysem of equaions has n equaions in n variales. We ge he following from (5).

6 6 For each link e k = (i, j) in G, (y..., y n, γ)h k = c ij. Tha is, y i y j γd ij = c ij, if i n and j n, y i γd in = c in, if j = n, and (6) y j γd nj = c nj, if i = n. From he aove, we can see ha we can se he poenial of node n a any consan. In all compuaions ha follow, we shall se he poenial of node n equal o zero. Definiion 2: ) For link e k = (i, j), c(e k, γ) = γd ij + c ij is called he acive cos of link (i, j), 2) r(i, j) = y j y i + γd ij + c ij is called he reduced cos of link (i, j), 3) The reduced cos of w is given y r(w) = γ, and 4) The reduced cos of a pah p is defined as r(p) = r(i, j) r(i, j). (i,j) p + (i,j) p I can e seen from (6) ha for any link (i, j) in G r(i, j) = y j y i + γd ij + c ij =. (7) From (7) we also have ha for any pah p from i o j and any cycle W in G r(p) = y j y i + γd(p) + c(p) =, (8) r(w ) = γd(w ) + c(w ) =. Lemma 5: If G conains a cycle W, hen γ = c(w )/d(w ); Oherwise, γ =. Proof: If here is no cycle in G hen he slack variale w is a asic variale and he corresponding column [,...,, ] will e a column of B. Since he cos of he slack variale is zero, we ge from (5) ha γ =. Suppose ha G conains a cycle W. By (8), we ge γd(w )+c(w ) =. By Lemma 3, d(w ). So γ = c(w )/d(w ). Lemma 6: A link is eligile o ener he asis if is reduced cos is negaive and he slack variale is eligile o ener he asis if γ <. Proof: The proof follows from Sep 2 of he revised simplex mehod. Once we have compued he value of γ as in Lemma 5, he oher poenials y i s can e calculaed using equaion (8) and selecing he pah in G from node n o node i. Summarizing he aove, we have he following procedure for solving Y B = c B and calculaing he poenials. () Se he poenial of node n o zero. (2) Compue γ as in Lemma 5. (3) For each node i, le p i e a simple pah in G from node n o node i. If here are wo pahs in G due o he cycle, we will ge he same resuls no maer which pah is seleced. (4) Se c(p) = (u,v) p c + uv (u,v) p c uv and d(p) = (u,v) p d + uv (u,v) p d uv, where p + i and p i are he ses of forward and ackward links on p i, respecively, as we raverse he pah from node n o node i. Once he poenials are deermined, an enering variale, if i exiss, can e seleced as in Sep 2 of he revised simplex mehod. D. Solving he Sysem BV = a k We nex show how o solve he sysem of equaions BV = a k. We consider hree cases: Case a): G conains no cycle, ha is, G conains only n links and he slack variale w is a asic variale. The link e k = (i, j) is he enering variale. Case ): G conains a cycle (ha is, G has n links) and he enering variale is e k = (i, j). Case c): G conains a cycle (G has n links) and he enering variale is he slack variale. Soluions in all he hree cases are summarized in he following heorem. Theorem 2: a) If G conains no cycle and he enering variale is an in-arc e k = (i, j), hen he vecor V = (v..., v n ) defined elow is he desired soluion o BV = a k, where W is he new cycle formed y adding he in-arc e k and he orienaion of W is chosen o e he same as he direcion of e k., for i < n and he link corresponding o he ih column of B is on W and is orienaion agrees wih he cycle orienaion, for i < n and he link corresponding o v i = he ih column of B is on W and is orienaion disagrees wih he cycle orienaion d(w ), for i = n, oherwise (9) ) If G conains a cycle W and he enering variale is a link e k = (i, j), hen V = V p + d(w ) d(w ) V, is he soluion o BV = a k, where d(w ) and d(w ) are he delays of cycles W and W, respecively and V p and V are defined y he cycles W and W, respecively (See Lemma ). c) If G conains a cycle W and he enering variale is he slack variale w, hen V = V d(w ) is he soluion o BV = a k, where V is defined y cycle W (See Lemma ). Proof: Case a): G conains only n links, i.e., here is no cycle in G and he slack variale w is a asic variale, and he link e k = (i, j) is he enering variale. In his case, Hn,n B =, D,n where H n,n is associaed wih he (n ) links in G and n nodes and D,n is he vecor of (n ) componens (corresponding o he asic variales excep for w) of he las row of marix A. Le W denoe he new cycle formed y adding he in-arc e k = (i, j) and le he orienaion of W e chosen o e he same as he orienaion of he in-arc. By Lemma, i is easy o verify ha he vecor V = (v..., v n ) defined as in he heorem solves he sysem BV = a k. Case ): The asic variales are associaed wih n links and he enering variale is e k = (i, j). In his case, Hn,n B =, D,n where H n,n is associaed wih he n links and n nodes and D,n is he vecor of he n componens of he las row

7 7 of A corresponding o hese n links, and hk a k =. d ij We need o solve he sysem of equaions Hn,n hk V =. (2) D,n d ij Firs, le us consider H n,n V = h k. (2) Because here are n links in G, here is exacly one cycle, denoed y W. Therefore according o Lemma, V, H n,n V =. (22) Afer adding link e k = (i, j), we ge a new cycle W and le us choose he orienaion of his cycle o e he same as ha of e k. Then y Lemma, V V p V p =, (H n,n, h k ) =. (23) So, H n,n ( V p) = h k. (24) Because rank(h n,n ) = n, V p + uv, u R is he soluion space of (2). We can compue u as follows. D,n ( V p + uv ) = d ij. (25) Since D,n V = d(w ) and D,n ( V p) + d ij = d(w ), we ge from (25) d(w ) ud(w ) = and hence u = d(w )/d(w ). Therefore we have proven ha V = V p + d(w ) d(w ) V (26) is he desired soluion o BV = a k. Case c): The asic variales are associaed wih n links and he enering variale is he slack variale w. Following he argumens in Case (), we can show ha V = V d(w ) is he soluion o BV = a k. Here V is defined y he cycle W in G. E. A Pivo Rule and Srucure of Basic Feasile Soluions In his susecion we presen a pivo rule and sudy he srucure of sugraphs of asic soluions generaed y he simplex mehod. The sugraph G of he iniial asic feasile soluion has (n ) links and he nh variale in his asic soluion is he slack variale w >. A his iniial sep, γ = (Lemma 5). Define d(g ) = (u,v) G x uv d uv. By (7), d(g ) = w. Now one of he following wo possiiliies occurs in he nex pivo.. The simplex mehod consrucs a new spanning ree soluion wih he slack variale w remaining nonzero in he new soluion. 2. The simplex mehod consrucs a G ha conains one cycle W (formed y adding he in-arc) and w ecomes nonasic wih respec o his soluion. The cycle W canno e a direced cycle. If i were a direced cycle, hen he reduced cos of he enering link will e equal o he sum of he coss of he links in W. This sum is a posiive numer conradicing he requiremen ha he reduced cos of he enering link mus e negaive (Sep 2 of he revised simplex mehod). By Lemma 4, here will e exacly wo s- pahs in G. Also, he flow values on all he links in W mus e nonzero, for oherwise all he link flows will e eiher or making w nonzero and hence asic. Summarizing, when he firs ime a G wih a cycle is encounered, i will e necessarily of he form shown in Fig. 2.(). Flows on he links in he cycle will e or. The simplex mehod will selec he value of > in such a way ha d(g ) =. Though he cycle in he G encounered he firs ime afer iniializaion will no e a direced cycle, in a susequen sep, a G wih a direced cycle may e creaed. To achieve an efficien implemenaion of he simplex mehod, we would like o avoid generaing any G conaining a direced cycle. This can e achieved y he pivo rule P given nex. Pivo Rule P: Selec he slack variale w as he enering variale if i is eligile o ener. Theorem 3: If he pivo rule P is followed and he simplex mehod on he RELAX-TCSP prolem is iniialized as in Secion V-B, hen no asic soluion sugraph G will conain a direced cycle. Proof: Assume ha a G wih a direced cycle W is creaed and le e ij = (i, j) e he in-arc wih which his cycle is creaed. Suppose W = e ij e jj e jj 2..., e jk i and p ji is he direced pah from j o i in W. Since e ij is an in-arc and Y = (y, y 2..., y n, γ) is he poenial vecor, we have r(i, j) = y j y i + γd ij + c ij < and r(p ji ) = y i y j + γd(p ji ) + c(p ji ) =. Summing he aove, we oain γd(w ) + c(w ) <. Since d(w ) > and c(w ) >, γ <. This implies ha he slack variale is eligile o ener he asis u was no seleced. This is a conradicion. Theorem 3 implies ha pivo rule P along wih he ransformaion inroduced in Secion III guaranees ha G will ake only he srucures shown in Fig. 2.(a) and Fig. 2.(). Under hese condiions we are also guaraneed ha he values of he variales x uv will e resriced o he range x uv. F. An Ani-Cycling Sraegy A asic soluion in which one or more asic variales assume zero values is called degenerae. A degenerae asic soluion may resul in a pivo ha does no aler he asic soluion. Such pivos are called degenerae. Furhermore, a asic soluion generaed a one pivo and reappearing a anoher will lead o cycling. Since degenerae pivos do no resul in any improvemen of he soluions, hey are also a cause of inefficiency. We presen wo sraegies o handle degeneracy. The firs one o e presened in his susecion is he ani-cycling sraegy which is a variaion and exension of Cunningham s ani-cycling sraegy in [2], [4], [8]. The second sraegy o e presened in Secion VI is designed o avoid degenerae pivos compleely.

8 8 Definiion 3: Given a feasile asic soluion sugraph G and a node called he roo, we say ha he link (u, v) G is oriened oward (resp. away from) he roo if any pah in G from he roo o u (resp. v) passes hrough v (resp. u). A feasile asic soluion G wih corresponding flow vecor x is srongly feasile if every link (u, v) of G wih x uv = is oriened oward he roo. If he ou-arc (u, v) is no a link of he cycle in he asic soluion, hen G (u, v) conains exacly wo componens G (u) and G (v) such ha u G (u) and v G (v). If he roo is in G (v), link (u, v) is oriened oward he roo; oherwise i is oriened away from he roo. See Fig. 2.(a), () for examples of a srongly feasile G. We shall selec node as he roo node. Lemma 7: For any degenerae pivo, he ou-arc is no on he cycle of he curren G. Proof: A degenerae pivo does no aler he asic soluion. This means ha each variale has he same value in he curren asic soluion as well as in he asic soluion resuling from he degenerae pivo. The flow on each link in a cycle is non-zero. If a link on a cycle were o leave he asis, hen afer he degenerae pivo i would ecome non-asic wih zero flow. Bu ha would conradic ha he curren pivo is degenerae. If he ou-arc is no on he cycle in he curren G, hen he poenials can e updaed easily as descried nex (See Chaper 5..2 of [4]). Le T e he curren G and e = (u, v) and e = (u, v ) e he ou-arc and he in-arc, respecively. Le T = T e+e e he sugraph of he new asic variales. If e is no on he cycle in he curren G, he new poenial vecor Y associaed wih T can e oained as follows (noice ha γ does no change in his case): { y u yu + r = u v, for u T u (27) y u, for u T v. where r u v = c(e u v, γ) + y v y u and T u (T v ) is he componen of T e conaining u (v ). Theorem 4: If he sugraphs G s of feasile asic soluions generaed y he simplex mehod are srongly feasile, hen he simplex mehod does no cycle. Proof: Firs oserve ha in any sequence of degenerae pivos, he value of he slack variale will remain he same. So he leaving and enering variales can only e he links in he nework. Le G e a feasile asic soluion sugraph and e he roo. We define wo unique values for G : C(G ) = (u,v) E c uvx uv and W (G ) = u V (y y u ). Noice ha for a given G, he value of W (G ) is unique even hough he values of he poenials Y may no e unique. Consider wo consecuive asic soluions G i = G and G i+ = G i +e f, where e and f are he in-arc and ouarc, respecively. We firs show ha eiher C(G i+ ) < C(G i ) or W (G i+ ) > W (G i ). Indeed if he pivo ha generaes G i+ from G i is nondegenerae, hen C(G i+ ) < C(G i ). If i is degenerae, we have C(G i+ ) = C(G i ). In his case we shall prove W (G i+ ) > W (G i ). Here he in-arc e = (u, v) sill has zero flow in G i+. By Lemma 7, f is no a link on he cycle in G i, so he value of γ does no change. Because G i+ is srongly feasile, in G i+, link e mus e oriened oward he roo node, which implies ha node elongs o G (v) (he componen of G i f conaining v). Now we can oain he poenials using equaion (27). Since r uv = c(e uv, γ)+y v y u <, W (G i+ ) = W (G i ) G (u) r uv > W (G i ). If he simplex mehod cycles, hen for some i < j, G i = Gj. This means Gi = Gi+ = G j. Bu hen W (G i ) > W (Gi+ ) > > W (G j ) = W (Gi ) conradicing ha W (G i ) = W (Gj ). VI. A STRATEGY FOR AVOIDING DEGENERATE PIVOTS AND THE NETWORK SIMPLEX BASED (NBS) ALGORITHM In his secion we firs presen in Secion VI-A a sraegy for avoiding degenerae pivos. We hen show in Secion VI-B how o selec a leaving variale. In Secion VI-C we presen a complee descripion of he new Nework Based Simplex (NBS) algorihm and is complexiy analysis. We also show how o exrac an approximae soluion o he TCSP (hence he original CSP) prolem and esalish he performance ounds on he approximae soluion. A. Avoiding Degenerae Pivos In his secion we shall develop a sraegy which avoids performing degenerae pivos which is ased on he following pivo rule. Enhanced Pivo Rule P2: If here is a choice for selecing he enering variales, hen selec an enering variale in he following order of preference: a) The slack variale if i is eligile o ener. ) Eligile links whose ail nodes are on he direced s- pah(s) in he curren G. As we discussed in Secion V, rule a) aove guaranees ha every G is of one of he wo forms in Fig. 2.(a), (). Boh hese sugraphs of asic soluions are srongly feasile. Consider nex rule ). Suppose we can find an in-arc e = (u, v) according o rule ). Le W denoe he new cycle in G + e wih is orienaion defined as he direcion of e. I can e seen ha he flows on all links in W whose direcions disagree wih ha of W are nonzero and hus we can push posiive amoun of flow along he cycle unil he flows on some links of he s- pah (whose direcions disagree wih he orienaion of W ) reach zero. By removing one such link wih zero flow, we oain a new G. In fac, we can selec he ou-arc in such a way ha he resuling G is also srongly feasile (see nex susecion). This pivo will no lead o degeneracy. On he oher hand, if no such link is eligile o ener he asis (noe: in his case γ is nonnegaive), hen we have no opion u o perform a degenerae pivo. To avoid performing degenerae pivos we proceed as follows. Le P e he se of nodes on he s- pah(s) in he curren asis sugraph G. Assign coss o links in he nework as follows: Link cos c uv wih u / P and v P is se as c(e uv, γ) + y v > ; Oherwise c uv is se as c(e uv, γ)(see Fig. 3).

9 9 P c(e uv, γ) + y v v u x y c(e xy, γ) Node poenials in P do no change 6 5 in-arc s W' W (a) Fig. 3. Link coss for nework N. Now condense all he nodes in P ino a single node, say, R, and reverse he direcions of all he links. Le he resuling nework e called N. Noe ha none of he links wih oh is ends in P will e in N. Now use Dijksra s algorihm on N and oain he shores pah ree wih node R as he sar node. The links of G corresponding o he links of he shores pah ree of N and he links wih heir oh end nodes in P will e a new asis sugraph G (Noice ha his operaion preserves he srongly feasiiliy of G and will no change he value of γ). Le he shores disance value of he node u compued y he algorihm e d(u). Then we se he poenials of he nodes wih respec o G as fellows: For each node u / P, y u = d(u) and for all oher nodes (all he nodes in P ) he poenials are he same as in he previous G. Now, (u, v), u / P, y u = d(u) d(v)+c(e uv, γ) = y v +c(e uv, γ), which implies ha for all such links, r(u, v) = y v y u +γd uv +c uv and hose links whose ails are no in P are no eligile for choice as in-arc. Since he aove operaion does no affec he value of γ, w is no eligile eiher. Thus we can only consider arcs whose ails are in P (par () of enhanced rule P2). If we sill canno find an in-arc according o enhanced rule P2 afer he aove operaion, i implies ha we have go he opimal asic soluion since no enering variale is availale. We will show in he following secion how o choose a leaving variale using Theorem 2. B. Finding a Leaving Arc (Ou-Arc) Suppose he curren feasile asic soluion G is srongly feasile and link e = (u, v) is he in-arc. If G conains a cycle W, hen he flow can e decomposed ino exacly wo s- pahs. We define he ranching poin as he firs node on W as we raverse he pahs from node s o (see Fig. 2.()). In his susecion, e and e always denoe he in-arc and ou-arc, respecively. Claim : If he curren asic soluion G is srongly feasile and is no opimal, hen one of he arcs e inciden o he ranching node or he ail node of he in-arc e is eligile for choice as ou-arc and G + e e is sill srongly feasile. We prove he claim y enumeraing all possile cases and deermining he leaving variale in each case using Theorem 2 and Sep 4 of he revised simplex mehod. Le he cycle creaed y adding he in-arc e denoed y W wih is orienaion defined as ha of he in-arc. Case : Slack variale w is in he asic soluion (he curren G is a ree, γ = and w > ). This corresponds o Theorem 2 (a). According o Sep 4 of he revised simplex mehod, we need o consider only he enries of V ha are or d(w ) in-arc 7 W' s 2-3 () s W in-arc 3 4 (c) 6 W 4 W' Fig. 4. Find leaving variale: (a) su-cases 2., () su-case 2.2, and (c) su-case 2.3. if d(w ) >. Wihou loss of generaliy, assume d(w ) >. These enries correspond o he links of W ha lie on he s- pah of he curren G or he slack variale w. The corresponding enries in he curren asic soluion x B are for he links and is curren value for w. The minimum value of saisfying he consrain x B V will e deermined y he inequaliies and w d(w ). Thus he maximum value of will e min{, w/d(w )}. Since w = d(g ) is odd and d(w ) is even, w/d(w ). So, if w < d(w ), w will leave he asis. Oherwise, he links in W ha lie on some s- pah in he curren G are eligile o leave he asis. We shall selec he unique link e on he s- pah in G ha is inciden o he ail node of he in-arc. This guaranees ha he new G, denoed as G, is srongly feasile. Noice ha if w leaves he asis, w = in G. This means ha d(g ) =. In his case, G conains wo s- pahs p and p 2 wih flow and, respecively (see Fig. 2). The value of can e calculaed from he equaion: d(p ) + ( )d(p 2 ) =. Case 2: The asic soluion consiss of n links, i.e., here is a cycle W wih ranching poin a in he asic soluion. The slack variale w is eligile o ener he asis if γ <. Then according o par a) of pivo rule P2, we le w ener he asis and shall selec one of he wo links in he curren G ha are inciden on he ranching poin a o leave he asis. The choice can e made using Theorem 2 (c) of Secion V-D. Suppose γ >. An in-arc will creae a new cycle W. This corresponds o Theorem 2(). We need o consider hree sucases ha capure all possiiliies. Wihou loss of generaliy, we assume ha he orienaion of W is clockwise and he orienaion of W agrees wih he direcion of he in-arc.

10 Case 2. (Fig. 4.(a)): Possile ou-arcs: (, 2), (3, 5), and (3, 4). Here, (x 2, x 35, x 34 ) = (,, ) and hus he ouarc corresponds o he firs zero componen in he following formula as increases from. (,, ) (, d(w )/d(w ), d(w )/d(w )) = (, d(w )/d(w ), + d(w )/d(w )). Case 2.2 (Fig. 4.()): Possile ou-arcs: (, 2), (2, 7), and (2, 3). Link (7, 6) is no eligile for ou-arc for oherwise w in he nex asic soluion due o he propery of he ransformed nework. The ou-arc is decided y he following formula as in Case 2.. (x 2, x 27, x 23 ) (, + d(w )/d(w ), d(w )/d(w )) = (,, ) (, + d(w )/d(w ), d(w )/d(w )). Case 2.3 (Fig. 4.(c)): Possile ou-arcs: (2, 3), (2, 9), and (4, 5). The ou-arc corresponds o he firs zero componen in he following formula when increases. (x 23, x 29, x 45 ) ( d(w )/d(w ), d(w )/d(w ), d(w )/d(w )). C. NBS Algorihm, Complexiy Analysis, and an Approximae Soluion We now presen a complee descripion of he Nework Based Simplex (NBS) algorihm ha uses he sraegies developed in Secion VI-A and VI-B for he RELAX-TCSP prolem. We show in Secion VI-C. ha he algorihm is of pseudo-polynomial ime complexiy. In Secion VI-C.2 we show how o exrac from an opimum soluion o he RELAX- TCSP prolem a feasile soluion o he TCSP prolem and hence o he original CSP prolem and derive ounds on he deviaion of his soluion from he cos of he opimum soluion. ) Complexiy Analysis: Fac : If here is no cycle in he asic soluion sugraph, hen for each link e uv, he associaed flow x uv is eiher or. If here is a cycle W in G, x ij is or a leas / d(w ). Proof: If here is no cycle, he proof is rivial. Assume here is a cycle W. I can e seen ha he flow on links no on he wo pahs are and he flows on he pahs u no on he cycle is. Since here is a cycle, he flow can e decomposed ino wo pahs p and p 2. Consider flows on he cycle W. Suppose he flow on p and p 2 are and wih < <. Assume d(p ) d(p 2 ). Since d(p ) and d(p 2 ) are oh even and is odd, d(p ) and d(p 2 ). Also y Lemma 3, d(w ). So d(p ) d(p 2 ) ecause d(w ) = d(p ) d(p 2 ). We also have d(p ) + ( )d(p 2 ) = (d(p ) d(p 2 )) + d(p 2 ) =. So, min{d(p ), d(p 2 )} max{d(p ), d(p 2 )} and = ( d(p 2 ))/(d(p ) d(p 2 )). Hence /d(w ) ecause d(p 2 ) and d(w ) = d(p ) d(p 2 ) >. Similarly, we can prove ha /d(w ). Algorihm 2 NBS Transform he original nework as in Secion III Find an iniial feasile asic soluion as in Secion V-B loop if γ < hen Le slack variale w e he enering variale (rule (a) of Pivo rule P2 in Secion VI-A) else if an in-arc saisfying rule () of Pivo Rule P2 is availale hen Choose one of hem as he enering variale else Invoke Dijksra s algorihm on he acive coss of N o updae he poenials (See Secion VI-A). if an in-arc saisfying rule () of Pivo Rule P2 is availale hen Choose one of hem as he enering variale else Sop end if end if end loop Deermine a leaving variale as in Secion VI-B Updae he flows and he poenials as in Secion V-C Fac 2: If e uv is he in-arc and W and W are he newly creaed cycle and he old cycle (if i exiss), respecively, we have < y u y v γd uv c uv = γd(w ) + c(w ) { c(w = ), for γ = c(w )d(w ) d(w )c(w ) / d(w ), for γ. Proof: Suppose he cycle W is e e 2... e k where e = e uv. Since all he links u e uv on W are in he asic soluion, he reduced coss on all hese links u e uv are. So y u y v γd uv c uv = γd(w ) + c(w ). Recalling ha γ = c(w )/d(w ), if here exiss a cycle W in he asic soluion or γ = if no such cycle exiss, we ge he righmos equaliy. Since e uv is an in-arc, y u y v γd uv c uv >. Fac 3: Le e he maximal flow ha can e pushed on he new cycle W. Suppose ha e uv and x uv are a link and is flow in he asic soluion, respecively. Then he srices consrain on is given y x uv ( + d(w )/d(w ) ), and. Hence max min{, d(w ) + d(w ) } = d(w ) + d(w ). Proof: Firs assume here is a cycle W in he curren asic soluion. If we push flow on he new cycle W, according o Theorem 2 and Sep 4 of he revised simplex mehod, in he worse case, he flow on all links will e decreased y a mos ( + d(w )/d(w ) ). Proof follows if we recall ha x uv /d(w ). The proof is similar if here is no cycle in he asic soluion. Fac 4: Le T and T e wo consecuive feasile asic soluions in he simplex mehod and c(t ) denoe he cos of he flow associaed wih he asic soluion T. If c(t ) < c(t ) and D is he maximal link delay, hen c(t ) c(t ) = y u y v γd uv c uv /(2n 2 D 2 ).

11 Proof: Follows from c(t ) c(t ) = y u y v γd uv c uv and Facs 2 and 3. Theorem 5: NBS algorihm erminaes wihin 2n 3 D 2 C pivos, where n = V and D (resp. C) is he maximum link delay (resp. cos) and hence is ime complexiy is pseudopolynomial. Proof: Le T, T... T l e he sequence of consecuive feasile asic soluions. I suffices o show ha l 2(nD) 3. According o Fac 4, c(t ) c(t l ) l/(2(nd) 2 ) and c(t ) nc. This implies ha l 2n 3 D 2 C. Since each pivo requires O(m) operaions, he NBS algorihm is of pseudo-polynomial ime complexiy. Using similar argumens, he revised simplex mehod ha allows degenerae pivos u only uses he ani-cycling sraegy of Secion V-F can also e shown o e of pseudo-polynomial ime complexiy. 2) An Approximae Soluion o he TCSP / CSP Prolem and Performance Bounds: If he opimal asic soluion sugraph for he RELAX-TCSP prolem conains no cycle, hen clearly he s- pah in his sugraph is also he opimum soluion o he original CSP prolem. On he oher hand, if he opimal asic soluion graph conains a cycle, hen he opimum flow can e decomposed ino flows along wo direced s- pahs p and p 2 wih posiive flow along each pah. Lemma 8: If c(p2) c(p ), hen eiher c(p 2 ) c(p ) c(p ) and d(p 2 ) d(p ), where p is he opimal pah of he original CSP prolem or one of he wo pahs p and p 2 is opimal. Proof: Le < < and e he flows on p and p 2, respecively. We have I follows from (28) ha d(p ) + ( )d(p 2 ) =, (28) c(p ) + ( )c(p 2 ) c(p ). (29) min{d(p ), d(p 2 )} max{d(p ), d(p 2 )}. (3) By (29), c(p ) and c(p 2 ) canno oh e greaer han c(p ). So c(p 2 ) c(p ). If c(p 2 ) = c(p ) hen y (29), c(p ) c(p ) which implies p or p 2 is an opimal soluion. Assume c(p 2 ) < c(p ). Now min{d(p ), d(p 2 )} = d(p ), for oherwise p 2 will e a feasile soluion o he CSP prolem wih cos smaller han c(p ). So we have he required inequaliy d(p 2 ) d(p ). Also pah p is feasile for he original CSP prolem y Theorem. So c(p ) c(p ). Thus we have he required inequaliy c(p 2 ) c(p ) c(p ). I follows from he aove lemma ha he pah p is a feasile soluion o he TCSP prolem. We may use his as an approximae soluion o he original CSP prolem. We nex evaluae he qualiy of his approximae soluion. Theorem 6: Le p and p 2 e he wo pahs derived from he opimal soluion o he RELAX-TCSP prolem wih c(p ) Fig. 5. c(p 2 ), hen cos, delay s, 4, - 2 4, - 4, An example demonsraing ha he gap can e arirarily large. c(p ) c(p ) + ( c(p 2) c(p ) ) and d(p 2 ) + ( d(p ) ), where is he flow on pah p a erminaion and is he delay ound. Proof: From c(p ) + ( )c(p 2 ) c(p ), we oain c(p ) c(p ) c(p ) ( )c(p 2 ) c(p ) Because c(p ) c(p ), = c(p 2 ) c(p ). c(p 2 ) c(p ) c(p 2 ) c(p ) = + ( c(p 2) c(p ) ). Similarly, we can prove ha d(p 2 ) + ( d(p ) ). Using a special example elow, we can show ha no consan facor approximaion soluion ased on relaxaion approach (including NBS and LARAC algorihm) is possile (However, simulaions show ha he approximae soluion is very close o he opimum). For closing he gap eween he opimum value and he approximae value see [39]. Le OP T, OP T S, and denoe he opimal cos, he cos of he pah reurned y relaxaion mehod, and he delay upper ound. In Fig. 5, he solid links correspond o he asic variales in he opimal asis. Thus OP T S = 4. Since OP T = 4, OP T S OP T /OP T = ( 8)/4, where can e specified arirarily. VII. SIMULATION AND COMPARATIVE PERFORMANCE EVALUATION We compared our NBS algorihm wih he general purpose LP solvers, LARAC algorihm [6], parameric search ased LARAC algorihm [39] (denoed as PARA), and he LHWHM algorihm [22]. The LARAC algorihm has ime complexiy of O(m 2 log 4 m) [5] while he parameric search ased LARAC algorihm has eer complexiy, namely, O((m + n log n) 2 ) [39]. However, he complexiy resuls are derived using he wors scenario and hus hey may no e an accurae indicaor of he performance of algorihms on average asis. So we compared he four mehods using simulaions. We use hree classes of nework opologies: regular graphs (see [32]), Power-Law Ou-Degree graph [28], and H k,n

12 2 Waxman s random graph [34]. For a nework G(V, E), he nodes are laeled as, 2..., n = V. Nodes n/2 and n are chosen as he source and arge nodes. For he Power-Law Ou- Degree graph and Waxman s random graph, he hop numer of feasile s- pahs is usually very small even when he nework is very large. This will ias he resuls in favor of he LHWHM algorihm. So, for Waxman s random graphs, a link joining node u and v is added if u v < V /5 esides oher rules for generaing random graphs. We keep he original version of Power-Law Ou-Degree graph as in [28]. Even hough his kind of graphs favors he LHWHM algorihm, he comparison of he performance of he LARAC and NBS algorihms is sill an indicaor of he meris of NBS. The link coss and delays are randomly independenly generaed even inegers in he range from o 2. The delay ound is.2 imes he delay of he minimum delay s- pahs in G. The resuls are shown in Fig Experimens show ha NBS algorihm can usually find eer soluions han he LARAC algorihm y selecing he es feasile pah encounered during he execuion insead of he opimum pah o he RELAX-TCSP prolem. We also find ha for sparse graphs (Fig. 6.(c)), NBS akes more ime han he LARAC algorihm. However, when he nework is dense (large ou-degree, See Fig. 6.(d)), NBS eas LARAC. Basically, NBS algorihm is a neighor search algorihm in which a eer soluion is derived from he curren soluion. A each pivo, he NBS algorihm ries all he nodes in he s- pah in he curren asic graph in order o find an in-arc emanaing from a node in he pah. When he graph is dense, i is more likely ha an eligile in-arc can e found in fewer ries. On he oher hand, he LARAC algorihm invokes a series of Dijksra s shores pah algorihm. When he graph is denser, each sep in Dijksra s algorihm akes more ime since Dijksra s algorihm checks all he neighors of he currenly processed node. We also compared he NBS algorihm wih general purpose LP solvers: CPLEX 8. ( QSop (www2.isye.gaech.edu/ wcook/qsop), and CLP ( Among all he hree solvers, CPLEX is always he fases (his is no surprising ecause CPLEX is recognized as one of he es LP solvers). So we only repor he experimens wih CPLEX. In our experimens wih CPLEX, we have used he same graphs as aove. Using CPLEX package, we may choose differen opimizers such as he primal dual mehod, nework simplex ec. Our experimens show ha he CPLEX using he primal dual uses he leas ime and so our comparison is wih respec o his opimizer. Noice ha CPLEX can also rerieve he nework srucure underlying he CSP prolem. Bu we found ha his does no help decrease he running ime. Acually, i akes longer ime o find he opimal soluion if CPLEX is direced o use he special srucure of he neworks. The numerical simulaion resuls in Fig. 9 shows ha he NBS algorihm is much faser. VIII. SUMMARY In his paper, we sudied he QoS rouing prolem (or equivalenly he CSP prolem) from he primal perspecive Cos Time(ms) Time(ms) Time(ms) LHWHM NB S OP T Node 5 NBS-TIM E LARAC-TIM E P ARA -TIM E Ou-Degree NBS-TIM E LHWHM -TIME LARAC-TIM E P ARA -TIM E NBS-TIM E LHWHM -TIME LA RAC-TIM E P ARA -TIM E (a) () Node (c) Node (d) Fig. 6. Simulaion on regular graph: (a) qualiy of soluions on regular graph wih ou-degree = 6, () compuaional ime on regular graph wih numer of nodes = 2, (c) compuaional ime on regular graph wih ou-degree = 6, and (d) compuaional ime on regular graph wih ou-degree = 36. in conras o mos of he currenly availale approaches ha sudied he prolem from a dual perspecive. Specifically we applied he revised simplex mehod on he primal form of he RELAX-TCSP prolem. Several sraegies are employed o achieve efficien implemenaion of he revised simplex mehod. These sraegies include: explici formulas o solve he sysems of equaions needed o find enering and leaving variales, an ani-cycling sraegy, and a sraegy o avoid degenerae pivos. These resul in wo algorihms. One of hese allows degenerae pivos and uses an ani-cycling sraegy developed in his paper. The oher algorihm called NBS algorihm avoids degenerae pivos. We show ha oh algorihms are of pseudo-polynomial-ime complexiy. We have also shown how o exrac an approximae soluion o he original CSP prolem

13 3 Cos Ti me (ms ) Node (a) NBS-TIME LHWHM-TIME LARAC-TIME PARA-TIME OPT LHWHM NBS LRARC Node Fig. 7. Waxman s random graph: (a) qualiy of soluions on Waxman s random graph wih α =.6 and β =.9 and () compuaional ime. Fig. 8. Fig. 9. Time(ms) () Compuaional Time on Pow er-law Ou-Degree graph NBS-TIM E LHWHM -TIM E LA RA C-TIM E PARA-TIM E Node Power-law ou-degree graph. Running Time CPLEX-REGULAR CPLEX-RANDOM CPLEX-POWER NBS-REGULAR NBS-RANDOM NBS-POWER Node NBS and CPLEX comparison. from he opimum soluion o he RELAX-TCSP prolem and derive ounds on he qualiy of his soluion wih respec o he opimum soluion. Exensive simulaion resuls are presened o demonsrae ha our approach compares favoraly wih he LARAC algorihm and is faser on dense graphs. Also, our algorihm is faser han he general purpose LP solvers. Besides providing insighs ino he srucure of soluions produced, our approach ased on he primal simplex offers a framework for sudying oher classes of prolems such as he disjoin QoS pahs selecion prolem and he QoS rouing prolem wih muliple consrains. In [37] we have repored our resuls on he disjoin QoS pahs selecion prolem. In he case of muliple consrains he srucure of asic soluions may conain up o l cycles for a prolem wih l addiive consrains. To apply our approach o he case involving muliple consrains, we need o develop efficien mehods o solve he wo sysems of equaions sudied in Secion V. Our approach in cominaion wih he approach developed in [24] is expeced o lead o furher advances in his area. REFERENCES [] Privae nework-nework inerface specificaion version. (PNNI.). Technical repor, ATM Forum Technical Commiee, March 996. [2] R. K. Ahuja, T. L. Magnani, and J. B. Orlin. Neworks Flows. Prenice- Hall, NJ, USA, 993. [3] Y. Bejerano, Y. Breiar, A. Orda, R. Rasogi, and A. Sprinson. Algorihms for compuing QoS pahs wih resoraion. IEEE Trans. of Neworking, 3(3):648 66, 25. [4] D. P. Bersekas. Nework opimizaion: coninuous and discree models. Ahena Scienific, Belmon, Massachuses, USA, 998. [5] B. Blokh and G. Guin. An approximaion algorihm for cominaorial opimizaion prolems wih wo parameers. Ausralasian Journal of Cominaorics, 4:57 64, 996. [6] A. Chakraari and G. Manimaran. Reliailiy consrained rouing in QoS neworks. IEEE Trans. of Neworking, 3(3): , 25. [7] S. Chen and K. Nahrsed. On finding muli-consrained pah. In ICC, pages , 998. [8] V. Chváal. Linear Programming. W. H. Freeman, New York, 983. [9] E. Crawley, R. Nair, B. Rajagopalan, and H. Sandick. A framework for QoS ased rouing in he inerne. RFC 2386, Inerne Engineering Task Force, Novemer 997. fp://fp.ief.org/inerne-drafs/draf-ief-qosrframework-2.x. [] R. Guerin, S. Kama, A. Orda, and T. Przygienda. QoS rouing mechanisms and OSPF exensions. RFC 2676, Inerne Engineering Task Force, March 997. [] G. Handler and I. Zang. A dual algorihm for he consrained shores pah prolem. Neworks, :293 3, 98. [2] R. Hassin. Approximaion schemes for he resriced shores pah prolem. Mah. of Oper. Res., 7():36 42, 992. [3] A. Iwaa, R. Izmailov, B. Sengupa D.-S. Lee, G. Ramamurhy, and H. Suzuki. ATM rouing algorihms wih muliple QoS requiremens for mulimedia inerneworking. IEICE Trans. Commun., 8:999 6, 996. [4] J. M. Jaffe. Algorihms for finding pahs wih muliple consrains. Neworks, 4:95 6, 984. [5] Alpár Jüner. On resource consrained opimizaion prolems. in review, 23. [6] Alpár Jüner, Balázs Szviaovszki, Ildikó Mécs, and Zsol Rajkó. Lagrange relaxaion ased mehod for he QoS rouing prolem. In INFOCOM, pages , 2. [7] Turgay Korkmaz and Marwan Krunz. Muli-consrained opimal pah selecion. In INFOCOM, pages , 2. [8] Turgay Korkmaz and Marwan Krunz. A randomized algorihm for finding a pah sujec o muliple QoS requiremens. Compuer Neworks, 36(2/3):25 268, 2. [9] F. A. Kuipers, T. Korkmaz, M. Krunz, and P. Van Mieghem. An overview of consrain-ased pah selecion algorihms for QoS rouing. IEEE Commun. Mag., 4:5 55, 22.

14 4 [2] Gang Liu and K. G. Ramakrishnan. A*prune: An algorihm for finding k shores pahs sujec o muliple consrains. In INFOCOM, pages , 2. [2] D. Lorenz and D. Raz. A simple efficien approximaion scheme for he resriced shores pahs prolem. Oper. Res. Leers, 28:23 29, 2. [22] G. Luo, K. Huang, J. Wang, C. Hos, and E. Muner. Muli-QoS consrains ased rouing for ip and ATM neworks. In in Proc. IEEE Workshop on QoS Suppor for Real-Time Inerne Applicaions, 999. [23] Kur Mehlhorn and Mark Ziegelmann. Resource consrained shores pahs. In ESA, pages , 2. [24] Pie Van Mieghem and Fernando A. Kuipers. Conceps of exac QoS rouing algorihms. IEEE/ACM Trans. New., 2(5):85 864, 24. [25] Pie Van Mieghem, Hans De Neve, and Fernando A. Kuipers. Hop-yhop qualiy of service rouing. Compuer Neworks, 37(3/4):47 423, 2. [26] H. De Neve and P. Van Mieghem. Tamcra: A unale accuracy muliple consrains rouing algorihm. Compu. Commun., 23: , 2. [27] Ariel Orda and Alexander Sprinson. Efficien algorihms for compuing disjoin QoS pahs. In INFOCOM, pages , 24. [28] C. R. Palmer and J. G. Seffan. Generaing nework opologies ha oey power laws. In IEEE GLOBECOM, pages , 2. [29] Cynhia A. Phillips. The nework inhiiion prolem. In STOC, pages , 993. [3] R. Ravindran, K.Thulasiraman, A. Das, K. Huang, G. Luo, and G. Xue. Qualiy of services rouing: heurisics and approximaion schemes wih a comparaive evaluaion. In ISCAS, pages , 22. [3] S. Shenkar, C. Paridge, and R. Guering. Specificaion of guaraneed qualiy of service. RFC 222, Inerne Engineering Task Force, Sepemer 997. [32] K. Thulasiraman and M. N. Swamy. Graphs: Theory and algorihms. Wiley Inerscience, New York, 992. [33] Zheng Wang and Jon Crowcrof. Qualiy-of-service rouing for supporing mulimedia applicaions. IEEE Journal on Seleced Areas in Communicaions, 4(7): , 996. [34] B. M. Waxman. Rouing of mulipoin connecions. IEEE Journal on Seleced Areas in Commun., 6(9):67 622, Dec [35] Y. Xiao, K. Thulasiraman, and G. Xue. Equivalence, unificaion and generaliy of wo approaches o he consrained shores pah prolem wih exension. In Alleron Conference on Conrol, Communicaion and Compuing, Universiy of Illinois, pages 95 94, 23. [36] Y. Xiao, K. Thulasiraman, and G. Xue. The primal simplex approach o he QoS rouing prolem. In QSHINE, pages 2 29, 24. [37] Y. Xiao, K. Thulasiraman, and G. Xue. Consrained shores link-disjoin pahs selecion: A nework programming ased approach. Acceped y IEEE Trans. on Circuis and Sysems, 25. [38] Y. Xiao, K. Thulasiraman, and G. Xue. GEN-LARAC: A generalized approach o he consrained shores pah prolem under muliple addiive consrains. In ISAAC, pages 92 5, 25. [39] Y. Xiao, K. Thulasiraman, G. Xue, and A. Jüner. The consrained shores pah prolem: algorihmic approaches and an algera sudy wih generalizaion. AKCE J. Graphs. Comin., 2(2):63 86, 25. [4] G. Xue. Minimum-cos QoS mulicas and unicas rouing in communicaion neworks. IEEE Trans. On Commun., 5:87 827, 23. [4] G. Xue, A. Sen, and R. Banka. Rouing wih many addiive QoS consrains. In ICC, pages , 23. [42] Xin Yuan. Heurisic algorihms for muliconsrained qualiy-of-service rouing. IEEE/ACM Trans. New., (2): , 22. [43] M. Ziegelmann. Consrained shores pahs and relaed prolems. PhD hesis, Max-Planck-Insiu fr Informaik, 2. Ying Xiao received he B.S. degree in compuer science from he Nanjing Universiy of Informaion Science and Technology (formerly Nanjing Insiue of Meeorology), Nanjing, China, in 998, M.S. degree in compuer applicaion from he Souhwes Jiaoong Universiy, Chengdu, China, in 2, and Ph.D degree in compuer science a he Universiy of Oklahoma, Norman, in 25. His research ineress include graph heory, cominaorial opimizaion, disriued sysems and neworks, saisical inference, and machine learning. Krishnaiyan Thulasiraman received he Bachelor s degree (963) and Maser s degree (965) in elecrical engineering from he universiy of Madras, India, and he Ph.D degree (968) in elecrical engineering from IIT, Madras, India. He holds he Hiachi Chair and is Professor in he School of Compuer Science a he Universiy of Oklahoma, Norman, where he has een since 994. Prior o joining he Universiy of Oklahoma, Thulasiraman was professor (98-994) and chair ( ) of he ECE Deparmen in Concordia Universiy, Monreal. He was on he faculy in he EE and CS deparmens of he IITM during Dr. Thulasiraman s research ineress have een in graph heory, cominaorial opimizaion, algorihms and applicaions in a variey of areas in CS and EE: elecrical neworks, VLSI physical design, sysems level esing, communicaion proocol esing, parallel/disriued compuing, elecommunicaion nework planning, faul olerance in opical neworks, inerconnecion neworks ec. He has pulished more han papers in archival journals, coauhored wih M. N. S. Swamy wo ex ooks Graphs, Neworks, and Algorihms (98) and Graphs: Theory and Algorihms (992), oh pulished y Wiley Iner-Science, and auhored wo chapers in he Handook of Circuis and Filers (CRC and IEEE, 995) and a chaper on Graphs and Vecor Spaces for he handook of Graph Theory and Applicaions (CRC Press,23). Dr. Thulasiraman has received several awards and honors: Endowed Gopalakrishnan Chair Professorship in CS a IIT, Madras (Summer 25), Eleced memer of he European Academy of Sciences (22), IEEE CAS Sociey Golden Juilee Medal (999), Fellow of he IEEE (99) and Senior Research Fellowship of he Japan Sociey for Promoion of Science (988). He has held visiing posiions a he Tokyo Insiue of Technology, Universiy of Karlsruhe, Universiy of Illinois a Urana-Champaign and Chuo Universiy, Tokyo. Dr. Thulasiraman has een Vice Presiden (Adminisraion) of he IEEE CAS Sociey (998, 999), Technical Program Chair of ISCAS (993, 999), Depuy Edior-in-Chief of he IEEE Transacions on Circuis and Sysems I ( 24-25), Co-Gues Edior of a special issue on Compuaional Graph Theory: Algorihms and Applicaions (IEEE Transacions on CAS, March 988), Associae Edior of he IEEE Transacions on CAS (989-9, 999-2), and Founding Regional Edior of he Journal of Circuis, Sysems, and Compuers. Recenly, he founded he Technical Commiee on Graph heory and Compuing of he IEEE CAS Sociey. Guoliang Xue is a Professor in he Deparmen of Compuer Science and Engineering a Arizona Sae Universiy. He received he Ph.D degree in Compuer Science from he Universiy of Minnesoa in 99 and has held previous posiions a he Army High Performance Compuing Research Cener and he Universiy of Vermon. His research ineress include efficien algorihms for opimizaion prolems in neworking, wih applicaions o faul olerance, rousness, and privacy issues in neworks ranging from WDM opical neworks o wireless ad hoc and sensor neworks. He has pulished over papers in his area. His research has een coninuously suppored y federal agencies including NSF, ARO and DOE. He is he recipien of an NSF Research Iniiaion Award in 994, and an NSF-ITR Award in 23. He is an Associae Edior of he IEEE Transacions on Circuis and Sysems-I, an Associae Edior of he Compuer Neworks Journal, and an Associae Edior of he IEEE Nework Magazine. He has served on he execuive/program commiees of many IEEE conferences, including Infocom, Secon, Icc, Gloecom and QShine. He is a TPC co-chair of IEEE Gloecom 26 Symposium on Wireless Ad Hoc and Sensor Neworks.