Optimal Network Design with End-to-End Service Requirements

Transcription

1 ONLINE SUPPLEMENT for Optiml Networ Design with End-to-End Service Reuirements Anntrm Blrishnn University of Tes t Austin, Austin, TX Gng Li Bentley University, Wlthm, MA Prsh Mirchndni University of Pittsurgh, Pittsurgh, PA APPENDIX A: Proofs of Select Propositions Proposition 1: If the overlpping pth condition holds for ll Ki, N \ s, t Dim(Conv[NDSR] = K ( A N +1 + A. Proof of Proposition 1. Let K A A 1 1 1, A, K;, A; nd, then z (A.1.1 where with t lest one coefficient not eul to zero, denote n ritrry ineulity. To estlish Proposition 1, it is sufficient to show tht if the hyperplne z (A.1.2 K A A contins ll the fesile points of Conv[NDSR], then (A.1.2 is liner comintion of the flow conservtion eutions (2.2. We will show this y compring the left-hnd side epression of (A.1.2 for different fesile NDSR solutions to determine or relte the vlues of its coefficients. We first construct fesile NDSR solution, denoted y ( 0, z with elements {(,( z }, y routing ech commodity on fesile pth nd setting the design vriles for ll rcs to one. Now, construct nother solution ( 1, z 1 which differs from ( 0, z 0 only in the vlue of one design vrile for some rc ' A. Specificlly, set ( z' ( z' 1, ( z ( z for ll rcs A\{'}, nd 1 = 0. Since oth ( 0, z 0 nd ( 1, z 1 elong to Conv[NDSR]nd, y ssumption lie on the hyperplne z = 0, A., we get ' 0 K A A. Since we chose rc ritrrily, A1

2 Now, construct solution ( 2, z 2 from ( 0, z 0 y rerouting the flow of commodity K on one of its lternte fesile pths while retining the originl vlues for ll the design vriles nd the flow vriles of ll commodities in K\. (The networ must contin t lest one lternte fesile pth since otherwise we cn fi the flow vriles on ll the rcs in the originl pth to one, which is not permitted. Since ( 0, z 0 nd ( 2, z 2 differ only in the flow of commodity, nd the choice of ws ritrry, we otin = constnt, sy, t, for ll fesile s - to-t pths p(s, t, K. p( s, t We will now show tht, for ll Ki, N \ s, t, is constnt, sy p( s, i fesile supths p(s, i. Given ny two fesile supths p s, i nd p s, i overlpping pth condition nd our result ri, t such tht p( s, i r( i, t i, for ll, the t imply tht there eists supth p( s, t t p( s, i r( i, t. Hence, fesile supths p(s, i for commodity from node s to node i, i s, nd 0. i for ll p( s, i Net, recll tht every rc elongs to some s -to-t fesile pth. Let p(s, j( e this fesile pth s supth contining rc s its lst rc. Then, j( ' i(, i.e.,,, j( i( A K. ' p( s, j( Now, ting liner comintion of flow conservtion constrints (2.2 using i s the multiplier for the flow conservtion constrint corresponding to node i nd commodity, we get (( j( i( ( t s j i A K nd s 0, this. Since,, ( ( K A K eution reduces to t K A K. The right-hnd side is constnt which we cn set to. Thus, we hve shown tht the hyperplne (A.1.2 z is liner K A A comintion of the K ( N 1 non-redundnt flow conservtion constrints, implying tht Dim(Conv[NDSR] = A ( K 1 K ( N 1 K ( A N 1 A. s Proposition 3: Given se cover C m for commodity nd metric m nd the ssocited m incomptile rc susets, the following Lifted Route Cover ineulity m C 1 (3.7 m C m A\ C is vlid nd cn strengthen formultion [NDSR] even fter dding the Incomptile r-arc ineulities (3.4. A2

3 Proof of Proposition 3: The pper discusses two methods to compute the lifting coefficients in the Lifted Route Cover ineulity (3.11. After some introductory discussions, we estlish the vlidity of these two methods in two seprte prts ( nd (. Recll tht, for ech rc C m m, is suset of rcs tht re mutully incomptile nd lso incomptile with rc. Let B = A\C m denote the rcs tht re not in the cover. For ech m m rc B, S { C : } denotes the set of cover rcs C m whose ssocited incomptile rc set contins rc. We refer to ny rc B with S = s n unrelted m rc with respect to cover C m m (for the given mutully incomptile rc susets. In the following discussions, we consider given fesile (nd integer NDSR solution tht routes commodity on suset of rcs B* B not in the cover C m (in ddition to routing the commodity on some cover rcs. We use the following oservtion in our proofs: Oservtion 1: For every pir of rcs 1, 2 B*, we must hve S 1 S2. Otherwise, if S 1 S2 contins n rc C m, then oth 1 nd 2 elong to the mutully incomptile set, contrdicting the fesiility of the given solution. m Define C' C m \ B* S s the remining rcs of the originl cover fter removing ll the rcs elonging to S, for ll B* (since rcs in S re incomptile with rc B* they cnnot crry commodity in the given solution. For this solution, sustituting, 1 nd 0 for ll S for every B*, nd 0 for ll B\B*in ineulity (3.7, we get: m C 1. C' B* By Oservtion 1, the susets S ecluded from C m re ll mutully eclusive. So, we cn rewrite the ove ineulity s: C' S 1. (A.3.1 C' B* B* We need to estlish the vlidity of this ineulity for the two lifting methods. ( The first lifting method computes the coefficients y rrnging the rcs of C m in order of m non-incresing nd non-decresing weights w (depending on whether the rc elongs to S, nd then compring the cumultive weights of these rcs with the weight of rc. For this proof, we will euivlently divide the coefficient into two integrl components nd such tht. m For ech rc B, if we rrnge the rcs in S in non-decresing order of weights w, let l v denote the weight of the l th 0 rc in this ordered list, nd define v 0. Then, define ' m{ 0,1,2,..., : l l m l S v } l ' 0 w. Let T S denote the suset of the first (i.e., lowest weight rcs in the ordered list of rcs in S, whose totl weight is covered y the weight of A3

4 rc. By these definitions, T. For convenience, we define = 0 for ll unrelted rcs B; for these rcs, S T. m m m For ech rc B, define e m{ w w,0} s the ecess weight of rc if S m rc s weight is greter thn or eul to the totl weight of ll the rcs in S ; otherwise, e is m m zero. Note tht, for unrelted rcs, e w. For every rc B, nd for l = 1, 2,, ( C m l S, let u denote the weight of the l th heviest rc in the suset C m \S of the cover rcs, nd set 0 u 0 Define: l m l ' m m{ l 0,1.2,..., C S : u e } if T S l ' 1 for ll B. 0 otherwise Thus, is the fewest numer of the cover rcs not in T (or S whose totl weight is covered y the ecess weight of rc. Oserve tht = 0 if = T < S. Since nd T S, the Lifted Cover ineulity (A.3.1 for the given solution (with commodity routed on B* B using the first lifting method reduces to: C' ( S T 1. (A.3.2 C' B* B* We wish to show tht this ineulity does not eliminte ny fesile NDSR solution tht routes commodity on the non-cover rcs B*. m e Oservtion 2: Since the ecess weight is greter thn or eul to the totl weight of t m lest rcs in C \ T it lso euls or eceeds the totl weight of t lest of the heviest of rcs in C" C m \ B* T(nd lso in C' C m \ B* S. Conseuently, the totl ecess m weight e B* of ll the rcs in B* euls or eceeds the sum of weights of t lest of the heviest rcs in C' or C". B*, nd let C* C' Define ( S T B* B* denote the suset of rcs of C' tht crry commodity in ny fesile solution tht routes commodity on rcs in B*. Ineulity (A.3.2 sttes C* C' 1. This reuirement clerly holds if < 0 since C* C'. Suppose > 0. In this cse, we show tht, if C* C', the totl weight of ll the rcs in the solution crrying commodity must eceed the weight limit W m. The totl weight of ll the rcs in the routing solution for commodity is: m m m m m w w w e w. (A.3.3 B* C* B* T B* C* Now, if C* C' C' ( S T, then the sum of the lst two terms in the B* B* right-hnd side of (A.3.3 euls or eceeds the sum of the weights of ll the rcs in " m m C C \ B* Tsince euls or eceeds the weight of the heviest rcs in e B* B* A4

5 C" nd C" contins only C' ( S T rcs. Therefore, B* w w w w w W, m m m m m m B* C" B* T m C \ B* T m C where the lst ineulity follows from the fct tht C m is cover. This finding contrdicts the fesiility of the given solution. Therefore, C* C' 1, nd ineulity (A.3.2 is vlid. ( For the second lifting method, for ech rc B, we set the lifting coefficient s follows: m m S if w w T( S 1 otherwise. Given fesile NDSR solution tht routes commodity on the suset of non-cover rcs B* B, let B' { B*: S 0} denote the suset of rcs in B* whose incomptile suset S is non-empty nd weight euls or eceeds the weight of ll rcs in S. Similrly, let B" { B*: S 1 0}. Then, for this solution, the Lifted Cover ineulity (A.3.1 reduces to: S ( S 1 C' S 1 i.e.,,. (A.3.4 C' B' B" B* C' S B" 1 C' B*\( B' B" Clerly, this ineulity holds if B*\ B ' is non-empty, i.e., either B " or B* \( B' B". So, we cn focus on the cse with B* B ', i.e., ll rcs in B* hve weight greter thn or eul to the totl weight of ll rcs in the corresponding incomptile sets S. In this cse, (A.3.4 reduces to C' C ' 1, specifying tht t lest one rc of C' must e omitted in ny fesile routing of commodity tht includes ll the rcs of B*. Suppose fesile solution routes commodity on ll the rcs of C' (in ddition to the rcs of B*. Then, the totl weight of ll the rcs tht crry commodity is: m m m m m m w w w w w W, C' B* m \ * m C B S B* S C where the lst ineulity follows from the fct tht C m is cover. This finding contrdicts the fesiility of the given solution. Therefore, ineulity (A.3.1 is vlid. Figure 1 in the tet of the pper provides n emple to demonstrte tht the Lifted Cover ineulity (3.7 cn strictly tighten formultion [NDSR]. Proposition 5: When the overlpping pth condition holds for commodity t node v, the Lifted Turn ineulity (3.12 is fcet of Conv[NDSR]. Proof of Proposition 5: We first define I ( v SI AH ( SI nd O ( v A (\ v IA ( SI (consistent with our nottion A5

6 for the more generl SFF ineulity in Section 3.2.2, nd rewrite (3.12 s: Define Suppose. (3.12 I ( v O ( v L = {(, zconv([ndsr]: (, z stisfies (3.12 t eulity}. z ˆ (A.5.1 ' ' ˆ ' K A A is generl ineulity tht holds t eulity for ll (, z in L. The coefficients in this epression ' 1 '1 1 re ll rel numers:, A, K;, A; nd, with t lest one non-zero coefficient. By chrcterizing these coefficients, we will show tht this ineulity reduces to constrint (3.12. Lemm 5.1. The coefficients 0, A. Proof. Construct fesile routing 0 tht stisfies (3.12 t eulity, nd set ll the design vriles to one, i.e., z 0 = 1, to otin fesile NDSR solution. Now, for ny rc ' A, construct solution ( 1, z y setting ( z ( z 1, eeping the vlues of ll other vriles ' in ( 0, z 0 the sme. Since ( 1, z 1 lso stisfies ineulity (3.12 t eulity, we get ' 0, which in turn implies tht 0, A. ' Lemm 5.2. Ineulity (A.5.1 is euivlent to ˆ. (A.5.2 A Proof. Since 0, A (from Lemm 5.1, z 0. Now, for ech commodity ', A construct nother solution ( 2, z 2 y re-routing commodity ' from its current pth, sy p ( s', t' in solution ( 0, z 0 to nother fesile pth, sy pth p ( s', t'. (An lternte fesile pth must necessrily eist since otherwise we cn fi the flow of commodity ' on ll the rcs of the originl pth to one, which is not permitted. In constructing ( 2, z 2 ' 2, we set ( 0for rcs ' 2 p ( s', t' nd ( 1for p ( s', t', ut eep ll other elements of ( 0, z 0 the sme. Ineulity (3.12 holds t eulity for oth ( 0, z 0 nd ( 2, z 2. So, y hypothesis, ineulity ' ' ' 0 ' ' 2 ' (A.5.1 holds t eulity, nd we get ˆ ˆ ( ˆ ( ˆ. Since p( s', t' A A p( s', t' we chose commodity ' nd its flow pth p ( s', t' ritrrily from the set of fesile pths of ', ' we see tht ˆ is constnt for ny fesile pth ps ( ', t ' of commodity ' nd for p( s', t' every commodity ' K \{ }. Thus, gives the desired result. ' ' ˆ is constnt, sy 0 ' K\{ } A. Setting ˆ 0 Lemm 5.3. Ineulity (A.5.2 is euivlent to 0, where ( ˆ A ( s, nd ˆ otherwise A A6

7 Proof. Multiplying the flow conservtion constrints ineulity (A.5.2 gives A ( s ( ˆ ˆ 0, A ( s A\ A ( s A 1 y, nd dding to i.e., 0. (A.5.3 Oservtion. Lemm 5.3 implies tht, for ny fesile pth p( s, t from node s to node t tht stisfies (3.12 t eulity, we must hve 0. p( s, t Lemm 5.4. Let p( s, v denote fesile supth from s to node v tht uses n rc from I (. v Then some constnt, sy,. p( s, v Proof. Since the flows on rcs in I ( v cnnot e fied to zero, there must e t lest one fesile supth from s to node v contining rcs from I ( v. If there is only one such supth, then the Lemm holds. Otherwise, consider ny two pths p ( s, v nd p ( s, v from s to node v, oth contining n rc from I ( v. By the overlpping pth condition, there eists supth rvt (, from node v to t tht is comptile with oth p'( s, v nd p''( s, v. Therefore, y the oservtion following Lemm 5.3, 0. p'( s, v p''( s, v, implying tht p'( s, v r( v, t p''( s, v r( v, t Lemm 5.5. (. e e ea I ( v O ( v Given fesile NDSR solution, consider the flow e of commodity on ny rc ( ( ei v O v IO ( v. Define ( e s the flow on rc e tht lter flows through rc I ( v. Similrly, ( e ( 0 ( e denotes the flow on rc e tht lter flows through rc in O ( v (tht does not go through ny rc in IO ( v. Thus, ( e ( e ( e. (A.5.4 e 0 I ( v O ( v Sustituting these vriles in the left-hnd side of (A.5.3, we get ee e 0 ( e ( e ( e ea ea\ IO ( v I ( v O ( v I ( v O ( v e0 ( e e( e e( e (A.5.5 ea\ IO ( v I ( v ea\ IO ( v O ( v ea\ IO ( v. I ( v O ( v In this series of steps, the first eution follows from simply sustituting the vlue of e from A7

8 (A.5.4 in (A.5.5. The second eution follows from rerrnging terms, nd the third step follows from Lemms 5.3 nd 5.4. Using Lemm 5.5, ineulity (A.5.3 reduces to 0. (A.5.6 I ( v O ( v Since (3.12 is vlid ineulity, the term within prentheses in the left-hnd side of (A.5.6 is non-positive. Hence, is non-negtive. If is zero, we get vcuous ineulity. So, must e positive. Dividing oth sides of (A.5.6 y, we see tht (A.5.3, nd hence (A.5.1, reduces to 0, which the sme s (3.12. Hence the proposition holds. I ( v O ( v Proposition 6: For ny collection of Gen-OR nd Gen-IF reltionships, if is even for ll rcs A nd u is positive odd integer, then the Gen OR-IF ineulity IOR z 2 u 2 ' ' IOR A IIF A A IOR is vlid nd tightens formultion [NDSR].. (3.13 Proof of Proposition 6. The Gen-OR ineulities re. For ech flow vrile ineulities re ' '' I A A IF ' u I A OR nd the Gen-IF in these constrints, consider the forcing constrints, z. By definition, is the numer of times tht rc ' " ppers in the sets A A for ll = 1, 2,, Q. Therefore, dding the Gen-OR, Gen-IF, nd forcing constrints, we get Q 2 z u 1 ' A A IOR. Dividing oth sides of this ineulity y 2, nd noting tht the left hnd side of the ineulity is integer, nd z / 2 is integer, we otin the Gen OR-IF ineulity (3.13. Figure 2 in the tet of the pper provides n emple to demonstrte tht the Gen OR-IF ineulity cn strictly tighten formultion [NDSR]. A For the proof of Proposition 7, we will ssume our rc nd commodity indices re modulo where we define (mod to eul. We inde the rcs in A F nd commodities in K F so tht the pirs of rcs nd (+1, in round roin fshion, hve n OR reltionship for commodity, i.e., the reltionships R OR(,{, 1},1 for = 1, 2,,, hold. Recll tht these reltionships, where = Q > 3 is odd. 1 1 led to the _OR ineulity (3.16, ( ( 1/2 z 1 A8

9 Let A F = { 1, 2,, } nd K F = { 1, 2,, } denote the rcs nd corresponding commodities tht induce the _OR ineulity (3.16. Let N F = {i( 1, j( 1, i( 2, j( 2,, i(, j( } denote the set of nodes incident to the rcs in A F. We refer to G F : (A F, N F s fcet-defining grph for ineulity (3.16. Figure A.1 illustrtes possile structure of this grph. In this figure, nodes N F nd rcs A F re shown in old; the hed nodes nd the til nodes of the fcet rcs, A F, re distinct nd do not overlp with ech other or with the source nd sin nodes. However, in generl, these nodes need not e distinct. The wvy lines in the figure indicte pths in G tht my possily include nodes not shown in the figure. Figure A.1: Fcet-defining grph for OR ineulity The proof will reuire us to construct fesile solutions tht stisfy ineulity (3.16 t eulity. We denote the design vrile vlues for fesile NDSR solution s its design solution, nd note tht the sme design solution might permit multiple fesile routes for the sme commodity tht ll stisfy ineulity (3.16 t eulity. Proposition 7: If the overlpping pth condition holds in G A F for ech commodity KF t nodes i( nd i( +1, nd if every commodity K hs t lest one fesile pth in, then the corresponding _OR ineulity (3.16, with 3 nd odd, is fcet-defining for Conv[NDSR]. Proof of Proposition 7: Define L = {(, zconv([ndsr]: (, z stisfies (3.16 t eulity}. Let G A F ˆ z (A.7.1 K A A represent generl ineulity tht holds t eulity for ll (, z in L. The coefficients in this epression re ll rel numers:, A, K;, A ; nd nd t lest one coefficient is non-zero. Lemm 7.1. Model [NDSR] hs 3 design solutions tht stisfy ineulity (3.16 t eulity. A9

10 Any fesile NDSR solution tht stisfies (3.16 t eulity must set either (1/2 or (/2 design vriles in A F to one. If fesile solution selects design vriles from A F, with < (1/2, then the left hnd side of ineulity (3.16 is t most 2since every rc ppers ectly twice on the left hnd side, nd the right hnd side is t lest + (1/2 > 2 Similrly, if fesile solution selects design vriles from A F, with > (1/2, then the right hnd side of ineulity (3.16 is + (1/2 which is t lest 1, nd the left hnd side is most since the sic OR ineulities, z 1 K F imply tht every commodity in K F cn flow over t most one of the two corresponding rcs in (3.16. With these restrictions on the numer of design vriles chosen in ny fesile solution, there re ectly 3 possile comintions. We will descrie three such solutions for ech, = 1, 2,, nd show tht they ll stisfy (3.16 t eulity. The first solution corresponding to sets the design vriles for rcs, +2,, + 1 to one, ll other design vriles in A F to zero, nd ll design vriles in A\ A F to one. We then route commodity on fesile pth contining rc ut not on rc +1 ; the pth my contin ny of the other rcs in A F for which we hve set the design vrile to one, or ny rc in A\ A F. Such solution eists ecuse we cnnot fi the flow of ny commodity on n rc to zero, nd ecuse the sic OR ineulity 1 holds for ech commodity in ineulity ( Similrly, we route commodities +1 nd +2 on rc +2 ut not on rcs +1 nd +3, respectively. We continue to ssign the remining commodities in K F to rcs in A F in similr fshion, ending with commodities + 2 nd + 1 flowing on rc + 1, ut not on + 2 nd + 1, respectively. Finlly, while commodities in K\ K F cn flow on ny of the chosen design rcs in this solution, we pic pth for ech in K\ K F tht does not use ny of the rcs in A F. By ssumption, such pth eists for ech in K\ K F. The solution we hve constructed stisfies ineulity (3.16 t eulity. We refer to the design solution outlined ove s the OddZ1( solution since it is one of the two solutions tht set n odd numer of design vriles in A F to one nd stisfy ineulity (3.16 t eulity. The second design solution, OddZ2( is the sme s solution OddZ1( ecept tht it sets the design vrile vlue t rc +1 to one (insted of rc nd routes commodity on this rc. The third design solution corresponding to sets the design vriles for rcs +1, +3,, + 2 to one, ll other design vriles in A F to zero, nd ll design vriles in A\ A F to one. Commodities nd +1 flow on rc +1 (ut not, respectively, on nd +2. These commodities cn flow on ny of the other rcs of A F for which we hve set the design vrile to one. We continue in this fshion until we ssign commodities + 3 nd + 2 to flow on rc + 2, nd commodity + 2 (ut not respectively on + 3 nd + 1. Commodity + 1 does not flow on + 1 or + (the two rcs for which it ppers in ineulity (3.16, ut it cn flow on the other rcs of A F s well s rcs in A\ A F. Finlly, while commodities in K\ K F cn flow on ny of the chosen design rcs in this solution, we pic pth for ech in K\ K F tht does not use ny of the chosen rcs in A F. By ssumption, such pth eists for ech in K\ K F. This solution A10

11 lso stisfies ineulity (3.16 t eulity. We refer to this solution s the EvenZ( solution since it sets n even numer of design vriles in A F to one. Tle A.1 summrizes these solutions, focusing on just the vriles tht pper in ineulity (3.16. Arc inde 1 Tle A.1. Solutions tht stisfy Ineulity (3.16 t eulity Solution OddZ1( Solution OddZ2( Solution EvenZ( Design Vriles Flow Vriles Design Vriles Flow Vriles Design Vriles Flow Vriles z z 1 1 z z 1 1 z z z z 1 1 z Lemm 7.2. The coefficients 0, A\ A. F Proof. Pic ny of the 3 fesile solutions in Lemm 7.1, nd denote it s solution ( 0, z 0. Modify ( 0, z 0 to get ( 1, z 1 y incresing the design vrile vlue y one for rc A\A F nd eeping ll other solution vlues the sme, tht is, set ( z' ( z' 1, ( z ( z for ll rcs A\{'}, nd 1 = 0. Since ( 1, z 1 lso stisfies ineulity (3.16 t eulity, ' 0. Since rc ws chosen ritrrily, 0, A\ A. F Lemm 7.3. Let K \ KF. The coefficients A must e such tht ll fesile s -to-t pths p(s, t tht stisfy (3.16 t eulity hve the sme length using s the rc lengths, i.e., is constnt, sy. p( s, t Proof. Consider solution ( 0, z 0. Suppose commodity K \ KF flows on pth p( s, t in ( 0, z 0. Note tht y construction, p ( s, t does not use ny rcs from A F. Now, otin solution ( 2, z 2 y rerouting on some other fesile s -to-t pth, p ( s, t (which my or my not intersect with A F tht is permitted y the design vriles z 0. Since the only difference etween ( 0, z 0 nd ( 2, z 2 is the flow pth of commodity, we get define s. which we p( s, t p( s, t Net, pic ny other solution ( 3, z 3 defined in Lemm 7.1. If the flow pth p ( s, t for commodity K \ KF is different from pth p ( s, t, then construct ( 4, z 4 from ( 3, z 3 y 1 A11

12 rerouting on pth p ( s, t. Since oth ( 3, z 3 nd ( 4, z 4 stisfy (3.16 t eulity, we get. Now, otin solution ( 5, z 5 y rerouting on some other fesile p( s, t p( s, t s -to-t pth (which my or my not intersect with A F tht is defined y the sme design vriles z 3. The sum of the coefficients on oth these pths must e the sme nd eul to. Since the choice of the solution ( 3, z 3 (from the solutions defined in Lemm 3.1 ws ritrry, we hve estlished Lemm 7.3. We oserve tht there my e other fesile solutions of model [NDSR] which route commodity K \ KF on rcs of A F tht do not correspond to ny of the 3 solutions in Lemm 7.1. But these solutions will not stisfy (3.16 t eulity nd hence will not elong to L. Lemm 7.4. Ineulity (A.7.1 is euivlent to z. (A.7.2 KF A AF Proof. Since 0, A (from Lemm 7.2, z 0. By Lemm 7.3, A. So, setting ˆ gives the desired result. K\ KF A K\ KF K\ KF Lemm 7.5. For commodity KF, let p ( s, t ( p ( s, t denote ny fesile pth tht goes through rc ( +1 nd uses design solution OddZ1( (OddZ2(, nd let pth p ( s, t denote ny fesile pth tht either does not intersect A F, or uses only rcs from +2, +4,, + 1. Then. (A t p ( s, t p( s, t p( s, t Proof. Let ( 6, z 6 denote solution OddZ1(. Let pˆ( s, t denote the pth for commodity in solution ( 6, z 6. Let P ( denote the set of fesile pths for commodity tht ll use rc nd stisfy the design solution OddZ1(. (The crdinlity of P ( cn e greter thn one. Then, is constnt since ll pths in ( p, pp ( pˆ ( s, t (3.16 t eulity. P stisfy Net, consider pth p ( s, t tht either does not intersect A F, or uses only rcs from +2, +4,, + 1. (If commodity uses ny other rcs of A F, then (3.16 will not e stisfied t eulity. Reroute commodity from pˆ( s, t to pth p ( s, t to otin ( 7, z 7. (Note tht the design solution corresponding to this solution is EvenZ(+1. Since oth ( 6, z 6 nd ( 7, z 7 stisfy (3.16 t eulity, they oth elong to L. Since ineulity (3.16 holds t eulity for ll these solutions, the first eulity (A.7.3 follows. Now, reroute commodity in solution ( 7, z 7 to ny fesile pth for commodity tht goes A12

13 through rc +1. Note tht the solution uses the design solution in OddZ2( nd stisfies (3.16 t eulity; this leds to second eulity in (A.7.3. Lemm 7.6. Let commodity KF nd ps (, i (, or 1, denote ny fesile supth from node s to node i ( tht elongs to solution tht stisfies (3.16 t eulity. Then some constnt, sy, i (. p( s, i( Proof. If there is only one fesile supth from s to node i (, then the Lemm immeditely holds. Suppose the Lemm is not true. Thus, there eist two pths p( s, i( nd p( s, i( from s to node i (, such tht. By the overlpping pth p( s, i( p( s, i( condition, there eists supth ri ((, t from node ( p( s, i( nd p( s, i(. Now, the condition p( s, i( r( i(, t p( s, i( r( i(, t i to t tht is comptile with oth implies tht p( s, i( p( s, i(. But this contrdicts the oservtion following Lemm 7.5. Lemm 7.7. Let commodity KF. Let p(( j, t, or, denote ny fesile 1 supth from node j( to node t. Then some constnt, sy, j( t i( p( j(, t. Proof. Follows since Lemm 7.5 nd 7.7 show tht i( t. Lemm 7.8.,. p( j(, t Proof. For ny, = 1, 2,,, consider solution OddZ1(+1 with commodities nd + 1 flowing on rc. Net, reroute the flow of commodity from rc to rc +1 (eeping the design solution OddZ1(+1. Let p( s, t denote the originl pth for commodity nd p ( s, t the new pth. Both these solutions stisfy (3.16 t eulity, nd since the only difference etween them is the flow of commodity, Lemm 7.5, we get 1 which implies the Lemm. Lemm 7.9. t 1 KF A KF.. Thus, using p( s, t p( s, t Proof. Given fesile solution tht stisfies (3.16 t eulity, consider the flow for rc, or 1. Define ( ( ( s the flow on rc tht lter flows through rc 1 A13

14 ( 1. Also ( 0 ( is the flow on rc tht does not go through rcs or 1. Therefore, ( ( ( 0. 1 For ech KF, we get, ( 0 ( ( ( A A\{, 1} 0 ( ( ( A\{, 1} A\{, 1} A\{, 1} 0 ( A\{, 1} A\{, 1} A\{, 1} ( ( ( t 0 i( t i( i( 1 t i( t 0 ( t t 1 1 t. 1 In this development, the first eulity follows rerrngement of terms, the second eulity follows from the definition of ( nd (, the third eulity follows from Lemms 7.6, 1 7.7, nd 7.8, the fourth eulity follows from simplifying terms nd the lst eulity follows from the oservtion tht ( Thus, t 1 KF A KF Now, sustituting the vlue of e from Lemm 7.9 in ineulity (A.7.2 implies KF A z 1 t. (A.7.4 KF AF KF Since (3.16 is vlid for Conv[NDSR], nd the design vriles re unounded, must e negtive. Setting ( 1/2, nd dividing (A.7.4 y, we get ineulity KF (3.16. Hence, we hve estlished Proposition 7. t Proposition 8: For ny collection of Gen-Cut nd Gen-IF reltionships, if is even for ll rcs A nd l is positive odd integer, then the Gen Cut-IF ineulity Icut 2 2. (3.17 z l I '' IF A A I Cut is vlid nd tightens formultion [NDSR]. Proof of Proposition 8. By definition of Gen-Cut reltionship, we now tht for ech A14

15 I Cut, z l ' A. Moreover, for ech IIF, '' ' the forcing constrints z hold for ech vrile included in the collection of reltionships. Adding these three sets of constrints Gen-Cut, Gen-IF nd forcing results in the ggregte ineulity 2 l z l /2 z /2 The untity l / 2 is frctionl, ut ICut or I '' IF A ICut A I '' IF A is integer. Hence, we cn round up l / 2 to l /2. ICut ICut A. We lso now tht A. I '' IF A ICut A is integer. Since is even, the right hnd side Figure 4 in the tet of the pper provides n emple to demonstrte tht the Gen Cut-IF ineulity cn strictly tighten formultion [NDSR]. Proposition 9: For ny collection of Gen-IF nd Gen-OR or Gen-Cut reltionships, with corresponding Gen OR-IF ineulity or Gen Cut-IF ineulity g hz, if > 1 for t lest one rc A' nd commodity K', the following ineulity is vlid nd tighter: g 2 hz z. (3.19 A' K' A' Proof of Proposition 9. We first prove the result for the Gen OR-IF ineulity. We re given collection { R1, R2,..., R Q } of Q Gen-OR nd Gen-IF ineulities, where the th reltionship ' ' " R is either Gen-OR reltionship OR(, A, u or Gen-IF reltionship IF(, A, A. Let I OR nd I IF respectively denote the susets of indices corresponding to Gen-OR nd Gen-IF reltionships in the collection. For this collection, the originl Gen OR-IF ineulity (3.13, corresponding to g hz, is s follows: z 2 u 2. (3.13 I ' ' OR A I IF A A I OR Let A' nd K' respectively denote the susets of rcs nd commodities tht re involved in the Q reltionships of collection, nd let denote the numer of reltionships of tht involve rc nd commodity (i.e., numer of reltionships R such tht = nd rc elongs to " or A. Let 1, 2, nd 2 respectively denote the numer of reltionships tht re OR reltionships, IF reltionships with elonging to the IF set, nd IF reltionships with elonging to the THEN set. Tht is: ' ' " 1 A { }, 2 A { }, nd 3 A { }. (A.8.1 IOR : IIF : IIF : ' A Note tht 1 2 2, nd (A.8.2 A15

16 the prmeters in constrint (3.13 re relted to the vlues vi the following eution: for ll A'. (A.8.3 K' The coefficients g nd h in the condensed version g hz re relted to the vlues s follows: g 1 2 nd h /2 for ll A', K'. (A.8.4 K' For ech A' nd K', define 2. To develop ineulity (3.25, we egin y dding ll the Gen-OR ineulities ' A u, for I OR, nd Gen-IF ineulities ' " A, for I A IF in the given collection. Collecting together ll the coefficients for ech vrile, for A' nd K', in the ggregte ineulity, we get: ( u. (A.8.5 A' K' IOR Net, for ech A' nd K', we dd ( 2 (> 0 times the forcing constrints to ineulity (A.8.5 to get the new ineulity: ( ( 2 z u, A' K' A' K' IOR which simplifies (using (A.8.2 nd (A.8.3 to: z u. (A.8.6. A' K' A' K' IOR Oserve tht the coefficients of ll the vriles in the left-hnd side of ineulity (A.8.6. re even. Further, the coefficients of the z vriles in the right-hnd side re lso even since is even for ll rcs A'. Hence, dividing ineulity (A.8.6 y two, gives the ineulity: 1 2 ( /2 z u 2, (A.8.7 A' K' A' K' IOR which is the sme s the tighter ineulity (3.19 corresponding to the Gen OR-IF collection (fter we sustitute the epressions in (A.8.4 for the coefficients g nd h. Using similr rguments, we cn show tht ineulity (3.25 lso holds for the Gen Cut-IF collection. z A16

17 APPENDIX B: Seprtion procedures for some ineulity clsses This Appendi summrizes the seprtion procedures for the Incomptile Arc nd Node Cliue ineulities, Incomptile r-arc ineulities, Lifted Turn ineulities nd Stem ineulities. The seprtion procedures for other ineulities re discussed in the min tet of the pper. Let { } nd z { z } denote the vlues of the routing nd design vriles in the optiml solution to the LP reltion of the current NDSR model t ny itertion. We see one or more ineulities in ech clss tht cn eliminte this solution. We dd to the current model ll such ineulities tht re not dominted y other ineulities generted in this itertion. B.1 Incomptile Arc Cliue (IAC nd Incomptile Node Cliue (INC ineulities: For ech commodity, we first consider ll cliues consisting of t most incomptile rcs (where is smll numer, e.g., two or three. For ech such strting cliue, we grow the cliue y seuentilly dding other rcs, in order of non-decresing order of, tht re incomptile with ech of the eisting rcs in the current cliue. This procedure, when it termintes, yields miml cliue. If the current LP vlues for the rcs in this cliue violte the IAC ineulity, we dd this ineulity to the model. We use similr procedure to identify nd dd violted node cliue ineulities (in this cse, we dd nodes to the cliue in non-decresing order of their degree in the Node Incomptiility grph for commodity. B.2 Incomptile r-arc ineulities: For ech commodity, the procedure first genertes vrious susets of r rcs tht re r-rc incomptile for some metric. In our implementtion, we consider r = 3 (recll tht r = 2 corresponds to the IAC ineulities. The susets re chosen so tht they do not contin ny mutully incomptile rcs (otherwise, the corresponding r-arc ineulity for the suset will not e effective. For every suset A', we consider ech of the remining rcs in non-decresing order of their current LP vlue, nd dd rc to A' if it preserves the r-rc incomptiility property of the suset. In this cse, the corresponding flow vrile ppers in the left-hnd side of (3.6 with coefficient of r or one depending on whether or not the new rc is incomptile with ll previously chosen rcs of A'. B.3 Lifted Turn ineulities: For ech commodity nd every rc A, we first determine its incomptile downstrem neighors IA tht hve positive flow in the current LP solution. We then identify rcs tht re hed-coincident with nd re lso incomptile with the sme downstrem neighors IA in order to lift the ineulity. If the LP solution violtes the corresponding Inound Turn ineulity, we dd it to the model. A similr procedure genertes Outound Turn ineulities. B.4 Stem ineulities: To identify violted Stem ineulities, insted of deleting rcs from G (s we did for the SFF ineulity, we grow the sugrph G', sed on the LP vlues. For commodity nd ny rc of G tht is not incident to the destintion, we initilly include only node j( in the sugrph G' nd formulte the Inound Stem ineulity for this sugrph (with the flow vriles on ll downstrem neighors tht re comptile with rc in the right-hnd side. We then ttempt to itertively etend G' so s to reduce the right-hnd side vlue for the A17

18 current LP solution. For every outlet (tht is G'-comptile with rc in non-incresing order of current flow, we determine whether including in G' nd replcing the flow on y the flows on its G'-comptile downstrem neighors in the right-hnd side of the current Stem ineulity will reduce the right-hnd side vlue for the current LP solution. If the right hnd side is lower, we etend G' y dding, nd repet the procedure with other outlets of G'. The procedure stops when no further improvement is possile. If the finl ineulity is violted y the current LP solution, we dd it to the model. A similr procedure identifies nd dds violted Outound Stem ineulities. A18