MIT Sloan School of Management

MIT Sloan School of Managmnt Working Papr 4465-03 Rvisd: Jun 2003 Original: April 2001 SENSITIVITY ANALYSIS FOR SHORTEST PATH PROBLEMS AND MAXIMUM CAPACITY PATH PROBLEMS IN UNDIRECTED GRAPHS Ramkumar Ramaswamy, Jams B. Orlin, Nilopal Chakravarty 2003 by Ramkumar Ramaswamy, Jams B. Orlin, Nilopal Chakravarty. All rights rsrvd. Short sctions of txt, not to xcd two paragraphs, may b quotd without xplicit prmission, providd that full crdit including notic is givn to th sourc. This papr also can b downloadd without charg from th Social Scinc Rsarch Ntwork Elctronic Papr Collction: http://ssrn.com/abstract=489804

Snsitivity Analysis for Shortst Path Problms and Maximum Capacity Path Problms in Undirctd Graphs Ramkumar Ramaswamy Infosys Tchnologis Limitd 3 rd Cross, Elctronics City Bangalor 561 229 India ramkumarr@infy.com Jams B. Orlin MIT E40-147 Cambridg, MA 02139 jorlin@mit.du Nilopal Chakravarti Intgratd Dcision Systms Consultancy 1 Jalan Kilang Timor Singapor 159303 nilotpal@idsc.com.sg March 2001 Rvisd, Dcmbr 2003 Abstract. This papr addrsss snsitivity analysis qustions concrning th shortst path problm and th maximum capacity path problm in an undirctd ntwork. For both problms, w dtrmin th maximum and minimum wights that ach dg can hav so that a givn path rmains optimal. For both problms, w show how to dtrmin ths maximum and minimum valus for all dgs in O(m + K log K) tim, whr m is th numbr of dgs in th ntwork, and K is th numbr of dgs on th givn optimal path. Ky words: snsitivity analysis, shortst path problm, bottlnck shortst path, maximum capacity path problm. 1

1. Introduction. Lt G = (N, E) b an undirctd graph with n nods and m dgs, and dsignatd sourc nod s and sink nod t. This papr addrsss snsitivity analysis qustions concrning th shortst s-t path (SP) problm in G and th maximum capacity s-t path (MCP) problm in G. For ach (i, j) E, lt c ij dnot ithr th lngth of (i, j) or th capacity of (i, j) dpnding on whthr w ar solving th shortst path or th maximum capacity path problm. Suppos that P is a shortst s-t path in G. For ach dg E, th lowr SP tolranc of dg is th minimum non-ngativ valu that th lngth of dg can tak (with all othr lngths staying fixd) so that P rmains an optimal path. Similarly, th uppr SP tolranc of dg is th maximum valu in R that th lngth of dg can tak so that P rmains an optimal path. W show that th problm of finding all uppr and lowr tolrancs of dgs in E is computationally quivalnt to th Minimum Cost Intrval Problm which is as follows. For ach i = 1 to r, lt [a i, b i ] dnot an intrval with ndpoints in {1,..., q}, and lt d i b th associatd cost. For ach k = 1 to q, idntify a minimum cost intrval [a i, b i ] containing k. W provid an O(r + q log q) algorithm for th Minimum Cost Intrval Problm, and provid an O(m + P log P ) algorithm for finding all uppr and lowr tolrancs of dgs in E. Rlatd snsitivity analysis problms for tr solutions hav bn considrd initially by Shir and Witzgall (1980). Exampls of subsqunt rsarch in th sam dirction includ an O(mα (m,n)) 1 algorithm for snsitivity analysis of th minimum cost spanning tr problm by Tarjan (1984) and an O(m) algorithm for snsitivity analysis of minimum spanning trs and shortst path trs in planar graphs by Booth and Wstbrook (1994). W rfr th radr to Gal (1995), Grnbrg (1998) and to Gal and Grnbrg (1997) for xtnsiv rfrncs to a varity of snsitivity analysis problms in combinatorial optimization. W furthr considr th snsitivity analysis problm for th maximum capacity path problm. For a ntwork with dg capacitis, th capacity of a path is th minimum capacity of an dg on th path. Lt Q b th maximum capacity s-t path in G with rspct to capacity vctor c. For ach dg E, th lowr (rsp., uppr) MCP tolranc of th dg is th minimum (rsp., maximum) valu in R {-, }. that th capacity of dg can tak so that Q rmains a maximum capacity path. W show that th problm of finding all uppr and lowr tolrancs of dgs in E can b solvd in O(m + Q log Q ) tim. Morovr, th problm of finding all tolrancs is narly computationally quivalnt to th Minimum Cost Intrval Problm. Th most vital SP dg problm is th problm of finding an dg whos dltion from G incrass th lngth of th shortst path from s to t as much as possibl. Th most vital MCP dg problm is th problm of finding an dg whos dltion from G dcrass th maximum capacity of a path from s to t as much as possibl. Th most vital SP dg is th dg with th largst uppr tolranc, and th most vital MCP dg is th dg with th lowst MCP tolranc. Algorithms for th most vital dg problms on shortst paths includ Bar-Noy t al (1995), Malik t al (1989), Vnma t al (1996). Algorithms for finding th most vital dgs in minimum cost spanning trs hav bn dvlopd by Hsu t al (1991), Hsu t al. (1992), Iwano and Kato (1993), and Banrj and Saxna (1997). Th most vital dg problm for minimum cost spanning trs is not dirctly rlatd to th problm of idntifying dg tolrancs, although both ar fundamntal qustions of snsitivity analysis. 1 α(m, n) is th invrs Ackrmann function, and is VERY slowly growing in m and n. 2

A diffrnt approach for dtrmining uppr SP-tolrancs in O(m + n log n) tim for dgs in a shortst path P was indpndntly dvlopd by Hrshbrgr and Suri (2001). Thy incorrctly claimd to hav solvd th snsitivity analysis problm for dirctd graphs, and pointd out thir rror in Hrshbrgr and Suri (2002). This papr xtnds prvious rsults in th unpublishd work of Ramaswamy (1994), who gav O(n 2 ) algorithms for th uppr and lowr tolranc problms considrd hr. In Sctions 2, 3, and 4, w considr snsitivity analysis for th shortst path problm. In Sction 5, w addrss th Minimum Cost Intrval problm; w show its quivalnc to th snsitivity analysis problm for SP and valuat its computational complxity. In Sctions 6, and 7, w considr th maximum capacity path problm. W offr a summary and conclusions in Sction 8. 2. Uppr and Lowr Tolrancs for Combinatorial Optimization Problms. In this sction, w giv procdurs for calculating th uppr and lowr tolrancs of variabls for combinatorial optimization problms with linar costs as wll as for problms with bottlnck (or maximin) costs. W first considr linar objctivs. Combinatorial Optimization Problms with Linar Objctivs. Considr a combinatorial optimization problm X whr costs ar assumd to b non-ngativ. Lt P(c) dnot th instanc of X min (cx: x F), whr F {0, 1} n, and whr c 0. W assum that X is closd undr substitution of linar objctivs. That is, th instanc P(d) = min (dx: x F) is also an instanc of X so long as c and d hav th sam numbr of componnts and d 0. Lt z(c) dnot th optimal objctiv valu for P(c). Suppos that x is an optimal solution to P(c). W ar intrstd in how much th cost cofficints of c can chang with x rmaining optimal. To this nd, for ach indx i and for ach k R, w lt ik, c dnot th vctor drivd from c as follows:, ik, k if j = i; c j = c j, if j i.. For ach componnt i, w dfin th lowr tolranc α i := min { k: k 0 and x is optimal for, P( c ik )} to b th last non-ngativ valu that th cost of componnt i can tak so that x rmains optimal. Similarly, w dfin th uppr tolranc β i to b max { k: x ik, is optimal for P( c )}. If x is ik, optimal for P( c ) for all finit valus of k, thn βi :=. Th nxt thorm charactrizs th uppr and lowr tolrancs of x in trms of optimal solution valus for rlatd problms. Th thorm is asily stablishd, and w omit th proof. x i = 1, thn α i = 0, and i, 0 x = 0, thn β i =, and α = ( ). Thorm 1. Lt x b optimal for P(c), and lt i {1, 2,, n}. If β i, i,0 i = z( c ) c x. If i i cx z c 3

Th Shortst Path Problm W now rturn to th shortst path problm on an undirctd graph G = (N, E). Lt n= N and lt m E. For ach dg E, c dnots th lngth of dg. W assum that c 0 for all dgs. = W prmit costs to b infinit. If S is a subst of dgs, thn c (S) = S c. W lt G k, dnot th k, graph G = (N, E) in which th cost vctor c is rplacd by c. d k, Lt P dnot a shortst path from nod s to nod t in G, and lt d(s, t) = c( P ). In gnral, lt k,,0 ( s, t) b th lngth of th shortst s-t path in G. W not that if P, thn c ( P ) = c( P ) c. If w intrprt Thorm 1 in trms of tolrancs for th shortst path problm, w gt th following corollary. Corollary 1. Lt P b a shortst path in G = (N, E). If P, thn α = 0, and, β = d (,) s t c(p ) + c. If P, thn β = and α, = cp ( ) d 0 ( st, ). By Corollary 1, th lowr tolranc of an dg P is 0, and th uppr tolranc can b calculatd by solving a singl shortst path problm. Also by Corollary 1, th uppr tolranc of an dg P is infinity, and th lowr tolranc can b calculatd by solving a singl shortst path problm. Thus, w could find all of th dg tolrancs by solving at most m shortst path problms. In th nxt sctions, w will show how to find all dg tolrancs in O(m + P log P ) tim, which is narly a factor of m improvmnt in running tim. W will also show that th problm of computing all tolrancs rducs to th Minimum Cost Intrval Problm, as dfind in Sction 1. Combinatorial Optimization Problms with Bottlnck Objctivs. For a givn n-vctor c, and a 0-1 n-vctor x, lt cmin ( x) = min{ ci : xi = 1}. In this subsction, th vctor c dnots a vctor of capacitis, and c min (x) is th capacity of solution x. Considr th following instanc of a bottlnck combinatorial optimization problm Y: max (c min (x): x F), whr F {0, 1} n, and whr c j R {-, } for ach j = 1 to n. W assum Y is closd undr substitution of linar objctivs, and w lt B(c) dnot th instanc max (c min (x): x F). (Hr, w prmit all linar objctivs, with positiv and ngativ cofficints.) Lt v(c) dnot th optimal objctiv valu for B(c). Lt x b an optimal solution to B(c). Analogously to bfor, w lt th lowr tolranc α i and th uppr tolranc β i b dfind as follows: α i = min { k : x, is optimal for B ( c ik )}. β i = max { k : x, is optimal for B( c ik )}. If x, is optimal for B ( c ik ) for all ngativ valus of k, thn αi = -. If x, is optimal for B( c ik ) for all positiv valus of k, thn βi =. 4

Th following thorm is analogous to Thorm 1. It can b provd in a straightforward mannr, and w omit th proof. Thorm 2. Lt x b optimal for B(c), and lt i {1, 2,, n}. If 1. ( i, α i = vc ). If 2. If x is optimal for B ( c i, ), thn βi =. 3. If x is not optimal for B ( c i, i, ), thn βi = c ( min x ) x i = 0, thn th following ar tru: 4. α i =. i, 5. If vc ( ) = c ( x ), thn βi =. min i, 6. If vc ( ) c min ( x), thn β > i = c ( x ) min. x i = 1, thn th following ar tru: Th Maximum Capacity Path Problm W now rturn to th Maximum Capacity Path Problm. W lt from s to t in G. P dnot a maximum capacity path Th following corollary is a translation of Thorm 2 to th MCP problm. Corollary 2. Lt P b a maximum capacity path for G = (N, E). If P, thn th following ar tru:, 1. α = vg ( ) and 2. If P is a maximum capacity path for G,, thn β = ; 3. If P is not a maximum capacity path for G,, thn β is th minimum capacity of an dg of P \. If P, thn th following ar tru: 4. α = ; 5. If P is a maximum capacity path for G,, thn β = ; 6. If P is not a maximum capacity path for G,,thn, β = c ( P ) min. 3. Lowr S-P tolrancs. In this sction, w show how to dtrmin lowr S-P tolrancs for all dgs in O(m) tim. W first giv psudo-cod for th algorithm, and subsquntly stablish its corrctnss and running tim. Algorithm 1. Comput Lowr S-P Tolrancs. bgin for ach P, α := 0; dtrmin th shortst path lngth d(s, i) in G from s to i for all i N; dtrmin th shortst path lngth d(j, t) in G from j to t for all j N; for ach dg = (i, j) P, α = c(p ) - min (c(p ), d(s, i) + d(j, t), d(s, j) + d(i, t)); nd 5

Bfor stablishing th corrctnss of Algorithm 1, w introduc som mor notation. Lt T s dnot a tr of shortst paths from nod s to all othr nods, and lt T t b a tr of shortst paths to nod t. Lt P(s, i) dnot th path in T s from nod s to nod i. Lt P(j, t) dnot th path in T t from nod j to nod t. For any dg (i, j) E, lt W( i, j) = Psi (, ), ( ij, ), Pjt (, ), which is th s-t walk obtaind by concatnating P ( s, i), (i, j), and P( j,t). Thorm 3. Algorithm 1 corrctly computs th lowr S-P tolrancs for an undirctd ntwork G = (N, E), and can b implmntd to run in O(m) tim. Proof. By Corollary 1, it suffics to show that for dgs P, d,0 ( s, t ) = min (c(p ), d(s, i) + d(j, t), d(s, j) + d(i, t)), If thr is a shortst s-t path in G,0 that dos not contain dg, thn d,0 ( s, t ) = c(p ). If thr is a,0 shortst s-t path in G that dos contain dg, thn d,0 ( s, t) = min (d(s, i) + d(j, t), d(s, j) + d(i, t)). Computing d,0 (,) s t for all = (, i j) P rquirs only that w comput d( s, i) for all nods i and that w comput d( j,t) for all nods j. This rquirs only two shortst path computations on undirctd ntworks, which taks O(m) tim using th algorithm by Thorup (1997). 4. Uppr tolrancs and th Minimum Cost Intrval Problm In this sction, w giv an algorithm for computing uppr S-P tolrancs in G with rspct to a minimum lngth path P. W first giv psudo-cod for solving th uppr tolranc problm. W latr show that th bottlnck stp is quivalnt to th Minimum Cost Intrval Problm. Algorithm 2. Comput Uppr S-P Tolrancs. bgin for ach P, β := ; choos ε > 0 so that for any substs S and S of dgs, if c(s) < c(s ) thn c(s) + ε < c(s ); 2 for ach P, lt c = c + ε/n 2 ; for ach P, lt c = c + ε/n; lt T s b a tr of shortst paths from nod s to all othr nods with rspct to costs c ; lt T t b a tr of shortst paths to nod t from all othr nods with rspct to costs c ; for ach i N, lt P(s, i) dnot th path in T s from s to i; for ach j N, lt P(j, t) dnot th path in T t from j to t; for ach dg P, β := c c(p) + min (c(w(i, j): (i, j) E\P, and W(i, j)); nd W will soon stablish th corrctnss of Algorithm 2. Howvr, w first mak a brif commnt on th running tim. It might appar that thr ar four potntial bottlnck oprations. First of all, thr is th calculation of ε in th scond lin. Scond, thr is th calculation of th shortst path trs. Third, 2 Th purpos of ε is to prturb th problm so that shortst path P is uniqu and such that P T s T t 6

thr is th dfinition of W(i, j) for all (i,j) E\P. Fourth, thr is valuating for all in P th following: min{ c( W ( i, j)) : ( i, j) A, and W ( i, j)}. As for th calculation of ε, this can b accomplishd by choosing any positiv ε < 1 if all data ar intgral. If data is prmittd to b irrational, thn on can rprsnt th prturbation of costs implicitly as a scond componnt of th costs, and calculat th shortst path trs using lxicography. Th calculation of th shortst path trs using lxicography is O(m) using th tchniqu of Thorup (1997). W also do not dirctly dtrmin W(i, j). Instad, w show how to dtrmin for all in P min{ c( W(, i j)):(, i j) E\ P, and W(, i j)} by rducing it to th minimum cost intrval problm. This is accomplishd in Thorm 4 blow. Thus, it is th calculation of min{ c( W( i, j)) : ( i, j) E\ P, and W( i, j)} that is th bottlnck stp of Algorithm 2. W bgin proving th corrctnss of Algorithm 2 with th following lmma concrning th spanning trs T s and T t. Lmma 1. Suppos that T s, T t, P(s, j), and P(j, t) ar dfind as in Algorithm 2. Thn for ach dg P, P(s, j) P(j, t). Proof. W first not that for any two paths P and P in G, if c (P) c (P ), thn c(p) c(p ) by our choic of ε. So T s and T t ar shortst path trs with rspct to c. Morovr, P T s T t. Lt P(s, j) = P(s, i 1 ), Q 1, whr i 1 is th last nod of P on th path P(s, j). Lt P(j, t) = Q 2, P(i 2, t), whr i 2 is th first nod of P(j, t) on path P. Suppos P(s, i 1 ) P(i 2, t). In that cas, lt P (s, j) = P(s, i 2 ), Rv(Q 2 ), whr Rv(Q 2 ) is obtaind from Q 2 by visiting th dgs in opposit ordr. Similarly, lt P (j, t) = Rv(Q 1 ), P(i 1, t). Thn P (s, j) is an s-j path, P (j, t) is a j-t path, and c (P (s, j)) + c (P (j, t)) < c (P(s, j)) + c (P(j, t)), contradicting that P(s, j) and P(j, t) ar both shortst paths with rspct to c. Thorm 4. Algorithm 2 corrctly computs th uppr S-P tolrancs for an undirctd graph G = (N, E). Proof. By Corollary 1, for ach P, β =. So, w hncforth considr th cas that P. Also, by Corollary 1, it suffics to show th following: d, (,) s t = min { c( W( i, j)) : ( i, j) E\ P, and W( i, j)}, (1), If thr is no s-t path in G\, thn d (,) s t =, and th minimization in th right hand sid of (1) is ovr th null st, and so (1) is valid. W now considr th cas that P is som shortst s-t path in G\ with rspct to costs c. Sinc th lngth of th shortst s-t path is also th lngth of th shortst s-t walk, it follows that d, (,) s t min { c( W(, i j)):(, i j) E\ P, and W(, i j)} W nxt prov th rvrs inquality. 7

Lt S = { i N : P( s, i)}. Lt i dnot th last nod on P that is also in S, and lt j b th subsqunt nod on P. W know that nods i and j xist bcaus P is an s-t path, s S, and t S. By assumption, i S and j S, and so P(s, j). By Lmma 1, P(j, t). Bcaus P, it follows that (i, j). So, W(i, j). Path P is a concatnation of a path from nod s to nod i, dg (i, j), and, a path from nod j to nod t. Thrfor, c ( P ) = c( P ) d( s, i +c + d( j, t) = c( W( i, j)), complting th proof. ) ij Thorm 4 dos not gnraliz to dirctd graphs. W giv a countrxampl to th gnralization of Thorm 4 to dirctd graphs in Figur 1. It is th failur of Thorm 4 to gnraliz to dirctd graphs that, in our opinion, maks it mor difficult to comput dg tolrancs in dirctd graphs. s 1 1 1 2 1 1 5 3 5 Figur 1. A countrxampl to th dirctd vrsion of Thorm 4, whr = (1, 2). 1 t Also, Thorm 4 would not b tru if dg costs could b 0 and if w did not rquir paths to b shortst paths with rspct to th prturbd costs c. Considr an undirctd vrsion of Figur 1 whr ach dg has a cost of 0. If w do not rquir paths to hav th fwst numbr of dg, w can lt ach of P(s, 1), P(s, 2), P(s, 3), P(1, t), P(2, t), and P(3, t) contain dg = (1, 2). In this cas, th minimization in Thorm 1 would b ovr th mpty st. W now us Thorm 4 to transform th problm of computing uppr tolrancs for dgs in to th Minimum Cost Intrval Problm. W accomplish this in Algorithm 3. Algorithm 3. Transform SP Uppr Tolranc Problm to Min Cost Intrval Problm bgin rlabl dgs so that P = 1, 2,, K ; lt T s, T t, P(s, j), P(j, t) b dfind as in Algorithm 2; d(s, j) := c(p(s, j)) for j = 1 to n; d(j, t) := c(p(j, t)) for j = 1 to n; for ach i N do bgin if P(s, i) P =, thn a(i) := 1; ls choos a(i) so that a(i)-1 is th last dg of P that is also on P(s, i); if P(i, t) P =, thn b(i) := K; ls choos b(i) so that b(i)+1 is th first dg of P that is also on P(i, t); nd for ach dg (i, j) P with a(i) b(j), crat an intrval [a(i), b(j)] with cost c(w(i, j)) = d(s, i) + c ij + d(j, t); nd Givn th shortst path trs from Algorithm 2, w can asily comput a(i) in Algorithm 3 for all nods i in O(n) tim by scanning nods of T s in dpth first sarch ordr. If j is th prdcssor of i in T s, P 8

and if j P, thn a(i) is th indx of th dg following j on P. If j P, thn a(i) = a(j). Similarly, w can comput b(i) for all nods i in O(n) tim by scanning nods of T t in dpth first sarch ordr. b ( j) + 1 Bcaus P T s T t, it follows that all dgs prcding on P(j, t) ar also on P. a ( i) 1 in P(s, i) and all dgs succding Thorm 5. For ach dg (i, j) P with a(i) b(j), crat an intrval [a(i), b(j)] with cost c(w(i, j)) = d(s, i) + c ij + d(j, t) as in Algorithm 3. Thn th minimum cost of an intrval covring k is d k, (,) s t = min { c( W(, i j)):(, i j) E\ P, and W(, i j)}., Proof. By Thorm 3 and Algorithm 2, d k (,) s t = min{ c( W(, i j)):(, i j) E \ P and k W(, i j)}. W will complt th proof by showing that for any dg k, w hav k W(i, j) if and only if k [a( i), b( j)]. If k W(i, j), thn k P(s, i) or k P(j, t) or both. Hnc k < a(i) or k > b(j) or both, and thus k [a(i), b(j)]. Convrsly, if k W(i, j), thn k P(s, i) and k P(j, t). Hnc k > a(i)-1 and k < b(j) + 1, and thus k [a(i), b(j)]. Lt TOL(K, m) dnot th tim to find th uppr and lowr tolrancs for a shortst path problm on an undirctd graph with m dgs and with K dgs in th givn shortst s-t path. Lt INT(q, r) dnot th tim to solv th Minimum Cost Intrval Problm ovr r intrvals with ndpoints in {1,..., q}. Th nxt thorm stats th computational quivalnc of th Minimum Cost Intrval Problm and th problm of finding uppr and lowr tolrancs for a shortst path problm on an undirctd graph. Thorm 6. Suppos that m n. Thn TOL(K, m) = O(INT(K, m)), and INT(q, r) = O(TOL(q, r)). Proof. Thorms 3 and 4 show that computing th tolrancs of dgs not on P rquirs th computation of two shortst path problms, and this taks O(m) tim (Thorup (1997)). Thorms 1 and 5 show that th additional tim to comput tolrancs of dgs on P is O(INT(K, m)). Thrfor TOL(K, m) = O(INT(K, m)). Now suppos that r q, and considr th Minimum Cost Intrval Problm in which th intrvals ar [a(i), b(i)], with cost d(i). Without loss of gnrality, assum that thr is at most on intrval with ndpoints i and j for all i and j. If thr is an intrval [j, j], lt z j dnot its cost. Othrwis, lt z j =. W crat a tolranc problm as follows. W crat a graph G = (N, E), with q + 1 nods, whr th sourc nod is nod 1, and th sink nod is nod q+1. Lt P b th path 1, 2,..., q+1, and suppos that ach dg of P has a cost of 0. Lt i dnot th dg (i, i+1). For ach i = 1 to r with a(i) b(i), thr is an dg r+i = (a(i), b(i)+1) with a cost of d(i). If an intrval i is such that a(i) = b(i), w call it a zro lngth intrval. W do not crat dgs in G corrsponding to zro-lngth intrvals. Lt β j dnot th uppr tolranc of dg j in G. Th minimum cost of an intrval covring j is min( di ( ) : j [ ai ( ), bi ( )]), which w will show is qual to min { βj, z j }. If j [a(i), b(i)] and a(i) b(i) thn j P( 1, a( i)), and P( b( i) + 1, q+ 1), and so W ( a( i), b( i) + 1). Similarly, if j [a(i), b(i)], thn j j ithr j < a(i), and j P( 1, a( i)) or ls j > b(i), and P( b( i) + 1, q+ 1). So, j is containd in a nonzro lngth intrval [a(i), b(i)] if and only if W ( a( i), b( i) + 1). By Thorm 5, th cost of a minimum cost j j 9

nonzro lngth intrval covring j is th uppr tolranc of j in G. This stablishs that INT(q, r) = O(TOL(q+1, q+r)). Howvr, sinc th tolranc problm can b solvd in polynomial tim and sinc q r by assumption, it follows that TOL(q+1, q+r) = O(TOL(q, r)), and accordingly INT(q, r) = O(TOL(q, r)). 5. Solving th Minimum Cost Intrval Problm. Hr w provid an O(r + q log q) algorithm for solving th Minimum Cost Intrval Problm. Shigno and Uno (2002) indpndntly solvd th Minimum Cost Intrval Problm with th sam running tim. Lt F dnot a st of r intrvals. Lt c ij dnot th cost of th intrval [i, j] for ach [i, j] F. Th minimum cost of an intrval covring intgr l is g(l) = min { c ij : i l j}. Lt f k (j) = min { c ij : 1 i k }. Thn g(k) = min { f k (j) : k j q}. Morovr, for ach k and for ach j, f k (j) = min {f k-1 (j), c kj }. Th following algorithm computs th minimum cost of ach intrval. Algorithm 4. Th Minimum Cost Intrval Algorithm bgin for j =1 to q, f(j) = min {c 1j, }; g(1) := min{ f(j): j = 1 to q}; for k = 2 to q do bgin for j = k to q do f(j) = min {f(j), c kj }; g(k) = min{ f(j): j = k to q}; nd nd Th first for loop computs f 1 (j) for ach j. Whn th indx is k, th scond for loop computs f k (j) for ach j. Th corrctnss of th algorithm follows from th fact that minimum cost of an intrval covring intgr k is min {f k (j): j = k to q}. Th algorithm can b implmntd in O(r + q log q) using Frdman and Tarjan s (1984) Fibonacci Hap. At ach itration, w stor th valus of f( ) in th hap. To initializ th hap taks O(q) stps. To updat th hap in th lin f(j) = min {f(j), c kj } taks O(r) stps in total sinc ach intrval causs f to ithr stay th sam or dcras, and th dcras opration taks O(1) stps. Finally, thr ar q oprations of finding th min and q options of dlting f(k) at th nd of itration k. Each of ths oprations taks O(log q) tim using Fibonacci Haps. W stat our conclusions in th nxt thorm. Thorm 7. A Fibonacci Hap implmntation of Algorithm 4 solvs th Minimum Cost Intrval Problm in O(r + q log q) stps. 6. Uppr Tolrancs for th Maximum Capacity Path Problm In this sction, w rturn to th Maximum Capacity Path Problm on a ntwork G = (N, E), whr c dnots th capacity of dg E. 10

Th MCP problm ariss in svral domains. For xampl, on mthod for implmnting th augmnting path algorithm for th maximum flow problm is to snd flow along a path with maximum capacity. This was first analyzd by Edmonds and Karp (1972). Additional dtails can b found in Ahuja t. al., (1993). Th maximum augmnting path problm is mathmatically quivalnt to th bottlnck shortst path problm. An xampl of th bottlnck shortst path problm is th problm of finding a path from s to t such that th minimum rliability of an dg is maximizd. Lt P dnot a maximum capacity path from s to t in G. W nxt us th rsults of Corollary 2 to provid algorithms for computing th uppr tolrancs of all dgs in O(m) tim. Algorithm 5. Comput Uppr MCP tolrancs for dgs not in P bgin H := {a E: c a > c min (P)}; lt G H dnot th graph (N, H); lt S b th nods in th sam connctd componnt as s in G H ; lt T b th nods in th sam connctd componnt as t in G H ; for ach dg = (i, j) E\P do if i S and j T or if i T and j S, thn β = c min (P); ls β = ; nd Algorithm 6. Comput Uppr MCP tolrancs for dgs in P. bgin for ach P such that c c min (P), β = ; lt b any minimum capacity dg in P; lt H := {a E: c a > c min (P\ ) }; lt G H dnot th graph (N, H); lt S b th nods in th sam connctd componnt as s in G H ; lt T b th nods in th sam connctd componnt as t in G H ; for ach = (i, j) P such that c = c min (P) do; bgin if i S and j T or if i T and j S, thn β = c min (P\ ); ls β = ; nd nd Thorm 8. Algorithm 5 corrctly computs th uppr MCP tolrancs for dgs not in P. Algorithm 6 corrctly computs th uppr MCP tolrancs for dgs in P. Each algorithms can b implmntd to run in O(m) tim. Proof. For ach dg = (i, j) P, Algorithm 5 dtrmins whthr thr is an s-t path in G, whos capacity xcds cmin(p ), and sts β corrctly using th rsults of Corollary 2. For ach dg = (i, j) P with c > c min (P ), Algorithm 6 sts β =, as pr Corollary 2. W now considr ach dg = (i, j) P with c = c min (P ). Th capacity of P in G, is cmin(p \), which is th scond smallst capacity of an dg of P. Algorithm 6 dtrmins if P is a maximum capacity path in G, by chcking whthr thr is som path with capacity gratr than cmin(p \), and thn sts sts β corrctly using th rsults of Corollary 2. For Algorithms 5 and 6, dtrmining H and th connctd componnts taks O(m) tim, as dos computing c min (P \), complting th proof. 11

7. Efficint Computation of Lowr Tolrancs for th MCP Problm In this sction, w comput lowr tolrancs for dgs P. Our rsults rly on a clos connction btwn maximum capacity paths and th maximum capacity spanning tr. Th maximum capacity spanning tr (MST) problm for G is to find a spanning tr T for which c is maximum. W will rfr to an optimum solution as a maximum capacity spanning tr. T Th following lmma is asily stablishd, and is wll known. Lmma 2. Lt T dnot a maximum capacity spanning tr of G = (N, E). For any pair of nods i and j, th path in T from i to j is a maximum capacity path from i to j in G. Hncforth, w lt T dnot som maximum capacity spanning tr of G. W lt b(s, j) dnot th capacity of th path in T from s to j, and w lt b(j, t) dnot th capacity of th path in T from j to t. Th tr T can b calculatd in linar tim via a randomizd algorithm as pr th tchniqu of (Kargr t. al., 1995). Th bst dtrministic algorithm for computing th minimum cost spanning tr is du to Gabow t. al., (1984), and th running tim is O(m f ( nm, )), whr f ( nm, ) = ( i) min( i : log n m / n). Th valus b(s, j) and b(j, t) can b computd for all j in an additional O(n) tim. Th notation b was slctd sinc th maximum capacity of a path is th capacity of its "bottlnck" dg. Th following algorithm computs lowr tolrancs in G. W will subsquntly prov its corrctnss and show how its running tim rducs to that of finding a maximum capacity spanning tr and solving an instanc of th Minimum Cost Intrval Problm. Algorithm 7. Comput Lowr MCP Tolrancs bgin for ach P, α = - ; lt T b a maximum capacity spanning tr; for ach dg P \T, α = c min (P ); lt P(i, j) dnot th path in T from nod i to nod j; lt b(i, j) = c min (P(i, j)) dnot th capacity of th path from i to j; for ach P T, α = max{ (min(b(s, i), c ij, b(j, t)) : (i, j), and T + (i, j) is a tr }; nd Thorm 9. Th algorithm Comput Lowr MCP Tolrancs corrctly computs th lowr tolrancs of capacitis in th ntwork G = (N, E), givn a maximum capacity s-t path P. Proof. For ach dg P, α = - by Corollary 2. Also by Corollary 2, for ach dg P, α is th maximum capacity of a path in G,-, which is dnotd as vc (, ). If P and T, thn by Lmma 2, P(s, t) is a maximum capacity path in G, and thus it is also a maximum capacity path in G,-. It follows that α = c min (P ). If P T and if thr is no path in G\ from s to t, thn Algorithm 7 corrctly sts α to - as thr is no arc (i, j) for which T + (i, j) is a tr. 12

Finally w considr dgs P T such that thr is a path in G\ from s to t. For ach dg (i, j) T, lt W(i, j) = P(s, i), (i, j), P(j, t), which is a walk from s to t containing dg (i, j). If W(i, j), contains dg, thn c ( min W (, )), i j =. Othrwis cmin ( W( i, j)) = min( b( s, i), cij, b( j, t)). Sinc W(i, j) contains a path from s to t whos capacity is at last as grat as vc (, ) min{b(s, i), c ij, b(j, t)) : W(i, j) dos not contain dg }. c ( W( i, j)) min, it follows that Morovr, W(i, j) dos not contain dg prcisly whn T + (i, j) is a tr. Thrfor, α max{ (min(b(s, i), c ij, b(j, t)) : T + (i, j) is a tr }. W now prov th rvrs inquality in th cas whn P T and thr is a path from s to t in G\. Lt P b a maximum capacity s-t path in G,-. Lt S dnot th nods in T \ in th sam componnt as s. Thn N\S ar th nods of T \ in th sam componnt as t. Thn P contains som dg (i, j) with i S, and j S. Also, P. So T + (i, j) is a tr. Morovr,, c ( P ) = c ( P ) min( b( s, i), c, b( j, t)), with th lattr inquality following from th fact that th path min min ij in P from s to i has capacity at most b(s, i) from Lmma 2, and th path in P from j to t has capacity at most b(j, t). W conclud that α = c min (P ) max{ (min(b(s, i), c ij, b(j, t)) : T + (i, j) is a tr }, and so th thorm is provd. W now us Thorm 9 to transform th problm of computing tolrancs for dgs in P to th Maximum Cost Intrval Problm. This is quivalnt to th Minimum Cost Intrval Problm xcpt that w want to dtrmin th maximum cost intrval containing k for ach k = 1 to q. Our transformation is vry similar to th on containd in Algorithm 3. Lt T b th maximum capacity spanning tr. For ach nod i N, rcall that P(s, i) is a maximum capacity path from s to i in T, and P(j, t) is a maximum capacity path from j to t in T. W assum that th dgs of E ar ordrd so that P(s, t) P consists of th dgs 1, 2,..., r in th ordr that thy appar on th path P(s, t). For ach nod i, if P has no dg in common with P(s, i), thn lt a(i) = 1. Othrwis, lt dnot th highst indx dg of P(s, t) P that is also on P(s, i). If P a ( i) 1 has no dg in common with P(j, t), thn lt b(j) = r. Othrwis, lt b ( j) + 1 dnot th last indx dg of P(s, t) P that is also on P(j, t). W can comput a(i) and b(i) for all i N in O(n) tim in a similar mannr to th way indics ar computd for Algorithm 3. Lmma 3. Considr th Maximum Cost Intrval Problm, dfind as follows: For ach dg (i, j) P, with a(i) b(j), thr is an intrval [a(i), b(j)] with cost minimum cost of an intrval covring k is c ( W( i, j)) min. Thn th k, vc ( )= max (c min (W(i, j) : k W(i, j) ). Proof. Essntially th sam as th proof of Thorm 4. 13

W lt TOLCAP(K, m) dnot th tim to find tolrancs for th maximum capacity path problm, whr K is th numbr of arcs of th givn path. Lt MST(n, m) b th tim to solv a minimum cost spanning tr problm on n nods and m dgs. 3 Thorm 10. Suppos that m n. Thn TOLCAP(K, m) = O(INT(K, m) + MST(n, m)). Morovr, INT(r, q) = O(TOLCAP(r, q)). Proof. Th proof that TOLCAP(K, m) = O(INT(K, m) + MST(n, m)) rlis on th fact that on can comput tolrancs by solving a maximum cost intrval problm and by finding a maximum cost spanning tr. Th othr dtails ar th sam as in th proof of Thorm 6. Now suppos that r q, and considr th Maximum Cost Intrval Problm in which th intrvals ar [a(i), b(i)], with cost d(i). Th proof rlis on th sam construction as in th proof of Thorm 6, xcpt that hr P is a path 1, 2,..., q + 1, whr ach dg of th path has a capacity of M > d max. 8. Summary and conclusion In this papr w hav considrd snsitivity analysis qustions for th shortst s-t path (SP) and maximum capacity s-t path (MCP) problms and prsntd algorithms for answring ths qustions that ar far suprior to succssiv roptimization. Tabl 1 summarizs our contribution. Tabl 1. Summary of rsults Problm Complxity 1 Minimum [Maximum] cost intrval INT(r, q) = O(r + q log q) Shortst path snsitivity 2 Lowr tolrancs of dgs O(m) 3 Uppr tolrancs of dgs O(INT(n, m)) = O(m + n log n) Maximum capacity path snsitivity 4 Lowr tolrancs of dgs O(MST(n, m) + INT(n, m)) = O(m + n log n) 5 Uppr tolrancs of dgs O(m) Som opn qustions includ th following. (1) What is th computational complxity for th snsitivity analysis qustions addrssd in this papr if on prmits ngativ cost dgs in th SP, but no ngativ cost cycl? (2), What is th computational complxity for th snsitivity analysis qustions addrssd in this papr if considrs dirctd rathr than undirctd graphs? (Som partial rsults hav rcntly bn obtaind by Hrshbrgr t al. (2003)) and (3) Ar thr suprior algorithms for solving th Minimum Cost Intrval Problm? Acknowldgmnts 3 Th maximum capacity spanning tr is mathmatically quivalnt to th minimum cost spanning tr problm, and so has th sam running tim. 14

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