1 Introducton Effcent and speedy recovery of electrc power networks followng a major outage, caused by a dsaster such as extreme weather or equpment f

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1 Effcent Recovery from Power Outage (Extended Summary) Sudpto Guha Λ Anna Moss y Joseph (Seff) Naor z Baruch Scheber x Abstract We study problems that are motvated by the real-lfe problem of effcent recovery from a wde scale electrc power outage caused by a major dsaster such as a hurrcane or an equpment falure. In most of these cases an optmzed schedulng of the workforce s requred snce the work crews on hand are not enough for mmedate recovery of the whole network. We model two varants of ths problem: the budgeted problem, and the mnmum weghted latency problem. We consder the problems for the general case as well as for two specal cases: trees, and bpartte networks. All but one of the problems (the budgeted tree problem) are NP-Hard and the algorthms gven for them are approxmaton algorthms. For the budgeted tree problem we gve an optmal soluton. Interestngly, the budgeted problem for bpartte networks s exactly the budgeted maxmum set cover problem, for whch we gve the best rato approxmaton algorthm. Λ Computer Scence Department, Stanford Unversty, Stanford, CA E-mal: sudpto@cs.stanford.edu. Supported by IBM Cooperatve Fellowshp, an ARO MURI Grant DAAH and NSF Award CCR , wth matchng funds from IBM, Schlumberger Foundaton, Shell Foundaton, and Xerox Corporaton. Part of ths work was done whle ths author vsted IBM T.J. Watson Research Center. y Computer Scence Department, Technon, Hafa 32000, Israel. E-mal:annaru@cs.technon.ac.l. z Bell Laboratores, Lucent Technologes, 600 Mountan Ave., Murray Hll, NJ On leave from the Computer Scence Department, Technon, Hafa 32000, Israel. E-mal: naor@research.bell-labs.com. Part of ths work was done whle ths author vsted IBM T.J. Watson Research Center. x IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heghts, NY E-mal: sbar@watson.bm.com.

2 1 Introducton Effcent and speedy recovery of electrc power networks followng a major outage, caused by a dsaster such as extreme weather or equpment falure, s one of the man challenges facng electrc power utlty companes. These companes are tacklng the problem n two levels. In the plannng level, they are tryng to desgn more relable and robust networks, and n the operatonal level, they are attemptng to manage the recovery optmally, n partcular to manage ther mantenance workforce n an effcent way. In ths paper we are motvated by the operatonal level problem. Most utlty companes use a Trouble Call Management System (TCMS), whch ncludes, along wth other components, an assgnment engne responsble for assgnng the trouble calls to the avalable work teams at any gven tme. One of the problems that needs to be resolved by ths assgnment engne s the prortzaton of the trouble calls, snce n most cases the avalable workforce does not suffce for mmedate recovery of the whole network. For example, a trouble call that nvolves a hosptal or any other emergency servce should receve hgher prorty than a call that nvolves a small number of resdental customers. State of the art TCMS products such as the ones offered by GE Harrs Energy Control Systems [LT96] and ASCADA [As] contan an assgnment engne. However, most utlty companes stll resort to manual assgnment and schedulng n cases of major outage. 1.1 The model and problems We model the nput by an undrected graph G(V;E), where the set of vertces V s parttoned nto three dsjont sets of vertces: generator vertces S, relay vertces R, and customer vertces C. As wll be clear later, we may assume wthout loss of generalty that the set S of generator vertces s a sngleton set. Denote the generator vertex by s. Each relay vertex r 2 R s assocated wth a cost, cost(r), and each customer vertex c 2 C has a weght (or proft), w(c). We assume that the graph G s connected. Note that the paths from the generator vertex s to a customer may consst of both relay vertces and other customer vertces. An outage occurs when at some pont of tme a subset F R of the relay vertces fal". The subgraph left after the removal of these vertces (.e., the graph nduced by V F )may contan several customer vertces that are dsconnected from the generator s. Our goal s to reconnect these customer vertces by recoverng some relay nodes from F. We consder two varants of ths problem. The frst varant s a one-tme" effcent assgnment of the workforce at hand, or maxmzng the total mportance of the customers recovered n a sngle day. That s, for a gven budget", the goal s to reconnect to the generator s a maxmum weght subset of the customers. The second varant requres a recovery of the whole network,.e., connectng all dsconnected customers, whch may be a lengthy process that takes several days. The goal s to mnmze the weghted latency suffered by the customers, where ths latency s scaled by the relatve mportance of each 1

3 customer. We consder these problems for general networks as well as for tree networks and for bpartte networks. These problems model the power outage recovery problem n a rather straghtforward way. The falures causng the power outage are modeled by the faled relay vertces. The cost of each such relay vertex represents the estmated number of reparmen (or other resources) requred to recover the correspondng falure n a day (or any other tme unt). Each customer vertex corresponds to a customer of the utlty company, and the weght of ths vertex corresponds to the mportance" of ths customer. Fnally, the budget corresponds to the total number of reparmen avalable n a day (or any other tme unt). Our modelng of the power outage recovery problem captures the maxmum coverage problem as a specal case. In ths problem, a set system wth weghts on the elements s gven, and the goal s to cover a maxmum weght subset of the elements usng, say, k sets. Ths problem has been studed extensvely n the lterature snce t models many applcatons arsng n crcut layout, job schedulng, faclty locaton, and other areas. (See [Ho97, pp ] and the references theren for examples of applcatons.) In fact, the budgeted problem for bpartte networks can be vewed as a budgeted maxmum coverage problem that ntroduces a more flexble model for the applcatons mentoned above. Consder, for example, the applacton of locatng k dentcal facltes so that the market share s maxmzed, ntroduced n [MZH83]. Namely, we are gven clents wth assocated profts, stuated n known locatons. A clent uses a faclty f the faclty s wthn a specfed dstance from the clent. The goal s to locate k facltes so that the total proft of the clents served by these facltes s maxmzed. A varant of ths aplcaton, consdered n [BLF92, BBL95], s known as optmal locaton of dscretonary servce facltes. In ths varant facltes gan profts assocated wth travel paths of customers; a faclty covers a path f t s located n one of the nodes on the path, or at some vertex close" to the path. Clearly, the above applcatons can be modeled by the unt cost maxmum coverage problem. However, n practce, the cost of constructng a faclty may depend on certan factors assocated wth the locaton of the faclty. For example, each canddate ste for constructng a faclty s assocated wth some cost, and one s assgned a lmted budget for constructng the facltes. Ths generalzaton of the problem of locatng facltes to maxmze market share s also dscussed n [MZH83]. The budgeted maxmum coverage problem s a model that allows us to handle ths type of applcatons. 1.2 Results All the problems that we consder, except for the budgeted tree problem, are NP-Hard. Therefore, we focus on polynomal tme approxmaton algorthms. Below, we detal the results acheved. 2

4 General networks: For general networks we present an O(log 2 n)-approxmaton algorthm for the budgeted reconnecton problem, where n s the number of customers (or, n realty, customer clusters) that are dsconnected from the generator s. Our result s a pseudo-approxmaton n the sense that the budget used by our algorthm can be at most double the budget used by the optmal soluton to whch our slouton s compared. For the mnmum weghted latency problem, we frst present a smple O(log 2 n)-approxmaton algorthm. Next, we extend ths algorthm to acheve an O(log n log log n) approxmaton factor. Our solutons nvolve solvng several nstances of the mnmum node weghted Stener tree problem. Ths s an NP-hard problem, and the frst approxmaton was gven by Klen and Rav [KR95]. To bound the performance of our algorthms, we need to bound the cost of the computed node weghted Stener tree as a functon of an optmal fractonal soluton whch s derved from a lnear relaxaton of the problem. However, the algorthm of [KR95] bounds the cost of the approxmate Stener tree as a functon of the optmal ntegral Stener tree. We gve a tghter analyss of the algorthm of Klen and Rav and bound ts performance as a functon of the optmal fractonal soluton. Ths proof also suggests an alternatve sphere growng" algorthm for the node weghted Stener tree problem. Tree networks: Here, the graph G s a tree where the generator s s the root of the tree. We present a polynomal tme optmal algorthm for the budgeted reconnecton problem n tree networks n whch the budget s polynomally bounded, and a strongly polynomal approxmaton scheme for arbtrary budgets. Ths s a bottom-up dynamc programmng algorthm. We apply these algorthms to derve a logarthmc factor approxmaton algorthm for the mnmum weghted latency problem. The latter problem can be shown to be strongly NP-Hard. Bpartte networks: Here, the graph G s a three layered graph: the frst layer s the generator s, the second layer s the set R of relayvertces whchwe assume s an ndependent set, and the thrd layer s the set C of customers. The budgeted problem n bpartte networks s a budgeted verson of the maxmum coverage problem. Ths s an NP-hard problem, and for the specal case of ths problem, where each set has unt cost, a (1 1)- e approxmaton algorthm was known, e.g., [Ho97, pp ]. Yet, no approxmaton algorthms were known for the budgeted, or general cost verson of the problem. We gve an (1 1 )-approxmaton algorthm for the budgeted maxmum coverage problem, and e argue that ths approxmaton factor s the best possble, unless NP DTIME(n log log n ). Smlar to the tree networks case, we apply ths algorthm to derve a logarthmc factor approxmaton algorthm for the mnmum weghted latency problem for bpartte networks. The rest of the paper s organzed as follows. In the next secton we gve a formal statement of the problems consdered. In Secton 3 we gve the approxmaton algorthms for general networks. In Sectons 4 and 5 we brefly consder tree and bpartte networks. We conclude wth some remarks and open problems. 3

5 2 Problem statement We gve here a formal statement of the problems. The budgeted problem: Gven a budget B, fnd a subset F 0 F, where cost(f 0 )» B, such that the weght of the customer vertces reconnected to s through F 0 s maxmzed. More formally, the weght of the customer vertces that are connected to s n the subgraph nduced by V (F F 0 ), but dsconnected from s n the subgraph nduced by V F, s maxmzed. The mnmum weghted latency problem: F 0 1 ;:::;F0 k F wth the followng propertes: Gven a budget B, fnd dsjont subsets 1. The cost of each F 0,1»» k, does not exceed the budget B. 2. All the customer vertces are connected to the generator vertex n the subgraph nduced by V (F [ k =1 F 0 ). For j = 1;:::;k, let C j be the set of customer vertces that are connected to s n the subgraph nduced by V (F [ j F 0 =1 ), but dsconnected from s n the subgraph nduced by V (F [ j 1 F 0 =1 ). Denote by w(c j) the total weght of the customer vertces n C j. Our goal s to fnd the subsets such that the sum kx j=1 j w(c j ) s mnmzed. Due to the fact that the problem requres an explct schedule of repars, we can assume that the cost of vertces are polynomally bounded. Ths assumpton however wll not be of concern to us n most of the problems dscussed below. It s not dffcult to see that for the above problems, we may ndeed assume wthout loss of generalty, that only a sngle generator vertex exsts, snce we can always coalesce all the generator vertces nto a sngle vertex. It can also be shown that we may assume wthout loss of generalty that the sets R and C are dsjont. To smplfy the presentaton, we make the followng assumptons, all wthout loss of generalty: (1) Ignore all relay vertces of cost more than B; (2) Treat all the customer vertces that reman connected after the removal of the set of faled relay vertces, as well as unfaled relay vertces, as relay vertces of zero cost; (3) Ignore the dstncton between relay vertces and customer vertces. Instead, assume that the weghts of the relay vertces and the costs of the customer vertces are set to zero. The budgeted problem can be vewed as the node verson of the k-mst problem wth edge costs. In the k-mst problem we are gven a graph wth weghts on ts nodes and a specfed node s. The goal s to fnd a maxmum weght tree spannng k nodes rooted at s. In a generalzaton of ths problem edges are assocated wth costs and nstead of the parameter 4

6 k we are gven a budget B. The goal s to fnd a maxmum weght tree rooted at s the cost of whch s bounded by B. The k-mst problem and ts generalzaton are dscussed n [?]. As mentoned earler, the budgeted problem for bpartte networks s dentcal to the budgeted maxmum coverage problem. In ths problem we are gven a collecton S of sets wth assocated costs defned over a doman of weghted elements, and a budget B. The goal s to fnd a subset S 0 Ssuch that the total cost of sets n S 0 does not exceed B, and the total weght of the elements covered by S 0 s maxmzed. Observe that any nstance of the budgeted problem for bpartte networks can be translated to an nstance of the budgeted maxmum coverage problem, as follows. Each relay vertex r 2 F corresponds to a set n S consstng of the elements that correspond to the customer vertces connected to r n G. The budget n both problems s the same. The translaton from an nstance of the budgeted maxmum coverage problem to an nstance of the budgeted problem for bpartte networks s analogous. For our algorthms we need to solve nstances of the node weghted Stener tree problem. In ths problem we are gven an undrected graph wth costs on the edges and on the vertces and a set of termnal vertces. The goal s to connect the termnals usng a mnmum cost Stener tree, where the cost s accumulated on both vertces and edges of the tree. 3 General networks 3.1 The budgeted problem Under our assumpton that there s no dstncton between relay vertces and customer vertces, the budgeted problem can be formulated as follows. Gven a graph G(V;E), and a specfed source s, fnd a connected subgraph T (V 0 ;E 0 )ofg that contans the source s, such that cost(v 0 )» B and w(v 0 ) s maxmzed. Clearly, we may assume that T s a tree. We show a polynomal tme algorthm that computes such a tree T whose weght s Ω(1= log 2 n) tmes the weght of an optmal tree whose cost s bounded by B. We allow our algorthm to use a budget of 2B. In ths sense our result s a pseudo-approxmaton. The cost functon cost( ) nduces a dstance metrc on the graph, where the cost of a path (n the metrc) s equal to the sum of the costs of the vertces belongng to t. Snce the avalable budget s B, no vertex whose dstance from s s more than B, can be connected to s n the optmal soluton. Thus, we can preprocess the graph and omt all vertces whose dstance from s s more than B. The algorthm conssts of three steps. 1. Compute an upper bound P on the weght usng a lnear program. 2. Fnd a good rato" tree T r (U; F );.e., a tree whose weght s a least P=4 and the rato of ts weght to cost s Ω(P=(B log 2 n)). 5

7 3. Fnd a subtree of T r whch has cost between B=2 and 2B that mantans the same rato of weght to cost. Clearly, the weght of ths tree s Ω(1= log 2 n) tmes the optmal weght. Computng an upper bound We formulate a mult-commodty flow problem whose optmal soluton s an upper bound on the weght. The LP relaxaton s as follows. max X X w X w X v w x f u wv» x v for v 2 V f u wv = X w x v c v» B 0» x v» 1 f u vw for v 2 V, v =2 fs; ug In the flow problem we have one commodty per each vertex and the goal s to maxmze the weghted flow from the source s. The budget constrant s mantaned usng the capactes. More formally, transform each edge nto two ant-parallel drected arcs. Let x v denote the capacty of every vertex. Let f u vw denote the flow of commodty u on the drected arc (v; w). The net outgong flow of commodty u from the vertex v has to be less than x v. The source shps x v unts of commodty v to every vertex v. Observe that anyntegral soluton of the lnear program corresponds to a possble outage recovery wthn budget B. Thus, the value of an optmal soluton to the LP relaxaton of the problem yelds an upper bound on the weght of an optmal (ntegral) soluton. Extractng a good rato tree Denote the LP optmum weght by P. Consder the set of vertces recevng non-zero flow from the source s n the optmal soluton. Frst, we can gnore the set of vertces that receve flow less than n2 1. Ths mght reduce the entre weght by at most a factor of (1 1 ). Ths n s because the weght of each of these vertces cannot exceed P, otherwse choosng such a sngle vertex v wth the cheapest path from s to v would have been a feasble soluton (snce the cost of the path does not exceed B) ofweght more than P, volatng optmalty. Note that we are only dsregardng the weght accrued by these vertces. We are not dsregardng the flow through these vertces or the cost assocated wth them. Partton the rest of the vertces recevng postve flow nto dsjont sets: the vertces n the set V receve flow from the source s n the nterval (2 ; 2 +1 ]. Clearly, there are at 6

8 most k = O(log n) sets, and the total weght accrued by the vertces n these sets s at least P (1 1 n ). We dstngush between two cases. Case 1: The total weght accrued by the vertces n V 1 [ [V log log n s at least P=4. In ths case, we fnd a node weghted Stener tree for the set of termnals V 1 [ [V log log n, where the cost of each vertex v s cost(v) and the cost of an edge e s zero f e 2 E and nfnty otherwse. Consder the fractonal relaxaton of the node weghted Stener tree problem. In ths relaxaton we need to assgn capactes to vertces and edges such that the total capacty ofany cut that separates the termnals from the source s at least one. (The cut may nvolve edges and vertces.) The goal s to mnmze the total cost whch s the sum of edge and vertex capactes multpled by ther cost. The soluton for our orgnal LP nduces a feasble soluton to ths relaxaton wth cost at most B log n, snce we need to multply each flow n our orgnal soluton by at most log n to get a soluton to the relaxaton. In the next subsecton we show how to fnd a node weghted Stener tree whose cost s O(log n) tmes the cost of an optmal fractonal soluton. We run ths procedure on our nstance, and we clam that ths tree s a good rato" tree snce t connects vertces whose total weght s at least P=4, and ts cost s O(B log 2 n). Case 2: The total weght accrued by the vertces n V 1 [ [V log log n s less than P=4. In ths case there exsts a set V, where >log log n, that accrues weght at least P=(2 log n). Note that the total weght of the vertces n V s at least 2 1 P=(2 log n) P=4. We fnd a node weghted Stener tree on the set of termnals V. Consder agan the fractonal relaxaton of the node weghted Stener tree problem. The soluton of our orgnal LP nduces a feasble soluton to ths relaxaton wth cost at most B 2, snce we need to multply each flow n our orgnal soluton by at most 2 to get a soluton to the relaxaton. In the next subsecton we show how to fnd a node weghted Stener tree whose cost s O(log n) tmes the cost of the optmal relaxed soluton. We clam that ths tree s a good rato" tree snce t connects vertces whose total weght s at least 2 2 P=(log n) and ts cost s O(B 2 log n). Extractng a good rato tree wth bounded budget In the prevous step we computed a tree whose weght s at least P and ts rato of weght 4 P to cost s fl = O( B log 2 ). We now show how to extract a tree of cost at most 2B and weght n Ω(flB) from ths tree. Orent all the edges n the tree towards s. Pck an arbtrary subtree n the tree rooted at any vertex. If, after deletng ths subtree: (1) the cost of the remanng tree s at least 7

9 B=2; and (2) the rato of the remanng tree s at least fl, then delete ths subtree. We stop when no such subtrees exst anymore. Let the resultng tree be T. It has weght at least flb=2. If the cost of T s at most 2B, then we are done. Suppose that the cost of T s more than 2B. We now look for a subtree T 0 of T wth rato less than fl such that no subtree of T 0 (f such exsts) has rato less than fl. It may be the case that such a subtree T 0 does not exst. We consder ths possblty n Case 1. Otherwse, we dstngush between Cases 2 and 3. Case 1: There s no such subtree T 0 of T. In ths case t must be that all subtrees n the tree T have a rato at least fl. In ths case fnd a subtree T 00 such that ts cost s more than B=2, yet the cost of subtrees rooted at the chldren of the root of T 00 s not. Such avertex must exst snce the weght oft s more than 2B. We have two possbltes dependng on the total cost of the chldren of the root of T 00. (a) If the total cost of the chldren s more than B, then pck subtrees rooted at the chldren of the root of T 00 arbtrarly, tll a set of subtrees havng cost between B=2 and B s accumulated. Snce each subtree has rato at least fl the total weght s at least flb=4. To obtan the desred tree connect these subtrees to s, ncurrng an addtonal cost of at most B. (b) If the total cost of the chldren s less than B (or f the root of T 00 has no chldren), then the cost of the tree T 00 must be at most 2B. Snce t has rato at least fl, ts weght s at least flb=2. Case 2: If there exsts a leaf of T 0 wth rato less than fl t must be the case that the rest of the tree has weght less than B=2. Otherwse, we would have deleted the leaf. Snce the leaf has cost at most B, the total cost of T s clearly less than 2B and weght more than flb=2. Case 3: The only remanng case s that there exsts an nternal vertex of v of T 0 such that the subtree rooted at v has rato less than fl and all the subtrees rooted at the chldren of v have rato at least fl. Snce all these subtrees have good rato, f any one of them has cost between B=2 and B we can connect t to s drectly at an addtonal cost of B to get the desred tree. If any such subtree of T 0 has cost more than B, we can use arguments smlar to Case 1 to get the desred tree. The only remanng case s when all the subtrees of T 0 have cost less than B=2. In ths case we can pck a set of them of cost between B=2 and B, once agan connect them to s payng an addtonal B. Notce that we can always pck such a subset snce the total cost of the subtrees of T 0 s at least B=2. Otherwse we would get that the cost of T s less than 2B because the cost of all but T 0 s at most B=2 (or otherwse we would have removed the tree T 0 ), and the cost of T 0 s at most B + B=2. 8

10 3.2 The node weghted Stener tree In ths subsecton we show how to fnd a node weghted Stener tree whose cost s O(log n) tmes the cost of an optmal soluton of the fractonal relaxaton of ths problem. Klen and Rav [KR95] gave an O(log n) approxmaton algorthm for ths problem. However, ther approxmaton s wth respect to the optmal ntegral soluton. We present a tghter analyss of the KR algorthm provng the approxmaton factor relatve to the fractonal optmal soluton as well. As a matter of fact our analyss also nduces an alternatve formulaton of the KR algorthm as a sphere growng" algorthm n the sense of the prmal-dual method for approxmaton algorthms [Ho97]. As n [KR95], we assume wthout loss of generalty that the cost of the termnals s zero. The KR algorthm works n teratons, and stops when all the termnals are connected. Iteraton starts wth a collecton of ff 1 connected components where each contans at least one termnal. These connected components were computed n prevous teratons. Intally each connected component conssts of a sngle termnal. Defne a spder as a graph wth at most one vertex of degree more than two. The th teraton of the KR algorthm s as follows. It fnds, among all spders that connect at least two components, a spder wth the mnmum rato of cost to number of components t connects. Ths spder s used to decrease the number of components. To prove the logarthmc factor on the approxmaton factor, Klen and Rav prove that ths rato can be no more than the value of an optmal soluton dvded by ff 1. We prove that the same clam but relatve toafractonal optmal soluton. Theorem 1 The rato of the spder found n teraton of the KR algorthm s no more than the value of the optmal fractonal soluton dvded by ff 1. Proof: To prove the theorem we show a dual feasble soluton such that ths rato s no more than the value of the dual feasble soluton dvded by ff 1. The dual problem s a packng problem n whch we need to pack all the cuts that separate the termnals. Snce, n our case, the cost of all edges n E s zero and the cost of the rest of the edges s nfnte, we may consder only vertex cuts. The packng constrant s that the total sum of the dual varables assgned to all cuts contanng a vertex v must be at most cost(v). Suppose that H s the best rato spder found n teraton and let h be the number of components connected by H. Defne the radus of H, denoted r H,tobecost(H)=h. To defne a dual feasble soluton we dentfy all the cuts for whch we are gong to assgn a non-zero dual varable, and then show that ths assgnment s feasble. Consder a connected component C at the start of teraton. For each vertex v outsde C we defne dst(c; v) to be the cost of the cheapest path that connects a vertex from C to v. (The cost of the cheapest path to v does not nclude the cost of ts two endponts.) Let v 1 ;v 2 ;::: be the vertces not n C ordered accordng to dst(c; v`) +cost(v`). Let 9

11 v k be the vertex wth the maxmum ndex such that dst(c; v k )+cost(v k ) <r H. We defne k +1 cuts that separate the component C from the rest of the graph. These are the cuts around" C. For 1» j» k the cut, cut(c; j), s the cut that conssts of all the vertces v` for whch dst(c; v`) < dst(c; v j )+cost(v j )» dst(c; v`)+cost(v`). The value of the dual varable assgned to ths cut s dst(c; v j )+cost(v j ) (dst(c; v j 1 )+cost(v j 1 )), where dst(c; v 0 ) and cost(c; v 0 ) are defned to be zero. The cut, cut(c; k + 1), s the cut that conssts of all the vertces v` for whch dst(c; v`) <r H» dst(c; v`)+cost(v`). The value of the dual varable assgned to ths cut s r H (dst(c; v k )+cost(v k )). Observe that the total sum of the dual varables assgned to all the cuts around C s r H. We need to show that the dual varables assgned to the cuts ndeed consttute a dual feasble soluton. Consder a component C and a vertex v` such that dst(c; v`)+cost(v`) < r H. We frst clam that ths vertex s not part of any cut around any other cut C 0. To prove ths assume that ths s not the case. Then, t must be dst(c 0 ;v`)» r H. But then the spder that connects C and C 0 has a cost strctly less than 2r H, contradctng the assumpton that H has the best rato. Let `0 be the maxmum ndex for whch dst(c; v`0) +cost(v`0) = dst(c; v`). By the defnton of dst(c; v`) such an ndex must exst. Note that the cuts around C that contan v` are the cuts cut(c; j), for `0 <k» `. It s easy to see the total sum of the dual varables assgned to these cuts s exactly cost(v`). All that remans to consder s a vertex v` such that dst(c; v`) < r H» dst(c; v`) + cost(v`). The total sum of the dual varables assgned to all the cuts around C that contan ths vertex s r H dst(c; v`)» cost(v`). Ths vertex may belong to cuts around other components C 0 as well. Ths may happen only f dst(c 0 ;v`) < r H < dst(c 0 ;v`) +cost(v`). To obtan a contradcton assume that v` s around all components C 2 C and that PC2C(r H dst(c; v`)) > cost(v`). Then, there must exst a spder connectng all the components n C whose cost s P C2C dst(c; v`) +cost(v`) < jcjr H, contradctng the assumpton that H has the best rato. Snce the total value of the cuts around each connected component s exactly r H and snce there are ff 1 such components, we conclude that there s a feasble dual soluton wth total value ff 1 r H. The theorem follows snce the cost of the spder s h r H The mnmum weghted latency problem We descrbe how to approxmate the mnmum weghted latency. Frst, we gve a smple O(log 2 n) approxmaton algorthm. Smlar algorthms acheve logarthmc factor approxmaton for tree and bpartte networks. Suppose that we know the optmal weghted latency, denoted by L Λ. In realty we wll try O(log n) values ncreasng by a factor of 1+ffl and consder the best schedule. Ths wll ntroduce an addtonal error factor not exceedng 1 + ffl. The algorthm progresses n teratons. In each teraton t connects a set S of nodes havng total weght W = w(s ). L Let W = w(v ) and W 0 = 0, and defne t +1 = Λ. W P j=1 W j 10

12 In teraton we consder the graph gven by coalescng all the vertces n fsg[ 1 j=1 S j to a sngle vertex, and solve the lnear program relaxaton for the budgeted problem descrbed above for ths graph wth budget 2Bt. The set S s the set of all vertces v that receve a flow of at least 1=4;.e. all vertces v for whch x v 1=4. We connect the vertces of S to the (coalesced) source s by solvng a node weghted Stener tree. Note that the cost of ths tree s O(log n) tmes the optmal value of the lnear program. Snce the total cost of the vertces n S s cbt log n, for some constant c, these vertces can be connected by dentfyng at most 2ct log n subsets of S, each of budget at most B. We contnue n ths manner untl the total weght of the vertces that reman dsconnected s less than L Λ =n 2. We connect the remanng vertces one by one ncurrng an addtonal weghted latency of at most L Λ =n snce at each tme unt we connect at least one vertex. We now bound the contrbuton to the bounded latency from the teratons descrbed above. Lemma 2 W +1 (1=3)(W P j=1 W j). P Proof: The weght W W j=1 j s the weght of the vertces that reman dsconnected after teraton. The optmal weghted latency of these nodes s at most L Λ, thus by Markov's nequalty P L the set of nodes whch complete by tme 2 Λ = 2t +1 s at W P j=1 W j least (1=2)(W W j=1 j). Thus the LP wll have a soluton whch gves at least that much weght. Consder W +1 whch s the total weght of the vertces for whch x v 1=4. Clearly, the maxmum weght that the LP P can acheve sw +1 +1=4(W P W j=1 j W +1 ). Snce ths must be at least 1=2(W W j=1 j), we get W +1 1=3(W W j=1 j). 2 Ths Lemma mples that number of teratons s O(log n). Now we clam the followng theorem. Theorem 3 The weghted latency s O(L Λ log 2 n). Proof: The reconnecton tme ncurred n teraton +1 so(t +1 log n). Thus, the contrbuton of ths tme to the weghted latency s no more than O (W P W j=1 j) t +1 log n. Ths s because ths tme s added to the latency of all vertces that complete after teraton. So the total contrbuton s bounded, up to a constant factor, by X (W X j=1 W j )t +1 log n = X = X (W X j=1 W j ) L Λ L Λ log n = L Λ log 2 n W P j=1 W j log n We acheved the O(log 2 n) approxmaton factor n the above algorthm by lower bound- 11 2

13 ng the total weght of the nodes connected n each teraton. The lower bound s computed from the estmate of the optmal latency L Λ. However, ths bound may not be so accurate. Itutvely, the reason for ths s the fact that we cannot dstngush between two extreme" cases that may result n the same weghted latency. In one case the number of teratons s small and the weght of the nodes connected n each teraton s roughly the same. In the other case the number of teratons s large but most of the weght s connceted n the frst teraton whle n the rest of the teratons only a small amount of weght s connected. To get a better approxmaton factor we try to use two bounds smultaneously: a lower bound on the total weght of the nodes connected n each teratons, as before; and a lower bound on the decrease n the resdual weghted latency after each teraton. We show that usng both bounds we can mprove the approxmaton factor to O(log log n log n). An O(log n log log n) algorthm: To refne our bounds we partton each teraton nto at most m sub-teratons. (The parameter m wll be computed later.) The output of the teraton s agan a set of vertces S that are to be connected. Before we descrbe the algorthm, let us set some notaton. Defne Φ +1 as an upper bound on the contrbuton to the optmal weghted latency of all nodes remanng after teratons. Set Φ 1 = L Λ. The algorthm descrbed below wll proceed, ether untl the total weght of all the remanng nodes to be connected s less than L Λ =n 2, or untl the end of teraton `, for the frst ` for whch Φ`+1 s less than L Λ =n 2. In both cases t s easy to see the latency ncurred by connectng all the remanng vertces (one by one) does not exceed L Λ =n. Defne 0 to be the set of all vertces that reman dsconnected after 1 teratons. Each teraton conssts of m sub-teratons. We descrbe sub-teraton j of teraton. The nput to ths sub-teraton s the set of vertces j 1. In ths sub-teraton we frst solve the (fractonal) budgeted problem relatve to the vertces n j 1 wth budget BffΦ w( j 1 )2 ; j 1 where ff 4 s a parameter to be computed later. To solve the budgeted problem we solve the LP gven n Secton 3.1 on an nstance of the orgnal graph n whch the weghts of the vertces not n j 1 are set to zero and the weghts of the vertces n j 1 are ther orgnal weghts n the graph. Defne j to be the set of nodes recevng a flow of 1 or more n the (fractonal) soluton 2 of the LP. If w( j )» ( ff 2 ff )w( j 1 ), teraton s termnated. Set S = j 1 and solve a node weghted Stener tree to connect the nodes n set S as n the prevous algorthm. In addton, set Φ +1 =Φ (1 2 j+1 ), and 0 +1 = 0 S. Otherwse (.e., f w( j ) > ( ff 2 ff )w( j 1 )) we dstngush between two sub-cases. f 12

14 j = m then agan teraton s termnated. Set S = m 1 and solve a node weghted Stener tree to connect the nodes n set S as n the prevous algorthm. In addton, set Φ +1 =Φ, and 0 +1 = 0 S. The only remanng subcase s j < m and w( j ) > ( ff 2 ff )w( j 1 ). In ths case we proceed to the next sub-teraton (j + 1) of teraton, and start t by solvng the budgeted problem relatve to the vertces n j wth budget BffΦ =(w( j )2j ), To prove the correctness of the algorthm, we frst prove nductvely that Φ s an upper bound on the weghted latency of the nodes n 0. Snce Φ 1 = L Λ, ths s true for the frst teraton. We assume that t s true for teraton. Note that Φ +1 6=Φ only when there s a1<j» m for whch w( j )» ff 2 ff w( j 1 ). Lemma 4 If w( j )» ff 2 ff w( j 1 ), then the contrbuton of j 1 to the weghted latency s at least Φ 2 j+1. Proof: The proof s by contradcton. To obtan a contradcton, suppose that the contrbuton of j 1 s less than Φ 2 j+1. Then, at least a weght of(1 1 ff )w( j 1 ) of the nodes n j 1 s completed wthn tme ffφ t = w( j 1 )2 : j 1 Thus, there exsts a soluton to the LP wth budget Bt (whch s the budget used n subteraton j) that achevesaweght larger than (1 1 ff )w( j 1 ). Now, the nodes n j receved aflow of more than 1 n the LP. Clearly, the maxmum possble weght obtaned by the LP 2 s at most w( j )+1 2 (w( j 1 ) w( j )): Thus, wemust have w( j )+1 2 (w( j 1 ) w( j ) 1 > 1 w( j 1 ) ff Rearrangng the expresson we get w( j ) > ff 2 ff w( j 1 ), a contradcton. 2 Observe that the fact that Φ s an upper bound on the contrbuton of the nodes n 0 to the weghted latency mples that we never termnate after the frst sub-teraton. To see ths, note that snce ff 4we termnate after the frst sub-teraton of teraton only f w( 1 )» 1 2 w( 0 ). However, ths mples that the value of the LP s bounded by 3 4 w( 0 ), whch n turn mples that the contrbuton of the nodes n 0 to the weghted latency s at least 3Φ 4 + 1Φ 2. Now we prove the followng bound on the approxmaton factor. Theorem 5 The above algorthm s an O (log n log log n) approxmaton algorthm for the mnmum weghted latency problem. Proof: For each j>1, we upper bound the contrbutons to the total latency of teraton n whch j was chosen to be connected. Let ths contrbuton be C. The frst observaton 13

15 s that f n teraton the algorthm chooses a certan j, for j>1, to be connected then, w( j ) ff 2 m w( 0 ff ). For each such, the nodes n j receved flow ofvalue at least 1 2 ffbφ w( j 1 )2 j 1 under the budget If we double the flow to every vertex and solve a node weghted Stener tree nstance on the nodes of j, the cost of the Stener tree s at most 2c log n tmes the above budget, where c s some constant (n our adaptaton of [KR95], c = 2). The reconnecton tme of the nodes n j s 1 tmes the cost of the tree. Thus, the contrbuton to the latency s B w( 0 )=B tmes the cost of the tree. Snce w( 0 )» ( ff ff 2 )m w( j ), and w( j )» w( j 1 ), we can upper bound the contrbuton C to the latency by C =2c ffφ log n w( 0) 4cffΦ 2 j 1 w( j 1 )» log n ff 2 j ff 2 m (1) Now, from the above equaton, for those teratons where m was not chosen, we can wrte Φ Φ +1 = Φ C 2 j 4cff m ff 2 log n Notce that for the teratons that m was ndeed chosen, the estmated P upper bound Φ remaned unchanged, that s Φ =Φ +1. Therefore, usng the fact that (Φ Φ +1 )» Φ 1, over the teratons such that m was not chosen, X C» Φ 1 4cff ff m log n: ff 2 We now consder the teratons for whch m was chosen. From equaton (1), the total contrbuton to the latency from these teratons s X C» X ff m 4cffΦ 2 m log n ff 2 By our observaton, at least ( ff 2 ff )m w( 0 ) weght s connected n each teraton. Thus, there are O(( ff ff 2 )m log n) P teratons total. Snce Φ n non-ncreasng, for these teratons where m was chosen, Φ can be bounded by O(Φ 1 ( ff ff 2 )m log n). Usng ths n the above equaton, the contrbuton from these teratons s at most, X C» O ψ ff! ffφ 1 2 ff 2 2m m log 2 n 14

16 Inspectng both the terms, snce ff=(ff 2) 1, the contrbuton to the total latency can be bounded by, O ψ ffφ 1 ff ff 2 2m max(log n; 2 m log 2 n) Settng ff = 2 log log n +2, and m = log log n, the above s O (L Λ log n log log n). Ths proves the theorem. 2! 4 Tree networks We present a polynomal tme algorthm that optmally solves the budgeted problem for polynomally bounded budgets n tree networks. For arbtrary budgets we present two strongly polynomal approxmaton schemes. Smlar to the (frst) algorthm for the mnmum weghted latency problem n general networks, we can apply these algorthms to obtan a logarthmc approxmaton algorthm for the mnmum weghted latency problem n tree networks. Ths latter problem can be shown to be strong NP-Hard. We frst consder polynomally bounded budgets. To obtan a polynomal tme algorthm that solves the problem optmally we use a bottom up dynamc programmng as follows. Consder a vertex v wth chldren v 1 ;v 2 ;:::;v m. Assume that for each chld v and every possble budget B 0» B, we have already computed the maxmum weght ganed from the subtree T rooted at v wth budget B 0, denoted T [v ;B 0 ]. We show how to compute T [v; B 0 ] for all B 0» B. Note that f v s a leaf then T [v; B 0 ]sv f ts cost s bounded by B 0 and ; otherwse. For =1;:::;m and B 0» B, let A[; B 0 ] denote the maxmum weght when the budget B 0 s dstrbuted over T 1 ;:::;T. Observe that f B 0 = 0 then A[1;B 0 ] s gven by the largest zero cost subtree rooted at v 1. A[1;B 0 ] = T [v 1 ;B 0 ] A[ +1;B 0 ] = max 0»j»B 0 fa[; B 0 j]+t [v +1 ;j]g T (v; B 0 ) = w(v)+a[m; B 0 cost(v)] The maxmum weght for the tree T s T [s; B]. The correctness proof follows by a smple nductve argument on the depth plus the degree at the root. The runnng tme of the algorthm s O(B 2 n). In case the budget s not polynomally bounded, we present two strongly polynomal tme approxmaton schemes. The frst scheme acheves the optmal weght by slghtly over 15

17 spendng the budget B. The second scheme acheves slghtly less than the optmal weght usng budget B. The frst scheme s obtaned by scalng the costs. Clearly, no vertex n the tree has cost more than the budget B. Round down the cost of every vertex to a multple of B=n 2. Dvde the rounded costs by B=n 2, and apply the above algorthm to compute the maxmum weght that can be acheved wth the rounded costs. The total costs whch are unaccounted for due to the roundng s at most B. Thus wth budget B(1+ 1 )we get the optmal weght. n n To obtan the polynomal tme approxmaton scheme that fnds the optmal weght wth budget B(1 + ffl), for any ffl, we round down the cost of every vertex to a multple of Bffl=n. The second scheme mantans the budget B and approxmates the weght by a factor of 1 ffl by scalng the weghts. We apply dynamc programmng based on the weghts of the vertces. Frst, assume that the weghts are polynomally bounded. Let T w [v; W] denote the mnmum budget requred to obtan a weght W from the subtree rooted at v. For a vertex v wth chldren v 1 ;v 2 ;:::;v m, let A w [; W ] denote the mnmum budget requred to obtan weght W from subtrees rooted at v 1 ;v 2 ;:::;v. We get A w [1;W] = T w [v 1 ;W] A w [ +1;W] = mn 0»j»W (A w [; W j]+t w [v +1 ;j]) T w [v; W] = cost(v)+a w [m; W w(v)] The optmalty of the above dynamc program can be proven by nducton smlar to the prevous dynamc program. In case the weghts are not polynomally bounded a strongly polynomal algorthm can be obtaned by roundng up every weght to the nearest multple of P ffl=n, where P s the optmal weght. The total weght of the fnal tree s at least P (1 ffl), whle mantanng the budget B. Note that P s not known. However, we can guess" the weght P wthn 1 ffl factor, and check each guess by runnng the above algorthm. The use of the estmate for P rather than P ncreases the approxmaton factor to (1 ffl) 2. 5 Bpartte networks We present a (1 1 ) approxmaton algorthm for the budgeted problem n bpartte networks. Agan, smlar to the (frst) algorthm for the mnmum weghted latency problem n e general networks, we can apply ths algorthm to obtan a logarthmc approxmaton algorthm for the mnmum weghted latency n bpartte networks. The problem of reconnectng a maxmum weght subset of the customers wth respect to a gven budget B n bpartte networks s equvalent to the budgeted maxmum coverage problem, defned as follows. A collecton of sets S = fs 1 ;S 2 ;:::;S m g wth assocated costs fc g m =1 s defned over a doman of elements X = fx 1 ;x 2 ;:::;x n g wth assocated weghts fw g n =1. The goal s to fnd a collecton of sets 16

18 S 0 S, such that the total cost of elements n S 0 does not exceed a gven budget B, and the total weght of elements covered by S 0 s maxmzed. A natural canddate for approxmatng ths problem s the greedy heurstc that pcks at each step a set maxmzng the rato of weght to cost. However, the greedy heurstc has an unbounded approxmaton factor. Consder, for example, two elements, x 1 of weght 1 and x 2 of weght p. Let S = fx g, = 1; 2, let c 1 = 1, c 2 = p +1, and let B = p +1. The optmal soluton contans the set S 2 and has weght p, whle the soluton pcked by the greedy heurstc contans the set S 1 and has weght 1. The approxmaton factor for ths nstance s p, and s therefore unbounded. We modfy the greedy heurstc as follows. We frst fnd a collecton of sets accordng to the greedy heurstc. Ths collecton s the frst canddate for the fnal output. The second canddate s a sngle set S t for whch W t s maxmzed. The modfed algorthm outputs the canddate soluton havng the maxmum weght. It can be shown that ths algorthm acheves an approxmaton factor of 1 2 (1 1 ) for the problem. (Due to space constrants e detals are omtted they can be found n [?].) We mprove the approxmaton factor by usng enumeraton. Let k be some fxed nteger. We consder all subsets of S of cardnalty k whchhave cost at most B, and we complete each such subset to a canddate soluton usng the greedy heurstc. Another set of canddate solutons conssts of all possble k or fewer subsets of S that have cost at most B. The algorthm outputs the canddate soluton havng the greatest weght. Agan, due to space constrants we do not gve here the proof of the followng theorem. (The proof can be found n [?].) Theorem 6 For k 3, the above algorthm acheves an approxmaton factor of (1 1 e ) for the budgeted maxmum coverage problem. 6 Remarks and open problems There are a lot of related open problems that may be consdered. Frst, can the approxmaton factors be mproved, or can t be shown (at least for some of the factors) that they are the best possble approxmaton factors. A bgger challenge s to try to make the model more accurate. Whle we abstract the problem, we gnore many factors that affect the outage recovery. For example, we consder only one resource (workforce) and gnore the rest of the requred resources. Although workforce s ndeed the scarcest resource other resources such as parts have tobetaken nto account. Another factor that s gnored s travel tme of the workforce. Here, we assume that each falure can be fxed wthn the gven tme unt and each reparman can work on any other falure n the next tme unt, gnorng the cases where travel tme from some falure locatons to others may be too long. 17

19 References [As] [BBL95] [BLF92] [Ho97] [KR95] [LT96] [MZH83] ASCADA: Software Solutons for Utltes", more nformaton on ther offerng can be found n prod.htm. O. Berman, D. Bertsmas, and R. C. Larson, Locatng Dscretonary Servce Facltes, II: Maxmzng Market Sze, Mnmzng Inconvenence", Operatons Research, vol. 43(4), pp , 1995 O. Berman, R. C. Larson, N. Fouska, Optmal Locaton of Dscretonary Servce Facltes", Transportaton Scence, vol. 26(3), pp , 1992 D. S. Hochbaum, Approxmaton Algorthms for NP-Hard Problems, PWS Publshng Company, P. Klen and R. Rav, A nearly best-possble approxmaton algorthm for nodeweghted Stener trees", J. of Algorthms, (19): , C. Letter and J. Tracey, Next Generaton trouble call management system heghtens utlty responsveness", Synergy Newsletter, GE Harrs Control Dvson, October N. Megddo, E. Zemel, and S. L. Hakm. The Maxmum Coverage Locaton Problem", SIAM Journal on Algebrac and Dscrete Methods, vol. 4(2), pp