D. Karger. A randomized fully polynomial time approximation scheme for all-terminal network reliability problem. Proceedings of the 27th ACM STOC, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....connectivity. The number of nodes, jV j, is denoted by n. A preliminary version of this paper has appeared in the Proc. of the 28th ACM S.T.O.C. 1996) pp.3. DepartmentofCombinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3L Supported in part by NSERC grant no.OGP013 (NSERC code OGPIN 007) email: jcheriyan dragon.uwaterloo.ca Department of Mathematics and Computer Science, UniversityofDenver, 23 S. Gaylord St. Denver CO 80208. Supported in part by NSF Research Initiation Award grant CCR 9210604. email: ramki cs.du.edu URL: http: www.cs.du.edu ramki ....

....of nodes, jV j, is denoted by n. A preliminary version of this paper has appeared in the Proc. of the 28th ACM S.T.O.C. 1996) pp.3. DepartmentofCombinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3L Supported in part by NSERC grant no.OGP013 (NSERC code OGPIN 007) email: jcheriyan dragon.uwaterloo.ca Department of Mathematics and Computer Science, UniversityofDenver, 23 S. Gaylord St. Denver CO 80208. Supported in part by NSF Research Initiation Award grant CCR 9210604. email: ramki cs.du.edu URL: http: www.cs.du.edu ramki We ....

[Article contains additional citation context not shown here]

D. Karger, "A randomized fully polynomial time approximation scheme for the all terminal network reliability problem," Proc. 27th ACM S.T.O.C.(T[O[C 11--17.

....or the edges of a graph (for a survey, see [21] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the MINIMUM CUT LINEAR ARRANGEMENT problem and has several applications such as VLSI design [2, 3, 35, 33] network relia bility [30], automatic graph drawing [44, 38] and information retrieval [15] Cutwidth has been extensively examined [17, 24, 25, 31, 35, 37, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi fied bandwidth [17, 18, 31, 34, 35] Briefly, ....

D. R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM J. Cornput., 29(2):492 514 (electronic), 1999.

....each of Bc, AB2 r and A B 0 as r . These latter conditions are needed for the analysis of the first two ranges. Range 1: 0 x Aandl k B see[1] E(Ck,x)dx (1 o(1 (3) 0 k=l Range 2: 0 x Aandk B see[1] r4k,x)dx=o( r ) 0 k=B Range 3: x A. We use aresult of Karger [4]. A cut ( u,v) E u ,v of Gis minimal if [ A. Karger proved that the number of minimal cuts is O(n2) We can associate each component of Gp with a cut of G. Thus and 1 r (fr = o = E(Ck, dx O 2 log n r =A k=l =A X We complete the proof by applying Lemma ....

D. R. Karger, A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem, Proceedings of the twenty-seventh annual ACM Symposium on Theory of Computing (1995) 11-17.

...., i.e. for the assignment OE, at least channels must be active in order at least one copy of all the messages to be delivered to the destination. Next, we adopt a time evolving representation of random channel subsets. Similar timeevolving representations of random edge subsets have been used in [1, 9] for proving lower bounds on graph reliability. We introduce non negative time t, and we define a time dependent random subset K(t) K representing the set of active channels by the time t. Initially, K(0) Each channel is given an arrival time chosen independently from the exponential ....

....the time it takes for the next channel arrival is independent of the previous channel arrivals and the time t. Therefore, once we have conditioned on the value of K(t) the time for the next arrival has probability 22 distribution function 1 Gamma e Gamman(t)t . In particular, it is shown in [9] that, since the time of arrival has no impact on which of the channels of K Gamma K(t) is the first to arrive, the time for the next arrival has the right exponential distribution, regardless of the values of K(t) we condition on. As Lomonosov observed [1] we can imagine that K(t) K oe ....

[Article contains additional citation context not shown here]

D.R. Karger, A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem, Proc. of the 27th ACM Symposium on Theory of Computing, 1995, 11--17. Journal submission available from http://theory.lcs.mit.edu/karger/papers.html.

.... the probability that two vertices in a probabilistic graph are connected are NP hard [8] Approximating the probability that a network is connected was shown to be NP hard, but Karger gave a randomized fully polynomial time approximation scheme for the probability that a given network will fail [3, 6]. This was done by enumerating the approximately minimum cuts in the network, and it was proved that the probability that the network will fail can be approximated using this enumeration. The problem of enumerating approximately minimum cuts has been dealt with in [1, 4, 5] Another objective ....

D. R. Karger, "A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem", Proc. of the 27th Annual ACM Symposium on the Theory of Computing, 1995, New York, pp. 11 - 17

....or the edges of a graph (for a survey, see [20] One of the most known problems of this type is the problem to compute the cutwidth of a graph. It is also known as the Minimum Cut Linear Arrangement problem and has several applications such as VLSI design [2, 3, 33, 31] network reliability [28], automatic graph drawing [42, 36] and information retrieval [14] Cutwidth has been extensively examined [16, 22, 23, 29, 33, 35, 49] and it appears to be closely related with other graph parameters like pathwidth, linear width, bandwidth, and modi ed bandwidth [16, 17, 29, 32, 33] Brie y, the ....

D. R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM J. Comput., 29(2):492-514 (electronic), 1999.

....has Pr[OE k ] 1 Gamma f ) Then, we show that OE is at least as reliable as any other assignment OE that fulfills the hypothesis. Wlog. we assume that MF(OE) We adopt a time evolving representation of random channel subsets. Similar representations of random edge subsets are used in [10, 8] for proving lower bounds on graph reliability. We introduce non negative time t and define a time dependent random subset M (t) M representing the set of active channels by the time t. Initially, M(0) Each channel is given an arrival time chosen independently from the exponential ....

....of the exponential distribution, the time it takes for the next channel arrival is independent of the previous arrivals and the time t. Therefore, once we have conditioned on the value of M (t) the time for the next arrival has probability distribution function 1 Gamma e Gammam(t)t (see also [8]) Thus, r (OE ) is exponentially distributed with mean 1=r. If at some time t, MF(OE(t) r and m(t) r, then the time for the next channel arrival, that may cause MF(OE(t) to become r Gamma 1, stochastically dominates an exponentially distributed random variable with mean 1=r. However, it ....

[Article contains additional citation context not shown here]

D.R. Karger. A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem. Proc. of the 27th ACM Symposium on Theory of Computing, pp. 11--17, 1995.

..... Then, we show that OE is at least as reliable as any other assignment OE that fulfills the hypothesis. Initially, we assume that MF(OE) Next, we adopt a time evoling representation of random channel subsets. Similar time evolving representations of random edge subsets have been used in [Lom74, Kar95] for proving lower bounds on graph reliability. We introduce non negative time t and define a time dependent random subset M(t) M representing the set of active channels by the time t. Initially, M(0) Each channel is given an arrival time chosen independently from the exponential ....

....the time it takes for the next channel arrival is independent of the previous channel arrivals and the time t. Therefore, once we have conditioned on the value of M(t) the time for the next arrival has probability distribution function 1 Gamma e Gammam(t)t . In particular, it is shown in [Kar95] that, since the time of arrival has no impact on which of the channels of M Gamma M(t) is the first to arrive, the time for the next arrival has the right exponential distribution, regardless of the values of M(t) we condition on. As Lomonosov observed [Lom74] we can imagine that M(t) M oe ....

[Article contains additional citation context not shown here]

D.R. Karger. A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem. Proc. of the 27th ACM Symposium on Theory of Computing, pp. 11-- 17, 1995. Journal submission available from http://theory.lcs.mit.edu/karger/papers.html.

....Then, we show that OE is at least as reliable as any other assignment OE that fulfills the hypothesis. Initially, we assume that MF(OE) 235 Next, we adopt a time evolving representation of random machine subsets. Similar time evolving representations of random edge subsets have been used in [12, 9] for proving lower bounds on graph reliability. We introduce non negative time t and define a time dependent random subset M(t) M representing the set of active machines by the time t. Initially, M(0) Each machine is given an arrival time chosen independently from the exponential ....

....the time it takes for the next machine arrival is independent of the previous machine arrivals and the time t. Therefore, once we have conditioned on the value of M(t) the time for the next arrival has probability distribution function 1 Gamma e Gammam(t)t . In particular, it is shown in [9] that, since the time of arrival has no impact on which of the machines of M Gamma M(t) is the first to arrive, the time for the next arrival has the right exponential distribution, regardless of the values of M(t) we condition on. As Lomonosov observed [12] we can imagine that M(t) M oe ....

[Article contains additional citation context not shown here]

D.R. Karger (1995), "A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem", Proc. of the 27th ACM Symposium on Theory of Computing, pp. 11--17.

....have a po; G7k timealgo; 0J to coo; itrigo007k6 . Recently,Alo) Frieze and Welsh [1] develo ed fully po;6)9;7k timerando00;7 appro ximatio schemesfo appro ximating the valueo the Tutte po)F0;J70 fo any dense graph G, whenever x, y # 1. This result was extendedto a general graph by Karger [18]. Hence this is especially usefulfo IMAI: INVARIANT POLYNOMIALS OF MATROIDS 331 calculating theappro ximate valueso the Tutte po6FFJ mials which have special meanings such as the number o fo7606G On theoe7= hand, the exactcot7( 0)7 o the Tutte p oG0J67kF still remains a challengingproleng ....

.... there have been pro oo manyappro ximatio algoiokF; such as theBall Pro van bo6( 4] which make useo the shelling po(FJ0J7k o a co(0;07k matro76 Recently, rando7kF9 fully po= G7kF9JF07 appro ximatio schemesfo coes7( J the netwo6 reliability have been develo ed byAlo0 Frieze, Welsh [1] and Karger [18]. Karger and Tai [19] rep implementatiom o thoo algoen 760 andsho w that the netwo6 o mo derate size upto 50to 60 vertices can be analyzedappro ximately by theirmetho ds. Fo r thewho6 netwoJ reliability research, see [9] 12] 15] 16] 32] The BDD basedapproe h can be appliedto this ....

D.R. Karger, "A randomized fully polynomial time approximation scheme for the all terminal network reliability problem, " Proc. 27th Annual ACM Symp. on Theory of Computing, pp.11--17, 1995.

....and the separators of a graph as one of the contributions of this work, and believe that our style of analysis may be amenable to related problems as well. The role of node and edge separators here suggests connections to problems such as approximating the failure probability of a network [7]. It is important to note, however, that the issues are quite di#erent at a technical level. Specifically, we are concerned with choosing a fixed set of nodes D from which we can detect any possible (adversarially chosen) failure of a particular type. Also, we allow here for node failures in ....

....concerned with choosing a fixed set of nodes D from which we can detect any possible (adversarially chosen) failure of a particular type. Also, we allow here for node failures in addition to edge failures; network reliability allowing node failures, on the other hand, is largely an open question [7]. Generalizations. We will actually prove the following strengthening of Theorem 1.1. Suppose that the nodes of our graph are initially partitioned into two classes: end nodes V 0 , and internal nodes V 1 . We are only allowed to place monitoring agents at nodes in V 0 , and we are interested in ....

D. Karger, "A randomized fully polynomial time approximation scheme for the allterminal network reliability problem," Proc. ACM Symp. on Theory of Computing, 1995.

....) Then, we will show that OE is at least as reliable as any other assignment OE that fulfills the hypothesis. Initially, we assume that MF(OE) Next, we adopt a time evoling representation of random bin subsets. Similar time evolving representations of random edge subsets have been used in [Lom74, Kar95] for proving lower bounds on graph reliability. We introduce non negative time t and define a time dependent random subset M(t) M representing the set of active bins by the time t. Initially, M(0) Each bin is given an arrival time chosen independently from the exponential distribution with ....

....the time it takes for the next bin arrival is independent of the previous bin arrivals and the time t. Therefore, once we have conditioned on the value of M(t) the time for the next arrival has probability distribution function 1 Gamma e Gammam(t)t . In particular, it is shown in [Kar95] that, since the time of arrival has no impact on which of the bins of M Gamma M(t) is the first to arrive, the time for the next arrival has the right exponential distribution, regardless of the values of M(t) we condition on. As Lomonosov observed [Lom74] we can imagine that M(t) M oe ....

D.R. Karger. A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem. Proc. of the 27th ACM Symposium on Theory of Computing, pp. 11-- 17, 1995. Journal submission available from http://theory.lcs.mit.edu/karger/papers.html.

....conditions are needed for the analysis of the rst two ranges. Range 1: 0 x A and 1 k B see [1] 1 r Z A x=0 B X k=1 E(C k;x )dx (1 o(1) n r (3) Range 2: 0 x A and k B see [1] 1 r Z A x=0 n X k=B E(C k;x )dx = o(n=r) Range 3: x A. We use a result of Karger [4]. A cut (S : S) f(u; v) 2 E : u 2 S; v = 2 Sg of G is minimal if j(S : S)j . Karger proved that the number of minimal cuts is O(n 2 ) We can associate each component of G p with a cut of G. Thus n X k=1 E(C k;x ) O 1 X s= n 2s= 1 x r s = O 1 X ....

D. R. Karger, A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem, Proceedings of the twenty-seventh annual ACM Symposium on Theory of Computing (1995) 11-17.

....sampling. For example, consider the network reliability problem, which is as follows. Given a network with edges which fail independently with different probabilities, compute the probability that the graph becomes disconnected. The approximation algorithm for network reliability due to Karger [6] uses direct Monte Carlo sampling in the case that the probability the network fails is above O(1=n 4 ) where n is the number of nodes in the network. This provides a lower bound on the probability of failure which may be far too low. By using the optimal sampling method this probability may be ....

David R. Karger. A randomized fully polynomial time approximation scheme for the all terminal network reliability problem. In Proccedings of the Twenty-Seventh Annual ACM Symposium on the Theory of Computing, pages 11--17, 1995.

....throughout (Theorem 3.6) The time per edge update is O(jEj (min(k; p n) log n)kn) This is improved to O(jEj min(k; p n)kn) in Section 6. Counting the number of k separators of a k node connected graph is a fundamental problem. For example, the recent approximation scheme of Karger [Kar 95] for estimating network reliability with respect to edge failures is based on counting (and generating) all the minimum cardinality edge separators in polynomial time. Karger s work raised the question whether this method extends to approximating network reliability w.r.t. node failures. We show ....

D. Karger, "A randomized fully polynomial time approximation scheme for the all terminal network reliability problem," Proc. 27th ACM S.T.O.C. (1995), 11--17.

....[78] established that calculating two terminal reliability, even when all edges have probability p = 1 2 is #P complete. Provan and Ball [64] proved the analogous result for allterminal reliability, and Provan [63] and Vertigan [80] established similar results for planar graphs. Recently, Karger [44, 45] established that, despite these very negative worst case complexity results, there does exist a fully polynomial approximation scheme. These complexity results establish that exact calculation of all of the coefficients (in any form) is #P hard. However, the calculation of some of the ....

D. Karger, A randomized fully polynomial time approximation scheme for the all terminal network reliability problem, Proc. Symp. Theory Comput. 1995.

....6 Charles J. Colbourn Fishman [15] gives a valuable treatment of four different Monte Carlo techniques. Buchsbaum and Mihail [4] describe a further technique based on Markov chains. Elperin, Gertsbakh, and Lomonosov [14] devise a different strategy based on graph evolution, which Karger [24, 25] uses to establish an important theoretical result on the estimation of all terminal reliability. For Monte Carlo techniques as well, restricting to probabilistic connectedness and other simple models of operation, the guarantees about algorithm performance and accuracy are typically stronger than ....

D. Karger, "A randomized fully polynomial time approximation scheme for the all terminal network reliability problem", Proc. Symp. Theory Comput. , 1995. Reliability Issues in Telecommunications Network Planning 15

....connected components. For k 4, our running time is within a factor of k of the fastest algorithm known for testing k node connectivity. One application of shredders is in increasing the node connectivity from k to (k 1) by efficiently adding an (approximately) minimum number of new edges. Jord an [JCT(B) 1995] gave an O(n 5 ) time augmentation algorithm such that the number of new edges is within an additive term of (k Gamma 2) from a lower bound. We improve the running time to O(min(k; p n)k 2 n 2 (log n)kn 2 ) while achieving the same performance guarantee. For k 4, the running time ....

....k shredders of a k node connected graph over a sequence of edge insertions deletions. The time per edge update is O(jEj (min(k; p n) log n)kn) Counting the number of k separators of a k node connected graph is a fundamental problem. For example, the recent approximation scheme of Karger [K 95] for estimating network reliability with respect to edge failures is based on counting (and generating) all the minimum cardinality edge separators in polynomial time. Karger s work raised the question whether this method extends to approximating network reliability w.r.t. node failures. We show ....

D. Karger, "A randomized fully polynomial time approximation scheme for the all terminal network reliability problem," Proc. 27th ACM S.T.O.C. (1995), 11--17.

....9, 2003 Lecturer: Eric Vigoda Scribe: Murali Krishnan Ganapathy In this lecture, we will consider the problem of calculating the probability that a given network fails, and give a FPRAS (Fully Polynomial Randomized Approximation Scheme) for the same. The work on reliability is Karger, see [1, 2], and the earlier work on #DNF is Karp and Luby [4, 3] 2.1 Problem Statement Fix an undirected graph G and 0 p 1. Let H be a graph obtained by deleting edges of G, independently with probability p. Let FAILG (p) denote the probability that the graph H obtained as above is disconnected. The ....

....X , otherwise set X t 1 = X t . Can we prove that this chain converges quickly to its stationary distribution An ecient method to generate (almost) uniform samples yields an ecient approximate counter. Another project is to devise a deterministic approximation algorithm for FAILG (p) Karger [2] showed that his approach can be eciently derandomized when FAILG (p) 1=n . However it is open how to derandomize the case FAILG (p) 1=n . The high level idea for small failure probabilities is the following. There is an ecient deterministic algorithm to list all small cuts via max ow ....

D. R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM Rev., 43(3):499-522, 2001. Reprint of SIAM J. Comput. 29 (1999), no. 2, 492-514.

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D. Karger. A randomized fully polynomial time approximation scheme for all-terminal network reliability problem. Proceedings of the 27th ACM STOC, 1995.

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D. R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM J. Comput., 29(2):492--514 (electronic), 1999.

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D. Karger. A randomized fully polynomial time approximation scheme for all-terminal network reliability problem. Proceedings of the 27th ACM STOC, 1995.

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