C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....u(vw) or u(vw) vw) 2E u(v ) R2) i ffl bad C i ffl C i : By a proof similar to that of Theorem 3.2 of [9] we can show that if we can satisfy the relaxed optimality conditions then we actually have an O(ffl) optimal flow. A complete version of the proof appears in [13]. Theorem 3.3 Suppose f , and ffl satisfy the relaxed optimality conditions and ffl 1=9. Then f is O(ffl) optimal, i.e. is at most a factor (1 9ffl) more than the minimum possible value. As we shall see in the next section, the relaxed optimality conditions will guide our algorithm. 4 ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, Cambridge, MA, August 1992.

.... 2 op1 would make a valid schedule possible. In section 6.1 we will analyze the ow scheduling problem in terms of ow graph problems, mainly variants of the multi commodity ow problem. In section 6. 2 we describe an approximation algorithm for the multi commodity ow problem proposed by Stein [22] and polished later by Klein [14] and Leighton [17] This algorithm is the basis of our ow scheduling algorithm, though we had to modify it because in our case the variables (commodities) may have more than one destination. The operand paths of those variables with more than one consumer are ....

....an approximate solution takes O(k 2 n 2 p km log (n 1 DU) time using a linear programming approach. Better running times can be achieved with combinatorial approximation methods. The foundations for these methods were laid by Shahrokhi and Matula [20] in the late eighties. Stein [22] Klein [14] and Leighton [17] have developed the method into a complete algorithm. While they concentrated on the problem of minimizing the congestion of the edges (also called the concurrent ow problem) the algorithm was later extended for the minimum cost multi commodity problem, 32 in which ....

C. Stein. Approximation Algorithms for Multicommodity Flow and Shop Scheduling Problems. PhD thesis, Massachusetts Institute of Technology, September 1992.

....0 i 2 ;j 2 ;l 2 ) P K i P k v k i 1 (p) j 1 ; v k i 2 (p) j 2 (v i;j ; v e ) v sr ; v t r G; D 4 1 Introduction Disjoint paths problems are fundamental, extensively studied NP hard problems. They have applications in areas such as telecommunications, VLSI and scheduling [7, 12, 13, 15]. Due to the rapid growth of high speed integrated networks that provide vast bandwidth and support heterogeneous applications, considerable effort has been made recently for disjoint paths, bandwidth allocation and related algorithmic problems. In this paper, we study the inapproximability of ....

C. Stein, "Approximation algorithms for multicommodity flow and shop scheduling problems ", Ph.D. thesis, MIT, Cambridge, MA, August 1992. 17

....has an associated cost and the goal is to find a flow of minimum cost that satisfies all the demands. Multicommodity flow arises naturally in many contexts, including virtual circuit routing in a communication network, VLSI layout, scheduling, and transportation, and hence was extensively studied [5, 7, 17, 11, 15, 16, 2, 13]. Department of Computer Science, Stanford University. Research supported by U.S. Army Research Office Grant DAAL 03 91 G 0102. y Department of Computer Science, Stanford University. Research supported by a grant from Stanford Office of Technology Licensing. z Department of Computer ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

....an associated cost and the goal is to find a flow of minimum cost that satisfies all the demands. Multicommodity flow arises naturally in many contexts, including virtual circuit routing in communication networks, VLSI layout, scheduling, and transportation, and hence has been studied extensively [7, 10, 14, 17, 12, 13, 18, 2, 16]. Since multicommodity flow algorithms based on general interior point methods for linear programming are slow [10, 19, 8] recent emphasis was on designing fast combinatorial algorithms that relied on problem structure. One successful approach has been to develop approximation algorithms. If ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

....on any machine. The bound we obtain is based on the fractional packing formulation which is the linear relaxation of the packing formulation. This approach was motivated by the packing approximation algorithm of Plotkin, Shmoys Tardos [17] along with an application to scheduling given by Stein [20]. This bound gives stronger bounds than those obtained by other linear programming relaxations but appears to be too time consuming. To improve on the packing bound we attempt to restrict the time in which operations are allowed to start processing. We call this interval a processing window. The ....

C. Stein. Approximation Algorithms for Multicommodity Flow and Shop Scheduling Problems. PhD thesis, MIT/LCS/TR-550, Laboratory for Computer Science, MIT, Cambridge, MA, 1992.

....flow with the minimum cost. Multi commodity flow problems arise in several practical applications such as design of communication and computer networks, virtual circuit routing in communication networks, VLSI layout, scheduling, and transportation, and hence has been extensively studied [32, 39, 77, 50, 71, 75, 16, 69, 80]. Though a natural extension of the single commodity flow problem, all known combinatorial algorithms developed for the single commodity case do not readily 6 CHAPTER 2. LINEAR PROGRAMMING APPROACH 7 extend to the multi commodity flow problem. This is because many of the properties like ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

....and by many others. Given the stable traffic matrix, we are able to calculate the worst case delay of packets. Whether a real demand traffic matrix is feasible under the stable traffic matrix is a multi commodity concurrent flow problem in nature. Much work on this problem is described in [6][7][8] 9] 10] 11] III. BASIC MODEL AND NOTATION We assume that an underlying routing subsystem determines a unique path between any pairs of hosts. For our purposes, we model the network at transport layer and assume that every host can communicate with every host. The network therefore can be ....

C. Stein, "Approximation Algorithms for Multicommodity Flow and Scheduling Problems," Ph.D. thesis, MIT, Cambridge, MA, August 1992

....between given source and sink, such that the total flow on each edge obeys capacity. The multicommodity flow problems arises naturally in many contexts, including virtual circuit routing in communication networks, VLSI layout, scheduling, and transportation, and hence has been extensively studied [4, 6, 16, 9, 14, 15]. Though a natural extension of the single commodity flow problem, all known combinatorial algorithms developed for the single commodity case do not readily extend to the multicommodity flow problem. This is because many of the properties like unimodularity of the constraint matrix in the linear ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

....solution to the flow, automatically adjusting it after every edge failure or demand change. Multicommodity flow arises naturally in many contexts, including virtual circuit routing in a communication network, VLSI layout, scheduling, and transportation, and hence was extensively studied [3, 4, 11, 12, 7, 9, 10]. Since it can be written as a linear program, one can use 1 We say that f(n) O (g(n) if f(n) O(g(n) log k n) for some constant k. interior point methods to solve it exactly. Unfortunately, the fact that the number of variables is large causes these methods to generate rather slow ....

C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

....flow. Multicommodity flow is a well studied problem [KP95,KP95a,KPP95] The basic multicommodity flow problem may be solved in polynomial time with linear programming methods [KV86] but the integer case is NP complete. However, there is a fast approximation algorithm due to Stein et al. [St92]. Tadpole s heuristic approach is heavily based on this approximation algorithm, but augmented with information from the space time graph. 3.6 Reduced Graphs, Shortest Paths, and Cost Functions The Stein algorithm begins by routing commodities independently of each other, and then iteratively ....

Clifford Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. Ph.D. Thesis; also Tech Report MIT/LCS/TR-550, MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, 1992.

....of conditional probabilities [34, 44] with the new construction to obtain two results. We get a much faster implementation of the sequential jobshop scheduling algorithm of Shmoys, Stein Wein [41] It is comparable in time complexity to the speedups due to Plotkin, Shmoys Tardos [33] and Stein [45] but importantly, the approximation bound it presents is better than the ones of [33] and [45] Here, we show that a problem can be derandomized directly, thereby avoiding the bottleneck step of solving a huge linear program. We also prove an exact partition result for set discrepancy, and ....

.... a much faster implementation of the sequential jobshop scheduling algorithm of Shmoys, Stein Wein [41] It is comparable in time complexity to the speedups due to Plotkin, Shmoys Tardos [33] and Stein [45] but importantly, the approximation bound it presents is better than the ones of [33] and [45]. Here, we show that a problem can be derandomized directly, thereby avoiding the bottleneck step of solving a huge linear program. We also prove an exact partition result for set discrepancy, and derive a polynomial time algorithm for it. The organization of the paper is as follows. Section 2 ....

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C. Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. PhD thesis, Laboratory for Computer Science, Massachusetts Institute of Technology, 1992. Available as MIT=LCS=TR \Gamma 550.

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C. Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. PhD thesis, MIT, Cambridge, MA, August 1992. Also appears as MIT/LCS/TR-550.

....a ae approximation algorithm with ae 5=4 would imply that P = NP. It remains an interesting open problem to close this gap. ffl Our algorithms, while polynomial time algorithms, are inefficient. Are there significantly more efficient algorithms which have the same performance guarantees Stein [19] has given an algorithm that directly finds a good approximate solution to the integer program (IP ) by using the framework of Plotkin, Shmoys, and Tardos [11] This yields an implementation of our algorithm that runs in O(n 2 m 2 2 n 3 2 log(m ) log(m ) time. Although this ....

C. Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. PhD thesis, MIT, Cambridge, MA, August 1992. Also appears as MIT/LCS/TR-550.

....under the rounding approach, providing further background as necessary. 1. 1 Packing Integer Programs Packing integer programs are a well studied class of integer programs that can model several NP complete problems, including independent set, hypergraph k matching [19, 1] job shop scheduling [23, 28, 33, 20] and many flow and path related problems. Many of these problems seem to be di#cult to approximate, and not much is known about their worst case approximation ratios. Following [30] a packing integer program (PIP) is defined as follows. Definition 1. Given A # [0, 1] m n , b # [1, #) m ....

C. Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. PhD thesis, MIT, Cambridge, MA, August 1992.

....0 w 0 2E u(v 0 w 0 ) v 0 w 0 ) R2) X i ffl bad C i ffl k X i=1 C i : By a proof similar to that of Theorem 3.2 of [9] we can show that if we can satisfy the relaxed optimality conditions then we actually have an O(ffl) optimal flow. A complete version of the proof appears in [13]. Theorem 3.3 Suppose f , and ffl satisfy the relaxed optimality conditions and ffl 1=9. Then f is O(ffl) optimal, i.e. is at most a factor (1 9ffl) more than the minimum possible value. As we shall see in the next section, the relaxed optimality conditions will guide our algorithm. 4 ....

C. Stein. Approximation algorithms for multicommodity flow and shop scheduling problems. PhD thesis, MIT, Cambridge, MA, August 1992. Also appears as MIT/LCS/TR-550.

....a ae approximation algorithm with ae 5=4 would imply that P = NP. It remains an interesting open problem to close this gap. ffl Our algorithms, while polynomial time algorithms, are inefficient. Are there significantly more efficient algorithms which have the same performance guarantees Stein [19] has given an algorithm that directly finds a good approximate solution to the integer program (IP ) by using the framework of Plotkin, Shmoys, and Tardos [11] This yields an implementation of our algorithm that runs in O(n 2 m 2 2 n 3 2 log(m ) log(m ) time. Although this ....

C. Stein, Approximation algorithms for multicommodity flow and shop scheduling problems, PhD thesis, MIT, Cambridge, MA, Aug. 1992. Also appears as MIT/LCS/TR-550.

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C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

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C. Stein. Approximation Algorithms for Multicommodity Flow and Scheduling Problems. PhD thesis, MIT, 1992.

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C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

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C. Stein. Approximation algorithms for multicommodity flow and scheduling problems. PhD thesis, MIT, 1992.

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