Nimrod Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7(3):97--107, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....SFM via the push relabel algorithm: minimizing f (X) w(X) for a positive vector w, and finding a lexicographically optimal base. The concept of a lexicographically optimal base was introduced by Fujishige [6] as a generalization of the lexicographically optimal flow earlier defined by Megiddo [16]. Before the existence of combinatorial polynomial time algorithms for general SFM, it was common to present algorithms that required minimizing submodular functions as having access to an oracle. This was done for two reasons: l) the ellipsoid method was viewed as a tool for proving polynomial ....

....time as a single push relabel submodular function minimization. By Lemma 3.1, the number of et visited by the algorithm is at most n. Finding a Lexicographically Optimal Base Another related application is finding a lexicographically optimal base. This concept was first introduced by Megiddo [16] for multi terminal network flow. Fujishige [6] generalized it to the framework of polymatroids. Let w 6 R v be a weight vector satisfying w(v) 0 for all v 6 V. For any base x 6 B(f) we denote by 0(x, w) the sequence of the numbers x(v) w(v) for v 6 V arranged in the increasing order. A base x ....

N. Megiddo, Optimal flows in networks with multiple sources and sinks, Math. Programming, 7 (1974), 97-107.

....that this is the definition of a tight set. The flow leaving A must be at a maximum for the remainder of the algorithm. Hence, once a set is tight, it remains tight for the remainder of the algorithm. Recall o(A) is the amount of flow that can leave set A in one unit of time. Lemma 4. 2 (Meggido [9]) o(A) o(B) o(A [ B) o(A B) A set function with this property is called submodular. A linear set function satsifies this inequality at equality. Since oe(A) P i2A oe i is linear, it is easy to see that o(A)T Gamma oe(A) is also submodular. For tight set A, o(A)T Gamma oe(A) 0 by ....

N. Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7:97--107, 1974.

....g, and a time bound T , maximizes the amount of flow leaving each terminal in the given order. The set of terminals may contain both sources and sinks: maximizing the amount of flow leaving a terminal is equivalent to minimizing the amount of flow entering the terminal. Minieka [16] and Megiddo [15] both noted that this objective is also equivalent to simultaneously maximizing the amount of flow leaving every high priority subset S i : fs 0 ; s 1 ; s i Gamma1 g; 0 i k. If T is integral, then the natural transformation of a lexicographically maximum discrete dynamic flow in time T ....

N. Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7:97--107, 1974.

....tight. Define a chain of sets to be a sequence of nested sets: each set is strictly contained in its successor. We need the following lemma. Lemma 4.2 The intersection and union of tight sets are tight. Proof: Recall o(A) is the amount of flow that can leave set A in one unit of time. Meggido [11] shows that o(A) o(B) o(A[B) o(A B) i.e. o is a submodular function. Since oe(A) P i2A oe i satisfies this inequality at equality, it is easy to see that o(A)T Gamma oe(A) is also submodular. For tight set A, o(A)T Gamma oe(A) 0 by definition. Since T is feasible if and only if o(A)T ....

N. Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7:97--107, 1974.

....no (s; t) paths of length less than , it follows that j[ k ]j = 0. Applying Lemma 4. 2 and observing j[ Gamma k Gamma1 ]j j[ Gamma k ]j , we obtain: j[ Gamma 0 0 ]j j[ Gamma k ]j fflj[ Gamma k ]j : 2 5 Lexicographic Maximum Dynamic Flows Minieka [11] and Megiddo [10] showed that in a static network with source set S = fs 1 ; s k g there exists a feasible flow such that the amount of flow leaving the sets of sources S i = fs 1 ; s i g is simultaneously maximum for every i. Such a flow is clearly a lexicographically maximum flow. This ....

....the network forever. This contradicts the fact that there is no flow left in the network after time T 0 . 2 The complexity of this algorithm is dominated by the k minimum cost flow computations, each of complexity M(m;n) where M(m;n) O( m log n) m n log n) 12] Minieka [11] and Megiddo [10] observed that a static lexicographic maximum flow problem with many sources S and many sinks T can be solved by separately solving two lexicographic maximum flow problems, one with a single supersink and many sources, and one with a supersource and many sinks. The two flows can be pasted together ....

[Article contains additional citation context not shown here]

Megiddo, N., Optimal Flows in Networks with Multiple Sources and Sinks, Mathematical Programming 7(1974)97--107.

....sink t and t is last in the given order. In this section we show how the algorithm in [10] can be extended to solve the more general problem. The quickest transshipment problem will be solved in Sections 4, 5 and 6 by reducing it to the problem considered in this section. Minieka [16] and Megiddo [14] showed that in a static network with terminal set S = fs 1 ; s k g there exists a feasible flow such that the amount of flow leaving the sets of terminals S i = fs 1 ; s i g is simultaneously maximum for every i. Such a flow is clearly lexicographically maximum. This observation ....

....only if o(A) v(A) for every subset A S. Next we show how to test feasibility in polynomial time. A function o( is submodular if it satisfies o(A) o(B) o(A B) o(A [ B) for every pair of subsets A and B. It is not hard to show that the o( function above is submodular (see Megiddo [14]) Theorem 4.2 A violated set in a dynamic transshipment problem (N ; v ; T ) can be found in O(2 k MCF ) time, in O(k 2 MCF log(nT U) time 2 , and also in strongly polynomial time. Proof: The O(2 k MCF ) time is immediate. To obtain the other two versions, consider the submodular ....

Megiddo, N., Optimal Flows in Networks with Multiple Sources and Sinks, Mathematical Programming 7(1974)97--107.

....g, and a time bound T , maximizes the amount of flow leaving each terminal in the given order. The set of terminals may contain both sources and sinks: maximizing the amount of flow leaving a terminal is equivalent to minimizing the amount of flow entering the terminal. Minieka [13] and Megiddo [12] both noted that this objective is also equivalent to simultaneously maximizing the amount of flow leaving every high priority subset S i : fs 0 ; s 1 ; s i Gamma1 g; 0 i k. If T is integral, then the natural transformation of a lexicographically maximum discrete dynamic flow in time ....

N. Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7:97--107, 1979.

No context found.

Nimrod Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7(3):97--107, 1974.

No context found.

Megiddo, N., Optimal Flows in Networks with Multiple Sources and Sinks, Mathematical Programming 7(1974)97--107.

No context found.

N. Megiddo. Optimal flows in networks with multiple sources and sinks. Mathematical Programming, 7:97--107, 1974.

No context found.

Megiddo, N., Optimal Flows in Networks with Multiple Sources and Sinks, Mathematical Programming 7(1974)97--107.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC