Gallo, G., Grigoriadis, M.D., & Tarjan, R.E. (1989), `A fast parametric maximum flow algorithm and applications', SIAM J. Computing 18, 30--55.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(1) and increases the vector c= that defines the capacities of the edges from s to D k in G. Hence in two successive iterations of Newton s method, the only change that G suffers is an increase in the capacities of some edges leaving the source vertex. The use of parametric flow techniques of [16] can perform all iterations of Newton s 15 method in the time required to perform a single maximum flow computation. Theorem 4.3. The robustness function of a transversal matroid M can be computed in =jEj 2) time, where E is the set of edges in its bipartite graph. Proof. We use the ....

G. Gallo, M.D. Grigoriadis and R.E. Tarjan, A fast parametric maximum flow algorithm and applications, SIAM J. Comput., 18 (1989), pp. 30--55.

....that are not edges of the graph. The densest subgraph problem can be solved exactly in polynomial time using flow techniques. One such algorithm is given by Lawler [8, Chapter 4] The currently best available time bound for the problem is O(mn log(n m) due to Gallo, Grigoriadis, and Tarjan [3]. It is obtained by reducing the densest subgraph problem to a parametric min cut problem and then solving it using a parametric max flow algorithm whose running time is the same as the running time of the non parametric max flow algorithm of Goldberg and Tarjan [5] Of more practical interest is ....

G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18:30--55, 1989.

....value of # 1, then proceed to the next step. 5. Set the prices of the goods in S to zero, remove these goods from the set of goods, and start again from Step 2. Step 4 in the above algorithm can be implemented using binary search over values of # or using a parametric network flow algorithm [7] to find the first event that occurs. Notice that Step 5 in the above algorithm is only for taking care of (pathological) cases where in the equilibrium some of the prices are zero. If, for example, we assume that each agent has a non zero utility for each good (i.e. u ij 0 for every i, j) ....

Giorgio Gallo, Michael D. Grigoriadis, and Robert E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18(1):30--55, 1989.

....calls by a factor of n. The resulting algorithm runs in O(n ff n 8) time. The push relabel framework was introduced by Goldberg and Tarjan [9] for the maximum flow problem. Subsequently, it was applied to polymatroid intersection by Fujishige and Zhang [7] Gallo, Grigoriadis, and Tarjan [8] extended the push relabel algorithm to solve monotone para metric maximum flow problems with no increase in time complexity. Iwata, Murota, and Shigeno [15] discussed an extension of the result in [8] to polymatroid intersection. They showed that a strong map sequence of submodular functions ....

....applied to polymatroid intersection by Fujishige and Zhang [7] Gallo, Grigoriadis, and Tarjan [8] extended the push relabel algorithm to solve monotone para metric maximum flow problems with no increase in time complexity. Iwata, Murota, and Shigeno [15] discussed an extension of the result in [8] to polymatroid intersection. They showed that a strong map sequence of submodular functions plays a similar role to that of a monotone parametric network. Analogously, we extend the push relabel algorithm for SFM to solve the parametric minimization problem for a strong map sequence. We then ....

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G. Gallo, M.D. Grigoriadis, and R. E. Tarjan, A fast parametric maximum flow algorithm and applications, SlAM J. Cornput., 18 (1989), 30-55. 10

....cv(P(qa) cv(P(qb) Return the result of running dynamic backtracking on A with the order q, qn on the propositional symbols. using decomposition, dynamic backtracking and c v ordering. To find a solution to such a problem we specialize and slightly modify the algorithm proposed by [11]. The original algorithm may yield an empty set of clauses, which we wish to avoid. The algorithm uses the push relabel max flow algorithm of Goldberg and Tarjan (see [5] the parametric max flow algorithm of [11] the fractional programming algorithm of [14] and the translation of [17] of the ....

....to such a problem we specialize and slightly modify the algorithm proposed by [11] The original algorithm may yield an empty set of clauses, which we wish to avoid. The algorithm uses the push relabel max flow algorithm of Goldberg and Tarjan (see [5] the parametric max flow algorithm of [11], the fractional programming algorithm of [14] and the translation of [17] of the selection problem into a flow problem. Together, they provide a general solution to zero one fractional programming, as shown in [11] Our combined algorithm is shown in Figure 7. Find MaxFraction takes as input a ....

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G. Gallo, M.D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18(1):30-55, 1989.

....universally quickest transshipment problem. The main contributions of this paper is to a) recognize that these problems can be solved by solving a parametric maximum flow problem on a suitably defined graph and b) generalize the parametric maximum flow algorithm of Gallo, Grigoriadis, and Tarjan [4] to fit the needs of this problem. The end result is that all the computations described in [11] can be performed in the same asymptotic time as one preflow push maximum flow computation on a network with nk vertices and (n m)k arcs. This improves the previous strongly polynomial run time by a ....

....of Goldberg and Tarjan [5] Ogier s approach leads to an O(k 2 mn log(kn 2 =m) algorithm to compute x . 2 A Faster Algorithm In this section, we discuss the main contribution of this paper which is a generalization of the parametric maximum flow algorithm of Gallo, Grigoriadis, and Tarjan [4] that speeds up the computation of W and the x , 2 W . We first review the parametric maximum flow algorithm, and the preflow push maximum flow algorithm on which it is based. 2.1 Parametric maximum flows Gallo, Grigoriadis, and Tarjan [4] present several algorithms based on a parametric ....

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G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18(1):30--55, 1989.

....known algorithm to solve this problem, and improves upon [5] by almost a factor of k. If there is just one sink, then it is known [5] that the minimum feasible time T can be found in strongly polynomial time using one call of the parametric maximum flow algorithm of Gallo, Grigoriadis, and Tarjan [9]. Thus, our rounding technique implies an improvementover the previous best algorithm by a factor of k [5] for this special case as well. The Gallo, et al. algorithm is based on the Goldberg and Tarjan [11] maximum flow algorithm and runs in the same asymptotic time. In addition, since there ....

G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18(1):30--55, 1989.

....is then used as a starting state for the next application of the parameter value. The parametric push relabel algorithm retains distance labels and flow as the parameter is increased from one value to the next. This algorithm was proved to have the same complexity as a single run of push relabel [GGT89]. Parametric LG retains the same branches in the normalized tree from one parameter value to the next, only some branches are then identified as strong whereas with the previous parameter they were weak. This parametric algorithm also has the same complexity as a single run of the LG algorithm ....

....of the algorithms, the number of groups of ore blocks that are profitable to mine is expected to be small. Notice that the running times of push relabel are larger than those for a single parameter application although the complexity bound for single parameter and multiple parameters is the same, [GGT89]. This is because in these runs the running time gets closer to the worst case complexity bound, whereas for single runs the algorithm typically performs much better than predicted by the complexity bound. One reason for the degraded performance is that in the parametric network, the capacities ....

G. Gallo, M. D. Grigoriadis and R. E. Tarjan, " A Fast Parametric Maximum Flow Algorithm and Applications," SIAM Journal of Computing, Vol. 18, No. 1 (1989) 30-55.

....ffl The Densest Subgraph (DS) problem concerns choosing a subset V 0 (of arbitrary size) such that the vertex induced subgraph has maximum average degree. This problem can be solved polynomially using flow techniques (cf. Chapter 4 of [Law76] The fastest algorithm known for DS is given in [GGT89] and runs in time O(mn log(n 2 =m) One may hope that some algorithmic techniques used in solving the DS problem can help approximate the DkS problem, but there seem to be major difficulties involved. Consider for example the case of regular graphs. The densest subgraph of a regular graph is ....

G. Gallo, M.D. Grigoriadis, and R.E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. on Comput., 18:30--55, 1989. 12

.... 0 , in N(v; G) We look for a vertex v and a subset V 0 N(v) that achieves the following minimum: min v ( min V 0 N(v) w v (V 0 ) e(V 0 ) It is important to note that the pair v; V 0 achieving this minimum can be found in polynomial time using flow techniques (cf. GGT 89] Given v and V 0 , one adds the edges connecting v and V 0 to the spanner. Note that in this way we 2 Gammahelp (or span) all the edges internal to V 0 , using low weight. This is done in iterations until the edges are exhausted. It follows from a proof similar to that in [KP 92] that ....

G. Gallo and M.D. Grigoriadis and R.E. Tarjan, A fast Parametric maximum flow algorithm and applications, SIAM J. on Comput, 18, 1989, 30-55

....= jE(U)j jU j : 2 The maximum density of the graph G is defined to be ae(G) max U V fae G (U)g: We call the problem of finding a subgraph of G with density ae(G) the maximum density problem. We recall the following fact, derivable, e.g. from [Law76] pp. 125 127, or alternatively from [GGT89]. Lemma 2.1 [Law76, GGT89] The maximum density problem can be solved polynomially using flow techniques. The fastest algorithm known for the maximum density problem is given in [GGT89] This algorithm runs in time O(mn log(n 2 =m) We make use of an alternative characterization of k ....

....2 The maximum density of the graph G is defined to be ae(G) max U V fae G (U)g: We call the problem of finding a subgraph of G with density ae(G) the maximum density problem. We recall the following fact, derivable, e.g. from [Law76] pp. 125 127, or alternatively from [GGT89] Lemma 2. 1 [Law76, GGT89] The maximum density problem can be solved polynomially using flow techniques. The fastest algorithm known for the maximum density problem is given in [GGT89] This algorithm runs in time O(mn log(n 2 =m) We make use of an alternative characterization of k Gammaspanners, given in the ....

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G. Gallo, M.D. Grigoriadis, and R.E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. on Comput., 18:30--55, 1989.

....the problem is that a vertex on the left is included, only if all its right side neighbors are included as well. Thus, it is not hard to see this problem can be solved via flow methods (combined with binary search procedure) The currently fastest algorithm for the selection problem is given in [GGT89] For the sake of completeness, we 18 Z S W S N (S ) 1 Z I 1 r I 1 l N (S ) 1 W W Z Figure 2: The sets in the bipartite graph G used by Procedure Neig. give a brief description of the flow algorithm that solves our special case of the selection problem, the 2 Neighborhood problem. ....

G. Gallo, M.D. Grigoriadis, and R.E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. on Comput., 18:30--55, 1989. 26

....the optimum, a surprising extension to Schulz s 2 approximation result. In Section 3, we provide a parametric extension to Sidney s decomposition, and show that a nest decomposition can be obtained by essentially solving a parametric minimum cut problem, which, by Gallo, Grigoriadis and Tarjan [3] (see also [12] requires about the time of a single maximum ow calculation. In Section 5 we show that, after this Sidney decomposition, the LP formulation can be solved as essentially a single dual minimum cost ow problem. Both network ow problems are on networks with the jobs as nodes (plus ....

.... 0 varies, we obtain a parametric minimum cut problem where the only arcs whose capacity varies with are adjacent to the source or the sink. This is precisely the setting for the parametric maximum ow algorithm, hereafter called the GGT algorithm, due to Gallo, Grigoriadis and Tarjan [3]. Indeed, the GGT algorithm produces, in about the time needed to compute a single maximum ow on a network N , a nested family of subsets ; H 0 H 1 : H k = N and a sequence of breakpoints 1 = 0 1 : k k 1 = 0 such that for all i = 0; k, H i maximizes f ....

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G. Gallo, M.D. Grigoriadis, and R.E. Tarjan, \A Fast Parametric Maximum Flow Algorithm and Applications", SIAM J. Computing 18 (1989), 30-55.

....Martel [10] and McCormick [12] determined the elimination number, i.e. the minimum number of remaining games a team must win in order to have any chance of finishing in first place. Their methods use di#erent extensions of the parametric maximum flow techniques of Gallo, Grigoriadis, and Tarjan [5]. McCormick [12] also showed that it is NP complete to determine whether a team is eliminated from finishing the season in t th place or better. Adler, Erera, Hochbaum, and Olinick [1] proposed an integer programming formulation to determine which teams are eliminated from finishing # A ....

....then they are either both eliminated or both not eliminated, regardless of their remaining opponents. Using our new ordering and binary search, we can find all eliminated teams with log n maximum flow computations. Using the parametric maximum flow techniques of Gallo, Grigoriadis, and Tarjan [5], we show how to determine all eliminated teams in the same complexity as a single maximum flow computation. It is also straightforward to determine all of the elimination numbers from our computation. We note that the new structural property was independently proved by Adler, Erera, Hochbaum, ....

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G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18:30--55, 1989.

....as a maximum flow problem on a bipartite network. Gusfield and Martel [4] shows that the minimum number of games a 3 given team must win in order to avoid elimination from first place can be found by solving a parametric maximum flow problem. By extending a result of Gallo, Grigoriadis and Tarjan [2] and using a binary search procedure, this paper proves a running time of O(n 3 n 2 log(nD) where n is the number of teams and D the number of games the team of interest has left to play, for finding this number. Recently, McCormick [8] has improved the time bound for solving this ....

G. Gallo, M.D. Grigoriadis, and R.E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18:30--55, 1989.

....which uses the ellipsoid method as a subroutine. The algorithm of Hajek and Ogier computes a fractional solution using O(n) maximum flow computations. When there is only one sink (or one source) we can speed up the search for by using the parametric flow ideas of Gallo, Grigoriadis, and Tarjan [6], to find each with one maximum flow. For this case, the parametric flow problem that we solve to find is simpler. The new network has only one new node, a super source the single sink obviates the need for a super sink. Now the arcs from super source to source i have capacity oe i Gamma f i ....

....interchangeable on A r for the interval [0; T r ) Hence, the output of the algorithm on the original problem with r sets is also a universally quickest evacuation. The algorithm has three stages: a) It first finds sets A i using the parametric flow algorithm of Gallo, Grigoriadis, and Tarjan [6], which also returns the corresponding time bounds T i . These A i are the minimum cuts in the corresponding maximum flow network parameterized by T i . b) This information is then used to construct a static flow which represents the initial flow rate. c) Finally, the dynamic flow is constructed ....

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G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. Comput., 18(1):30--55, 1989.

....in strongly polynomial time O(n 3 ) or in time O(n 8=3 log U) where U = max i (p i w i ) Proof: Computing the minimum cut in G for each 0 can be viewed as a parametric maxflow computation. There are at most n values of for which the minimum cut changes in the graph. Gallo et al. [7] show that it is possible to obtain all the distinct values of in time to do one maximum flow computation using the push relabel algorithm. Goldberg and Tarjan s [11] push relabel algorithm runs in O(nm log(n 2 =m) time. Recently, Goldberg and Rao [10] improved the maximum flow running time to ....

....algorithm runs in O(nm log(n 2 =m) time. Recently, Goldberg and Rao [10] improved the maximum flow running time to O(minfn 2=3 ; m 1=2 gm log(n 2 =m) log U) where U is the maximum capacity, and also showed that their bound applies for the parametric flow techniques of Gallo et al. [7]. The associated graph we construct has Omega Gamma n 2 ) edges, therefore the claimed bounds follow. 2 5 Integrality Gap of the Linear Ordering Relaxation In this section we show that the linear ordering relaxation of Potts [20] has a factor 2 integrality gap. The gap also applies to the half ....

G. Gallo, M.D. Grigoriadis, and R. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM J. on Comput., 18:30--55, 1989.

....the function gives a different vertex of the convex hull. The problem we need to solve is the parameterized flow problem, where the flow graph has edges emanating from the source with capacities which are linear in the parameter ff. This problem has been solved by Gallo, Grigoriadis, and Tarjan [GGT89] who give a polynomial time algorithm for the problem. It turns out that the minimum cuts identified by the algorithm have a nesting property, which implies that for an n vertex graph there are at most n Gamma 1 cuts. This is interesting because this implies that although there might be ....

G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18(1):30--55, 1989.

....1.26 (Flow Problems) ffl The maximum flow problem: Find a flow with maximum flow value. ffl The perfect flow sharing problem: In a network as specified in 1.24(6) find a flow with maximum flow value among all flows that are perfectly shared. This definition is equivalent to that given in [GGT]. It is useful to formulate certain relations between (s; t) cuts in a flow network in terms of order theory. Notation 1.27 On the set of all (s; t) cuts in a flow network let the partial order OE be defined by [X 1 ; Y 1 ] OE [X 2 ; Y 2 ] X 1 6 X 2 . In the remainder of this thesis all ....

....after a number of steps that is bounded from above by an exponential number multiplied by the total number of breakpoints of all discrete functions belonging to the input SDFinstance. 5 A solution to the Perfect Flow Sharing Problem 5. 1 Introduction A solution to this problem is given in [GGT]. This algorithm is developed from the general parametric maximum flow algorithm, which bases on the maximum flow algorithm by Goldberg and Tarjan ( GT] In this thesis we will introduce another algorithm, which makes use of a maximum flow algorithm, too, but that maximum flow algorithm may be ....

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Gallo, G., M. Grigoriadis, and R.E. Tarjan (1989): A Fast Parametric Maximum Flow Algorithm and Applications. SIAM J. of Computing 18, 30-55.

....of , the number of clusters can only increase as well, or stay the same. When implementing our algorithm we often need to apply a binary search like approach in order to determine the best value for , or make use of the nesting property . The nesting property has been used by Gallo et.al. [5] in the context of parametric maximum ow algorithms. A parametric network is de ned as a regular network G with source s and sink t, only the edge weights are linear functions of a parameter , as follows: 1) w (s; v) is a nondecreasing function of for all v 6= t. 2) w (v; t) is a ....

....holds for our graph G Lemma 3.5 For a source s in G a given on line sequence of parameter values 1 2 : max , yields a sequence S 1 ; S 2 ; S max of communities of s with respect to t, such that S 1 S 2 : Smax . Proof. This is a direct result of a similar lemma in [5]. In fact, in [5] it has been shown that for some s the total number of di erent communities S i is no more than n 2 and they can all be computed in time proportional to a single max ow computation, when a variation of the Goldberg Tarjan pre ow push algorithm [7] is employed. Thus, if we want ....

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G. Gallo, M. D. Grigoriadis and R. E. Tarjan. A Fast Parametric Maximum Flow Algorithm and Applications. SIAM Journal of Computing, Vol. 18, No. 1 (1989) 30-55. 10

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Gallo, G., Grigoriadis, M.D., & Tarjan, R.E. (1989), `A fast parametric maximum flow algorithm and applications', SIAM J. Computing 18, 30--55.

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Gallo, G., Grigoriadis, M. D., Tarjan, R. T. (1989). A fast parametric maximum flow algorithm and applications. Siam J. Comput. 18 (1), 30 -- 55.

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Gallo,G., Grigoriadis,M.D., and Tarjan,R. E., A fast parametric maximum flow algorithm and applications, SIAM Journal of Computing, 18(1):30--55, 1989.

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G. Gallo, M. D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing, 18(1):30--55, 1989.

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G. Gallo, M.D. Grigoriadis, and R. E. Tarjan. A fast parametric maximum flow algorithm and applications. SIAM Journal of Computing, 18(1):30--55, 1989.

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