D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log--space hard for P. Information Processing Letters, 8:96--97, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a linear time algorithm for linear programming in fixed dimension d (hereafter referred to as LPd) using an elegant multidimensional search technique. The same search technique, also known as prune and search, yielded optimal algorithms for other optimization problems (see Megiddo [27] and Dyer [12, 13]) Following Megiddo s linear time algorithm, there have been significant improvements in the constant factor (which was doubly exponential in d) due to Dyer [13] and Clarkson [4] Further progress was made, using random sampling, by Clarkson [5] This algorithm was later derandomized by Chazelle ....

D. Dobkin, R. Lipton and S. Reiss. Linear programming is log-space hard for P, Information Processing Letters 8, 1979, 96--97.

....The restricted version of LE is denoted [ Gamma1,1] LE. A.4.3 Linear Programming ( LP) Given: An integer n Theta d matrix A, an integer n Theta 1 vector b, and an integer 1 Theta d vector c. Problem: Find a rational d Theta 1 vector x such that Ax b and cx is maximized. Reference: [DLR79, Kha79, DR80, Val82a] Hint: LP is not in P , but is in FP by [Kha79] Reduce LI to LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log space ....

....1 vector x such that Ax b and cx is maximized. Reference: DLR79, Kha79, DR80, Val82a] Hint: LP is not in P , but is in FP by [Kha79] Reduce LI to LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log space equivalent by reducing LP to LI using rational binary search [Pap78, Rei78] to find the value of the maximum and an x that Part II: P Complete Problems ffl 63 yields it. However, it is not clear how to perform this ....

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D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log-space hard for P. Information Processing Letters, 8(2):96--97, February 1979.

....combinatorial optimization. It is the problem of optimizing a linear function c T x over a convex polyhedron fx : Ax b; x 0g, where x 2 R n , A is an m Theta n matrix and b; c 2 R n . The parallel complexity of this problem is, by now, well understood. Dobkin, Lipton, Reiss and Khachyan [4, 8] showed that (the general) LP was complete, in the strong sense, for P under logspace reductions. Later on, it was shown that even the problem of approximating the value of a general linear program is P complete [13, 12] Therefore, there is no fast parallel algorithm for solving LP or for ....

D. Dobkin, J.R. Lipton, and S. Reiss. Linear Programming is logspace hard for P. Information Processing Letters, 8:96--97, 1979.

....the reduction given by Itai [Ita78] The restricted version of LE is denoted [01,1] LE. A.4.3 Linear Programming ( LP) Given: An integer n2d matrix A, an integer n21 vector b, and an integer 12d vector c. Problem: Find a rational d 2 1 vector x such that Ax b and cx is maximized. Reference: [DLR79, Kha79, DR80, Val82a] Hint: LP is not in P , but is in FP by [Kha79] Reduce LI to LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log ....

....2 1 vector x such that Ax b and cx is maximized. Reference: DLR79, Kha79, DR80, Val82a] Hint: LP is not in P , but is in FP by [Kha79] Reduce LI to LP by picking any cost vector c, say c = 0, and checking whether the resulting linear program is feasible. Remarks: The original reduction in [DLR79] is from HORN, Problem A.6.2, to LP. In [DR80] LP and LI are shown to be log space equivalent by reducing LP to LI using rational binary search [Pap78, Rei78] to find the value of the maximum and an x that Part II: P Complete Problems ffl 63 yields it. However, it is not clear how to perform ....

[Article contains additional citation context not shown here]

D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log-space hard for P. Information Processing Letters, 8(2):96--97, February 1979.

....combinatorial optimization. It is the problem of optimizing a linear function c T x over a convex polyhedron fx : Ax b; x 0g, where x 2 R n , A is an m Theta n matrix and b; c 2 R n . The parallel complexity of this problem is, by now, well understood. Dobkin, Lipton, Reiss and Khachyan [4, 8] showed that (the general) LP was complete, in the strong sense, for P under logspace reductions. Later on, it was shown that even the problem of approximating the value of a general linear program is also P complete [13, 12] Therefore, there is no fast parallel algorithm for solving LP or for ....

D. Dobkin, J.R. Lipton, and S. Reiss. Linear Programming is logspace hard for P. Information Processing Letters, 8:96--97, 1979.

....Consequently, all these problems can be solved sequentially with only O( log m log 2 n) 2 log 2 n) space. It is also shown that if the underlying graph has bounded tree width and an underlying tree is given then the problem is in the class NC. 1. Introduction Dobkin, Lipton and Reiss [4] first showed that the general linear programming problem was (log space) hard for P. Combined with Khachiyan s result [10] that the problem is in P, this establishes that the problem is P complete (that is, log space complete for P) A popular specialization of the general linear programming ....

D. Dobkin, R. J. Lipton and S. Reiss, "Linear programming is log space hard for P", Information Processing Letters 8 (1979) 96-97.

....number of variables. In the one dimensional case this bound is optimal. If we take into account the operations needed for processor allocation, the time bound is O( log log n) d c ) where c is an absolute constant. 1. Introduction The general linear programming problem is known to be P complete [6] so it is interesting to investigate the parallel complexity of special cases. One important case is when the number of variables (the dimension) d is fixed while the number of inequalities n grows. Megiddo [11] showed that this problem can be solved in O(n) time for any fixed d. Clarkson [4] and ....

D. Dobkin, R. J. Lipton, and S. Reiss, "Linear programming is log space hard for P," Information Processing Letters 8 (1979) 96--97.

....a normal form game that is typically exponentially larger. In this case, we can prove a lower bound only in the game theoretic model and not in the accept reject model. Theorem 4.6 Value(pr; pr) is EXP hard. 9 Proof: Our proof is based on the proof that linear programming is P hard [DLR79]. This result can be used to show that solving two person zero sum games in normal (matrix) form is also P hard. The following is a long and very technical exposition showing how this P hardness result can be scaled up to the exponential case. Consider a deterministic exponential time Turing ....

.... jQjff) 0: Since there exists a strategy 1 guaranteeing a value of 0 against every clause, the value of the game in this case is at least 0. The other case, where w 62 L, is a little harder to see. It can be proved by equivalence to a certain linear system used by Dobkin, Lipton, and Reiss [DLR79], which in turn is equivalent to unit resolution in Horn clauses. We will sketch a direct proof. The proof asserts that any variable in Q must get probability at least 1 jQj . This clearly suffices. The proof is by induction on t, the time to which the proposition p[t; l; oe] refers. In the ....

D. Dobkin, R. Lipton, and S. Reiss, "Linear Programming is LOG-SPACE Hard for P," Information Processing Letters, 8 (1979), pp. 96--97.

....combinatorial optimization. It is the problem of optimizing a linear function c T x over a convex polyhedron fx : Ax b; x 0g, where x 2 R n , A is an m Theta n matrix and b; c 2 R n . The parallel complexity of this problem is, by now, well understood. Dobkin, Lipton, Reiss and Khachyan [4, 8] showed that (the general) LP was complete, in the strong sense, for P under logspace reductions. Later on, it was shown that even the problem of approximating the value of a general linear program is P complete [13, 12] Therefore, there is no fast parallel algorithm for solving LP or for ....

D. Dobkin, J.R. Lipton, and S. Reiss. Linear Programming is logspace hard for P. Information Processing Letters, 8:96--97, 1979.

....Randomized Rounding technique [RT87] or the same technique with more refined probability choices [GW94] If we would like to employ similar ideas to obtain parallel approximation algorithms, a lot of difficulties are encountered. Indeed, to begin with, we cannot solve in NC a given LP, unless P=NC [Kha79, DLR79]. Even we cannot find an approximate solution of a given LP in NC [Ser91, Meg92] The only known kind of parallel algorithm for LP is the Luby and Nisan s approximation scheme [LN93] for a subclass of LP, namely an LP with non negative coefficients and in packing covering form, referred to as ....

D. Dobkin, J.R. Lipton, and S. Reiss. Linear Programming is logspace hard for P. Information Processing Letters, 8:96--97, 1979.

....solvable if the line drawing is already labeled. The problem can in this case be reduced to an instance of Linear Programming (Sugihara [7] Unfortunately, this does not seem to be a biologically plausible approach to realizability, in that Linear Programming is P complete (Dobkin et al. [23]) and is therefore unlikely to be efficiently parallelizable. There are, however, some positive results about realizability of a labeled line drawing. Kirousis Papadimitriou [9] showed that the realizability problem for (labeled) planar projections of Manhattan scenes can be solved in time ....

....are of the implicational type (Kirousis [22] As for realizability, the situation is more delicate. The usual approach to realizability is by reduction to Linear Programming (Sugihara [7] but this is not a biologically plausible approach since Linear Programming is P complete (Dobkin et al. [23]) On the positive side, when the location of vanishing points is given, the problem becomes efficiently parallelizable and its solution is the intuitive one in terms of the ordering of surfaces. All of these results point to the importance, advocated for example by Kanade [27] of geometrical ....

D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log-space hard for p. Inf. Process. Lett., (9):96--97, 1979.

....Gaussian elimination with partial pivoting. Graphs [104] Maximum flow, lexicographically first maximal independent set, lexicographically first maximal path, lexicographically first depth first search ordering, high degree subgraph, minimum degree elimination order. Combinatorial optimisation [75, 154]: Linear programming, linear inequalities, first fit decreasing bin packing, nearest neighbour travelling salesman heuristic, two player game. Geometry [25] Plane sweep triangulation, visibility layers. Some simple examples of problems in P which are not currently known to be in NC or to be ....

D Dobkin, R J Lipton, and S Reiss. Linear programming is log-space hard for P. Information Processing Letters, 8:96--97, 1979.

....form generates a normal form game that is typically exponentially larger. In this case, we can prove a lower bound only in the game theoretic model and not in the accept reject model. Theorem 4.6 Value(pr; pr) is EXP hard. Proof: Our proof is based on the proof that linear programming is P hard [DLR79]. This result can be used to show that solving two person zero sum games in normal (matrix) form is also P hard. The following is a long and very technical exposition showing how this P hardness result can be scaled up to the exponential case. Consider a deterministic exponential time Turing ....

....1) jQjff) 0: Since there exists a strategy 1 guaranteeing a value of 0 against every clause, the value of the game in this case is at least 0. The other case, where w 62 L, is a little harder to see. It can be proved by equivalence to a certain linear system used by Dobkin, Lipton, and Reiss [DLR79], which in turn is equivalent to unit resolution in Horn clauses. We will sketch a direct proof. The proof asserts that any variable in Q must get probability at least 1 jQj . This clearly suffices. The proof is by induction on t, the time to which the proposition p[t; l; oe] refers. In the ....

D. Dobkin, R. Lipton, and S. Reiss, "Linear Programming is LOG-SPACE Hard for P," Information Processing Letters, 8 (1979), pp. 96--97.

....class is in fact a bit artificial and is defined having in mind the Nor False Gates problem. However, it is interesting because it fills a gap on the parallel approximability of LP. The parallel complexity of (subclasses of) LP can be summarized as follows: a) LP is hard to solve in parallel [DLR79] b) LP is hard to approximate in parallel [Ser91] c) the subclass of Positive LP can be approximated in parallel within any constant [LN93] d) the subclass of Positive LP cannot be exactly solved in parallel [TX98] e) the generalization of Positive LP in which are allowed also equality ....

D. Dobkin, J.R. Lipton, and S. Reiss. Linear Programming is logspace Hard for P. Information Processing Letters, 8:96--97, 1979.

....space as a function of the input size even if we instead use a Turing machine model with rational input. It is also shown that if the underlying graph has bounded tree width and an underlying tree is given then the feasibility problem is in the class NC. 1. Introduction Dobkin, Lipton and Reiss [7] first showed that the general linear programming problem was (log space) hard for P. Combined with Khachiyan s deep result [14] that the problem is in P, this establishes that the problem is P complete (that is, log space complete for An earlier version of this paper appeared in the ....

D. Dobkin, R. J. Lipton and S. Reiss, "Linear programming is log space hard for P," Information Processing Letters 8 (1979) 96--97.

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D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log--space hard for P. Information Processing Letters, 8:96--97, 1979.

No context found.

D. Dobkin, R. J. Lipton and S. Reiss, "Linear programming is log space hard for P", Information Processing Letters 8(1. 96-97.

No context found.

D. Dobkin, R. J. Lipton and S. Reiss, "Linear programming is log space hard for P," Information Processing Letters 8(1. 96--97.

No context found.

D. Dobkin, R. J. Lipton, and S. Reiss. Linear programming is log-space hard for P. Information Processing Letters, 8(2):96-97, Feb. 1979.

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