David P. Dobkin and Steven P. Reiss. The complexity of linear programming. Theoretical Computer Science, 11:1 1S, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....studied in the game theory literature [168] The name matrix game comes from the fact that the relevant parameters can be sum marized by a 1.411 x 1.421 matrix consisting of the immediate re;yard values. Solving a matrix game is known to be polynomially equivalent to solving a linear program [47]. As with alternating Markov games, I mainly consider the problem of finding minimax optimal policies. Once again, I consider only the discounted expected value criterion. It is possible to define a notion of undiscounted rewards for Markov games, but not all Markov games have optimal ....

....matrix, reward matrix, and the discount factor are all rational. This result is discussed in more detail in Section 5.5, but it is important to note this now because the following analyses makes use of it. As I mentioned earlier, solving matrix games is equivalent to linear programming [47]. This means that they can be solved exactly in polynomial time. Although an optimal policy for a matrix game can be stochastic, the probabilities and values are guaranteed to be rational if the transitions, rewards, and discount factor are rational. 106 5.4.2 Iterative Algorithms From Lemma 5.1 ....

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David P. Dobkin and Steven P. Reiss. The complexity of linear programming. Theoretical Computer Science, 11:1 1S, 1980.

....we first determine if v is a convex vertex. This is the case if and only if v is an extremal point in the set fv; w 1 ; wm g. This is equivalent to the problem of determining if v can be separated from fw 1 ; wm g by a plane, which in turn is equivalent to linear programming [7]. Therefore we can determine if v is convex by linear programming in linear time (see e.g. 8, 19, 27] If v is not a convex vertex, then v is not a local maximum for any direction, and we stop considering v. Otherwise, let h v be a plane that contains v and has w 1 ; wm to one side of ....

Dobkin, D., and S. Reiss, The Complexity of linear programming. Theoretical Computer Science, 11, pp. 1--18, 1980.

....degenerate configurations [12, Section 9. 4] The problem of deciding whether a given point p 2 P is an extreme point is equivalent to the problem of determining whether p 2 conv(P n fpg) This problem can be dualized into a linear programming problem that involves n constraints and d variables [9]. Before we give our algorithm, we review the concepts of linear programming in relation to the computation of extreme points. 2.1 Linear Programming and Extreme points A natural and frequently studied problem regarding convex polyhedra is the linear programming problem. We are given a set H of ....

D. P. Dobkin and S. P. Reiss. The complexity of linear programming. Theoretical Computer Science, 11:1-18, 1980.

....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 ....

....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 reduction in NC 1 . Since LP and LI are ....

D. P. Dobkin. and S. Reiss. The complexity of linear programming. Theoretical Computer Science, 11(1):1--18, 1980.

....we first determine if v is a convex vertex. This is the case if and only if v is an extremal point in the set fv; w 1 ; wm g. This is equivalent to the problem of determining if v can be separated from fw 1 ; wm g by a plane, which in turn is equivalent to linear programming [27]. Therefore we can determine if v is convex by linear programming in linear time (see e.g. 28, 57, 80] If v is not a convex vertex, then v is not a local maximum for any direction, and we stop considering v. Otherwise, let h v be a plane that contains v and has w 1 ; wm to one side of ....

Dobkin, D., and S. Reiss, The Complexity of Linear Programming. Theoretical Computer Science, 11, pp. 1--18, 1980.

....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 ....

....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 reduction in NC 1 . Since LP and LI are ....

D. P. Dobkin. and S. Reiss. The complexity of linear programming. Theoretical Computer Science, 11(1):1--18, 1980.

....we first determine if v is a convex vertex. This is the case if and only if v is an extremal point in the set fv; w 1 ; wm g. This is equivalent to the problem of determining if v can be separated from fw 1 ; wm g by a plane, which in turn is equivalent to linear programming [7]. Therefore we can determine if v is convex by linear programming in linear time (see e.g. 8, 16, 25] If v is not a convex vertex, then v is not a local maximum for any direction, and we stop considering v. Otherwise, let h v be a plane that contains v and has w 1 ; wm to one side of ....

Dobkin, D., and S. Reiss, The Complexity of Linear Programming. Theoretical Computer Science, 11 (1980), pp. 1--18.

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