Computationally Related Problems

Abstract

Abstract. We look at several problems from areas such as network flows, game theory, artificial intelligence, graph theory, integer programming and nonlinear programming and show that they are \. related in that anyone of these problems is solvable in polynomial time iff all the others are, too. At i present, no polynomial time algorithm for these problems is known. These problems extend the equivalence class of problems known as P-Complete. The problem of deciding whether the class of 1 languages accepted by polynomial time nondeterministic Turing machines is the same as that accepted 1 by polynomial time deterministic Turing machines is related to P-Complete problems in that these two classes of languages are the same iff each P-Complete problem has a polynomial deterministic solution.l In view of this, it appears very likely that this equivalence class defines a class of problems that cannot be solved in deterministic polynomial time.

Asymptotic Perturbation Bounds for Probabilistic Model Checking with Empirically Determined Probability Parameters

“…First, quadratic bounds are computed based on linear bounds (or condition numbers). Second, we observe that the constraint kỹk ¼ 1 in Problems (4) and (5) Alternatively, we can show that computing quadratic bounds is in the complexity class of functional NP [23].…”

Integer equal flows

“…As mentioned in Section 1, Sahni [16] showed that the maximum integer equal flow problem is NP-hard. Later, Garey and Johnson [10] observed that the modification of a construction by Even et al [7] shows that the problem is NP-hard even if the capacity of every arc is 1.…”

“…The equal flow problem was first studied by Sahni [16] as a generalization of the traditional network flow problem. Its setup is similar to a standard maximum flow problem: we are given a directed graph G = (N, A) with capacities u a for all a ∈ A, and a designated source node s and sink node t. In addition, we are also given sets R 1 , R 2 , .…”

“…Sahni [16] proved that the maximum flow version of this problem is NP-hard with a reduction from NonTautology. Later, Even, Itai, and Shamir [7] showed via a reduction from Satisfiability that the problem remains NP-hard even if the capacity of each arc is 1.…”

Reducibility as a tool to extend the power of approximation algorithms the minimization of boolean expressions

“…The studies made by Karp and successively those of Sahni [5] and Ullman [7] have more and more enlarged this équivalence class which constitutes the top of complexity for the NP-problems.…”