Digital currency is primarily designed on problems that are computationally hard to solve using traditional computing techniques. However, these problems are now vulnerable due to the computational power of quantum co...
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Digital currency is primarily designed on problems that are computationally hard to solve using traditional computing techniques. However, these problems are now vulnerable due to the computational power of quantum computing. For the postquantum computing era, there is an immense need to reinvent the existing digital security measures. Problems that are computationally hard for any quantum computation will be a possible solution to that. This research summarizes the current security measures and how the new way of solving hard problems will trigger the future protection of the existing digital currency from the future quantum threat.
Consider the supposedly simple problem of computing a point in a convex set that is conveyed by a separation oracle with no further information (e. g., no domain ball containing or intersecting the set, etc.). The aut...
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Consider the supposedly simple problem of computing a point in a convex set that is conveyed by a separation oracle with no further information (e. g., no domain ball containing or intersecting the set, etc.). The authors' interest in this problem stems from fundamental issues involving the interplay of (i) the computational complexity of computing a point in the set, (ii) the geometry of the set, and (iii) the stability or conditioning of the set under perturbation. Under suitable definitions of these terms, the authors show herein that problem instances with favorable geometry have favorable computational complexity, validating conventional wisdom. The authors also show a converse of this implication by showing that there exist problem instances that require more computational effort to solve in certain families characterized by unfavorable geometry. This in turn leads, under certain assumptions, to a form of equivalence among computational complexity, geometry, and the conditioning of the set. The authors' measures of the geometry, relative to a given reference point, are based on the radius of a certain domain ball whose intersection with the set contains a certain inscribed ball. The geometry of the set is then measured by the radius of the domain ball, the radius of the inscribed ball, and the ratio between these two radii, the latter of which is called the aspect ratio. The aspect ratio arises in the analysis of many algorithms for convex problems, and its importance in convex algorithm analysis has been well-known for several decades. However, the presence in our bound of terms involving only the radius of the domain ball and only the radius of the inscribed ball are a bit counterintuitive;nevertheless, we show that the computational complexity must involve these terms in addition to the aspect ratio, even when the aspect ratio itself is small. This lower-bound complexity analysis relies on simple features of the separation oracle model;if we instead assume
We study the complexity of query satisfiability and entailment for the Boolean Information Retrieval models WP and AWP using techniques from propositional logic and computational complexity. WP and AWP can be used to ...
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We study the complexity of query satisfiability and entailment for the Boolean Information Retrieval models WP and AWP using techniques from propositional logic and computational complexity. WP and AWP can be used to represent and query textual information under the Boolean model using the concept of attribute with values of type text, the concept of word, and word proximity constraints. Variations of WP and AWP are in use in most deployed digital libraries using the Boolean model, text extenders for relational database systems (e.g., Oracle 10g), search engines, and P2P systems for information retrieval and filtering.
We show that piecewise-linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point. Our e...
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We show that piecewise-linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point. Our examples are based on an example of Murty exhibiting exponential growth for Lemke's algorithm for the linear complementarity problem.
The problem of computing the meet over all paths(MOP) solution in a monotone data flow framework over an infinite meet semilattice is generally undecidable[1]. Hence, the maximum fixed point(MFP) solution, which is po...
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The problem of computing the meet over all paths(MOP) solution in a monotone data flow framework over an infinite meet semilattice is generally undecidable[1]. Hence, the maximum fixed point(MFP) solution, which is polynomial time computable on semi-lattices of finite height, is generally used in practice for program analysis questions in monotone data flow frameworks. However, we show that if the semi-lattice is finite, computing MOP solution is NL-complete with respect to log space reductions, which implies parallelizability and polynomial time computability. It is also shown that the problem of computing the maximum fixed point(MFP) solution is P-complete with respect to log space reductions, and hence not efficiently parallelizable, even when the flow graph is directed acyclic and the semilattice has just four elements. These results appear in contrast with the fact that when the semilattice is not finite, solving the MOP problem is significantly harder than MFP. (C) 2021 Elsevier B.V. All rights reserved.
A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certai...
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A priori estimation is presented for a computational complexity of the homotopy method applying to a certain class of strongly monotone nonlinear equations. In the present papers, a condition is presented for a certain class of uniquely solvable equations, under which an upper bound of a computational complexity of the Newton type homotopy method can be a priori estimated. In this paper, a condition is considered in a case of linear homotopy equations including the Newton type homotopy equations. In the first place, the homotopy algorithm based on the simplified Newton method is introduced. Then by using Urabe type theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, a condition is presented under which an upper bound of a computational complexity of the algorithm can be a priori estimated, when it is applied to a certain class of strongly monotone nonlinear equations. The presented condition is demonstrated by numerical experiments.
The dialectical frameworks (ADFs) have recently been proposed as a versatile generalization of Dung's abstract argumentation frameworks (AFs). In this paper, we present a comprehensive analysis of the computationa...
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The dialectical frameworks (ADFs) have recently been proposed as a versatile generalization of Dung's abstract argumentation frameworks (AFs). In this paper, we present a comprehensive analysis of the computational complexity of ADFs. Our results show that while ADFs are one level up in the polynomial hierarchy compared to AFs, there is a useful subclass of ADFs which is as complex as AFs while arguably offering more modeling capacities. As a technical vehicle, we employ the approximation fixpoint theory of Denecker, Marek and Truszczynski, thus showing that it is also a useful tool for complexity analysis of operator-based semantics. (C) 2015 Elsevier B.V. All rights reserved.
We show that for several solution concepts for finite n-player games, where n >= 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the...
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We show that for several solution concepts for finite n-player games, where n >= 3, the task of simply verifying its conditions is computationally equivalent to the decision problem of the existential theory of the reals. This holds for trembling hand perfect equilibrium, proper equilibrium, and CURB sets in strategic form games and for ( the strategy part of) sequential equilibrium, trembling hand perfect equilibrium, and quasi-perfect equilibrium in extensive form games of perfect recall. For obtaining these results we first show that the decision problem for the minmax value in n-player games, where n >= 3, is also equivalent to the decision problem for the existential theory of the reals. Our results thus improve previous results of NP-hardness as well as SQRT-SUM-hardness of the decision problems to completeness for there exists R, the complexity class corresponding to the decision problem of the existential theory of the reals. As a byproduct we also obtain a simpler proof of a result by Schaefer and Stefankovic giving there exists R-completeness for the problem of deciding existence of a probability constrained Nash equilibrium.
By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision prob...
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By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision problem on the existence of locally injective homomorphisms to Theta graphs with three distinct odd path lengths. (C) 2007 Elsevier B.V. All rights reserved.
We study the computational complexity of decision problems about Nash equilibria in m-player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the exist...
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We study the computational complexity of decision problems about Nash equilibria in m-player games. Several such problems have recently been shown to be computationally equivalent to the decision problem for the existential theory of the reals, or stated in terms of complexity classes,.R-complete, when m >= 3. We show that, unless they turn into trivial problems, they are there exists R-hard even for 3-player zero-sum games. We also obtain new results about several other decision problems. We show that when m >= 3 the problems of deciding if a game has a Pareto optimal Nash equilibrium or deciding if a game has a strong Nash equilibrium are there exists R-complete. The latter result rectifies a previous claim of NP-completeness in the literature. We show that deciding if a game has an irrational valued Nash equilibrium is there exists R-hard, answering a question of Bilo and Mavronicolas, and address also the computational complexity of deciding if a game has a rational valued Nash equilibrium. These results also hold for 3-player zero-sum games. Our proof methodology applies to corresponding decision problems about symmetric Nash equilibria in symmetric games as well, and in particular our new results carry over to the symmetric setting. Finally we show that deciding whether a symmetric m-player game has a non-symmetric Nash equilibrium is there exists R-complete when m >= 3, answering a question of Garg, Mehta, Vazirani, and Yazdanbod.
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