Inspired by the CONFIDANT protocol (Buchegger and Boudec in Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking & Computing, pp. 226-236, 2002), we define and study a basic reputation-ba...
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Inspired by the CONFIDANT protocol (Buchegger and Boudec in Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking & Computing, pp. 226-236, 2002), we define and study a basic reputation-based protocol in multihop wireless networks with selfish nodes. Its reputation mechanism is implemented through the ability of any node to define a threshold of tolerance for any of its neighbors, and to cut the connection to any of these neighbors that refuse to forward an amount of flow above that threshold. The main question we would like to address is whether one can set the initial conditions so that the system reaches an equilibrium state where a non-zero amount of every commodity is routed. This is important in emergency situations, where all nodes need to be able to communicate even with a small bandwidth. Following a standard approach, we model this protocol as a game, and we give necessary and sufficient conditions for the existence of non-trivial Nash equilibria. Then we enhance these conditions with extra conditions that give a set of necessary and sufficient conditions for the existence of connected Nash equilibria. We note that it is not always necessary for all the flow originating at a node to reach its destination at equilibrium. For example, a node may be using unsuccessful flow in order to effect changes in a distant part of the network that will prove quite beneficial to it. We show that we can decide in polynomialtime whether there exists a (connected) equilibrium without unsuccessful flows. In that case we calculate (in polynomialtime) initial values that impose such an equilibrium on the network. On the negative side, we prove that it is NP-hard to decide whether a connected equilibrium exists in general (i.e., with some nodes using unsuccessful flows at equilibrium).
Let F-q be a finite field of characteristic p with q = p(v) elements, g a primitive element, a not equal 0 an arbitrary element of F-q, and x = log g a the discrete logarithm of a to the base g. In this paper we consi...
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Let F-q be a finite field of characteristic p with q = p(v) elements, g a primitive element, a not equal 0 an arbitrary element of F-q, and x = log g a the discrete logarithm of a to the base g. In this paper we consider the discrete logarithm problem in the case when q equivalent to 1( mod 4) and put forward a deterministic algorithm computing the first k <= c log n digits x(0), x(1),..., x(k), k < n, in the binary expansion x = x(0) + x(1)2+ x(2)2(2) + - - - + x(n)2(n) of x in a polynomialtime.
A graph G is called 2-edge-Hamiltonian-connected if for any X subset of {x(1)x(2) : x(1), x(2) is an element of V(G)} with 1 <= vertical bar X vertical bar <= 2, G boolean OR X has a Hamiltonian cycle containing...
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A graph G is called 2-edge-Hamiltonian-connected if for any X subset of {x(1)x(2) : x(1), x(2) is an element of V(G)} with 1 <= vertical bar X vertical bar <= 2, G boolean OR X has a Hamiltonian cycle containing all edges in X, where G boolean OR X is the graph obtained from G by adding all edges in X. In this paper, we show that every 4-connected plane graph is 2-edge-Hamiltonian-connected. This result is best possible in many senses and an extension of several known results on Hamiltonicity of 4-connected plane graphs, for example, Tutte's result saying that every 4-connected plane graph is Hamiltonian, and Thomassen's result saying that every 4-connected plane graph is Hamiltonian-connected. We also show that although the problem of deciding whether a given graph is 2-edge-Hamiltonian-connected is NP-complete, there exists a polynomialtime algorithm to solve the problem if we restrict the input to plane graphs. (C) 2013 Elsevier Ltd. All rights reserved.
It is well known (cf. Krajicek and Pudlak ['Propositional proof systems, the consistency of first order theories and the complexity of computations', J. Symbolic Logic 54 (1989) 1063-1079]) that a polynomial t...
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It is well known (cf. Krajicek and Pudlak ['Propositional proof systems, the consistency of first order theories and the complexity of computations', J. Symbolic Logic 54 (1989) 1063-1079]) that a polynomialtime algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1((k)) and outputs a tautology tau(k) of size at least k such that P is not p-bounded on the set of all formulas tau(k). We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP not equal coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT boolean AND{0, 1}(k). In Find, given 1((k)) and a tautology alpha of size at most k(0)(c), one should output a size k tautology beta that has no size k(1)(c) P-proof from substitution instances of alpha. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2(O(k)) algorithm. Using a stronger hypothesis about the proof complexity of the Nisan-Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan-Wigderson generator as a proof system.
We give an explicit family of polynomial maps called centre unstable Henon-like maps and prove that they exhibit blenders for some parameter values. Using this family, we also prove the occurrence of blenders near cer...
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We give an explicit family of polynomial maps called centre unstable Henon-like maps and prove that they exhibit blenders for some parameter values. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.
We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems...
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We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call TREE CONTRACTION and PATH CONTRACTION, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in 0(n + m) and 0(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs. (C) 2013 Elsevier B.V. All rights reserved.
We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems...
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We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call TREE CONTRACTION and PATH CONTRACTION, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in 0(n + m) and 0(nm) time, respectively. As a contrast, both problems remain NP-complete when restricted to bipartite input graphs. (C) 2013 Elsevier B.V. All rights reserved.
Boolean function f is k-interval if - input vector viewed as n-bit number - f is true for and only for inputs from given (at most) k intervals. Recognition of k-interval fuction given its DNF representation is coNP-ha...
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Boolean function f is k-interval if - input vector viewed as n-bit number - f is true for and only for inputs from given (at most) k intervals. Recognition of k-interval fuction given its DNF representation is coNP-hard problem. This thesis shows that for DNFs from a given solvable class (class C of DNFs is solvable if we can for any DNF F ∈ C decide F ≡ 1 in polynomialtime and C is closed under partial assignment) and fixed k we can decide whether F represents k-interval function in polynomialtime. 1
A compression algorithm is introduced for multideterminant wave functions which can greatly reduce the number of determinants that need to be evaluated in quantum Monte Carlo calculations. We have devised an algorithm...
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A compression algorithm is introduced for multideterminant wave functions which can greatly reduce the number of determinants that need to be evaluated in quantum Monte Carlo calculations. We have devised an algorithm with three levels of compression, the least costly of which yields excellent results in polynomialtime. We demonstrate the usefulness of the compression algorithm for evaluating multideterminant wave functions in quantum Monte Carlo calculations, whose computational cost is reduced by factors of between about 2 and over 25 for the examples studied. We have found evidence of sublinear scaling of quantum Monte Carlo calculations with the number of determinants when the compression algorithm is used.
Grover’s quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand wha...
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Grover’s quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover’s algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ.
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