Extending a well-known property of np optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples ov...
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Extending a well-known property of np optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all np optimization problems admit a logical characterization. It is shown that any npoptimization problem can be stated as a problem in which the constraint conditions can be expressed by a Pi(2) first-order formula. The paper analyzes the weighted analogue of all syntactically defined classes of optimizationproblems that are known to have good approximation properties in the nonweighted case. Dramatic changes occur when negative weights are allowed.
The Quantum Approximate optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization *** consists of interleaved unitary transformations induced by two operat...
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The Quantum Approximate optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization *** consists of interleaved unitary transformations induced by two operators labelled the mixing and problem *** fit this framework,one needs to transform the original problem into a suitable form and embed it into these two *** this paper,for the well-known np-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem ***,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other *** a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s *** argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to np optimization problems.
In this paper, we consider systems that can be modeled as directed acyclic graphs such that nodes represent components of the system and directed edges represent fault propagation between components. Some components c...
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In this paper, we consider systems that can be modeled as directed acyclic graphs such that nodes represent components of the system and directed edges represent fault propagation between components. Some components can be equipped with alarms that ring when they detect faulty (abnormal) behavior. We study algorithms that attempt to minimize the number of alarms to be placed so that a fault at any single component can be detected and uniquely diagnosed. We first show that the minimization problem is intractable, i.e., np-hard, even when restricted to three level graphs in which all nodes have outdegree two or less. We present optimal algorithms for three special classes of graphs - tree structured graphs, single-entry single-exit series-parallel graphs and two level graphs. We then present a polynomial-time approximation algorithm for the general case which guarantees that the ratio of the number of alarms placed to the optimum required is within a factor that is logarithmic in the number of nodes in the graph. Moreover, by showing a reduction from the minimum dominating set problem to the minimum alarm set problem, we argue that this performance guarantee is tight to within a constant factor. Finally, we demonstrate the connection between the minimum alarm set problem and the minimum test collection problem, and prove similar results. (C) 2000 Elsevier Science B.V. All rights reserved.
In recent years the technological limits inherently present in the classical Turing paradigm of computation have sparked the development of innovative solutions based on quantum devices or analog-digital mixed approac...
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In recent years the technological limits inherently present in the classical Turing paradigm of computation have sparked the development of innovative solutions based on quantum devices or analog-digital mixed approaches often based on the time evolution of differential equations. Such promising machinery require accurate analysis to understand if and howthey will be able to perform better than classical approaches in solving hard optimizationproblems. Here we challenge two machines representative of the quantum annealing and differential equations approaches, namely D-Wave and Memcomputing by devising a benchmark of three well known hard optimizationproblems from the realms of number theory, optimal transport and optimal scheduling. We introduce the Mean First Solution Time, a novel metric for accurately comparing performances, and take as baseline the classical Gurobi solver. We show that performances of both solvers are heavily dependent on the selected set of internal parameters. Results shed lights on the advantages and current limits of each paradigm and give a perspective on possible future developments.
The Quantum Approximate optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimizationproblems, derived from an approximation to the Quantum Adiabatic Algori...
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The Quantum Approximate optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimizationproblems, derived from an approximation to the Quantum Adiabatic Algorithm (QAA). In solving combinatorial optimizationproblems with constraints in the context of QAOA, one needs to find a way to encode problem constraints into the scheme. In this paper, we propose and discuss sev-eral QAOA-based algorithms to solve combinatorial optimizationproblems with equality and/or inequality constraints. We formalize the encoding method of different types of con-straints, and demonstrate the effectiveness and efficiency of the proposed scheme by pro-viding examples and results for some well-known np optimization problems. Compared to previous constraint-encoding methods, we argue our work leads to a more generalized framework for finding, in the context of QAOA, higher-quality approximate solutions to combinatorial problems with various types of constraints.(c) 2022 Elsevier Inc. All rights reserved.
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