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A bounded-error quantum polynomial-time algorithm for two graph bisection problems

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Abstract

The aim of the paper was to propose a bounded-error quantum polynomial-time algorithm for the max-bisection and the min-bisection problems. The max-bisection and the min-bisection problems are fundamental NP-hard problems. Given a graph with even number of vertices, the aim of the max-bisection problem is to divide the vertices into two subsets of the same size to maximize the number of edges between the two subsets, while the aim of the min-bisection problem is to minimize the number of edges between the two subsets. The proposed algorithm runs in \(O(m^2)\) for a graph with m edges and in the worst case runs in \(O(n^4)\) for a dense graph with n vertices. The proposed algorithm targets a general graph by representing both problems as Boolean constraint satisfaction problems where the set of satisfied constraints are simultaneously maximized/minimized using a novel iterative partial negation and partial measurement technique. The algorithm is shown to achieve an arbitrary high probability of success of \(1-\epsilon \) for small \(\epsilon >0\) using a polynomial space resources.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria, Egypt

    Ahmed Younes

  2. School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK

    Ahmed Younes

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  1. Ahmed Younes
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Correspondence to Ahmed Younes.

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Younes, A. A bounded-error quantum polynomial-time algorithm for two graph bisection problems. Quantum Inf Process 14, 3161–3177 (2015). https://doi.org/10.1007/s11128-015-1069-y

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