55 research outputs found

    From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

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    The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

    Complexity of coalition structure generation

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    We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds.Comment: 17 page

    Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph from the Small Set Expansion Hypothesis

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    The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph GG, find a complete bipartite subgraph of GG with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph GG, find a balanced complete bipartite subgraph of GG with maximum number of vertices. - Minimum kk-Cut: given a weighted graph GG, find a set of edges with minimum total weight whose removal partitions GG into kk connected components. - Densest At-Least-kk-Subgraph (DALkkS): given a weighted graph GG, find a set SS of at least kk vertices such that the induced subgraph on SS has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP ⊈\nsubseteq BPP, no polynomial time algorithm gives n1−εn^{1 - \varepsilon}-approximation for MEB or MBB for every constant ε>0\varepsilon > 0. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum kk-Cut and DALkkS to within (2−ε)(2 - \varepsilon) factor of the optimum for every constant ε>0\varepsilon > 0. The ratios in our results are essentially tight since trivial algorithms give nn-approximation to both MEB and MBB and efficient 22-approximation algorithms are known for Minimum kk-Cut [SV95] and DALkkS [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

    Inapproximability of Combinatorial Optimization Problems

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    We survey results on the hardness of approximating combinatorial optimization problems
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