Structure of Polynomial-Time Approximation

Leeuwen, Erik Jan van; Leeuwen, Jan van

doi:10.1007/s00224-011-9366-z

Cited by 3 publications

(1 citation statement)

References 40 publications

(73 reference statements)

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“…Also in the top two panels, we note that the gradual flattening out of the SA convergence curve -in the sense that doubling work is much less effective at large w than at small w -is not unexpected in the context of NP-hard problems. The existence of an approximation method that achieves polynomial improvements in solution quality with only polynomial extra work in the worst case (called an FPTAS), would imply that P = N P [60].…”

Section: Classical Performance Driversmentioning

confidence: 99%

Milestones on the Quantum Utility Highway

McGeoch

Farré

2022

2022 IEEE/ACM 7th Symposium on Edge Computing (SEC)

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We introduce quantum utility, a new approach to evaluating quantum performance that aims to capture the user experience by including overhead costs associated with the quantum computation. A demonstration of quantum utility by a quantum processing unit (QPU) shows that the QPU can outperform classical solvers at some tasks of interest to practitioners, when considering computational overheads. We consider overhead costs that arise in standalone use of the QPU (as opposed to a hybrid computation context). We define three early milestones on the path to broad-scale quantum utility that focus on restricted subsets of overheads: Milestone 0 considers pure anneal time (no overheads) and has been demonstrated in previous work; Milestone 1 includes overhead times to access the QPU (that is, programming and readout); and Milestone 2 incorporates an indirect cost associated with minor embedding.We evaluate the performance of a D-Wave Advantage QPU with respect to Milestones 1 and 2, using a testbed of 13 input classes and seven classical solvers implemented on CPUs and GPUs. For Milestone 1, the QPU outperformed all classical solvers in 99% of our tests. For Milestone 2, the QPU outperformed all classical solvers in 19% of our tests, and the scenarios in which the QPU found success correspond to cases where classical solvers most frequently failed.Analysis of test results on specific inputs reveals fundamentally distinct underlying mechanisms that explain the observed differences in quantum and classical performance profiles. We present evidence-based arguments that these distinctions bode well for future annealing quantum processors to support demonstrations of quantum utility on ever-expanding classes of inputs and for more challenging milestones.

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Section: Classical Performance Driversmentioning

confidence: 99%

Milestones on the Quantum Utility Highway

McGeoch

Farré

2022

2022 IEEE/ACM 7th Symposium on Edge Computing (SEC)

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Shortcutting directed and undirected networks with a degree constraint

Tan

Leeuwen

2017

Discrete Applied Mathematics

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a b s t r a c tShortcutting is the operation of adding edges to a network with the intent to decrease its diameter. We are interested in shortcutting networks while keeping degree increases in the network bounded, a problem first posed by Chung and Garey. Improving on a result of Bokhari and Raza we show that, for any δ ≥ 1, every undirected graph G can be shortcut in linear time to a diameter of at most O(log 1+δ n) by adding no more than O(n/ log 1+δ n) edges such that degree increases remain bounded by δ. The result extends an estimate due to Alon et al. for the unconstrained case. Degree increases can be limited to 1 at only a small extra cost. For strongly connected, bounded-degree directed graphs Flaxman and Frieze proved that, if ϵn random arcs are added, then the resulting graph has diameter O(ln n) with high probability. We prove that O(n/ ln n) edges suffice to shortcut any strongly connected directed graph to a graph with diameter less than O(ln n) while keeping the degree increases bounded by O(1) per node. The result is proved in a stronger, parameterized form. For general directed graphs with stability number α, we show that all distances can be shortcut to O(α⌈ln n α ⌉) by adding only 4n ln n/α + αφ edges while keeping degree increases bounded by at most O(1) per node, where φ is equal to the so-called feedback-dimension of the graph. Finally, we prove bounds for various special classes of graphs, including graphs with Hamiltonian cycles or paths. Shortcutting with a degree constraint is proved to be strongly NP-complete and W [2]-hard, implying that the problem is neither likely to be fixed-parameter tractable nor efficiently approximable unless FPT = W [2].

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Milestones on the Quantum Utility Highway: Quantum Annealing Case Study

McGeoch,

Farré

2023

ACM Trans. Quantum Comput.

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We introduce quantum utility , a new approach to evaluating quantum performance that aims to capture the user experience by considering the overhead costs associated with a quantum computation. A demonstration of quantum utility by the quantum processing unit (QPU) shows that the QPU can outperform classical solvers at some tasks of interest to practitioners, when considering the costs of computational overheads. A milestone is a test of quantum utility that is restricted to a specific subset of overhead costs and input types. We illustrate this approach with a benchmark study of a D-Wave annealing-based QPU versus seven classical solvers, for a variety of problems in heuristic optimization. We consider overhead costs that arise in standalone use of the D-Wave QPU (as opposed to a hybrid computation). We define three early milestones on the path to broad-scale quantum utility. Milestone 0 is the purely quantum computation with no overhead costs, and is demonstrated implicitly by positive results on other milestones. We evaluate performance of a D-Wave Advantage QPU with respect to milestones 1 and 2: For milestone 1, the QPU outperformed all classical solvers in 99% of our tests. For milestone 2, the QPU outperformed all classical solvers in 19% of our tests, and the scenarios in which the QPU found success correspond to cases where classical solvers most frequently failed. This approach isolating subsets of overheads for separate analysis reveals distinct mechanisms in quantum versus classical performance, which explain the observed differences in patterns of success and failure. We present evidence-based arguments that these distinctions bode well for annealing quantum processors to support demonstrations of quantum utility on ever-expanding classes of inputs and with more challenging milestones, in the very near future.

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Structure of Polynomial-Time Approximation

Cited by 3 publications

References 40 publications

Milestones on the Quantum Utility Highway

Milestones on the Quantum Utility Highway

Shortcutting directed and undirected networks with a degree constraint

Milestones on the Quantum Utility Highway: Quantum Annealing Case Study

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