7 research outputs found
Approximate Approximation on a Quantum Annealer
Many problems of industrial interest are NP-complete, and quickly exhaust
resources of computational devices with increasing input sizes. Quantum
annealers (QA) are physical devices that aim at this class of problems by
exploiting quantum mechanical properties of nature. However, they compete with
efficient heuristics and probabilistic or randomised algorithms on classical
machines that allow for finding approximate solutions to large NP-complete
problems. While first implementations of QA have become commercially available,
their practical benefits are far from fully explored. To the best of our
knowledge, approximation techniques have not yet received substantial
attention. In this paper, we explore how problems' approximate versions of
varying degree can be systematically constructed for quantum annealer programs,
and how this influences result quality or the handling of larger problem
instances on given set of qubits. We illustrate various approximation
techniques on both, simulations and real QA hardware, on different seminal
problems, and interpret the results to contribute towards a better
understanding of the real-world power and limitations of current-state and
future quantum computing.Comment: Proceedings of the 17th ACM International Conference on Computing
Frontiers (CF 2020
Computing maximum matchings in temporal graphs
Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms
Computing maximum matchings in temporal graphs.
Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ ∈ ℕ is given. The requirement that a vertex cannot be matched twice in any Δ-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms
FPT-Algorithms for the l-Matchoid Problem with Linear and Submodular Objectives
We design a fixed-parameter deterministic algorithm for computing a maximum
weight feasible set under a -matchoid of rank , parameterized by
and . Unlike previous work that presumes linear representativity of
matroids, we consider the general oracle model. Our result, combined with the
lower bounds of Lovasz, and Jensen and Korte, demonstrates a separation between
the -matchoid and the matroid -parity problems in the setting of
fixed-parameter tractability.
Our algorithms are obtained by means of kernelization: we construct a small
representative set which contains an optimal solution. Such a set gives us much
flexibility in adapting to other settings, allowing us to optimize not only a
linear function, but also several important submodular functions. It also helps
to transform our algorithms into streaming algorithms.
In the streaming setting, we show that we can find a feasible solution of
value and the number of elements to be stored in memory depends only on
and but totally independent of . This shows that it is possible to
circumvent the recent space lower bound of Feldman et al., by parameterizing
the solution value. This result, combined with existing lower bounds, also
provides a new separation between the space and time complexity of maximizing
an arbitrary submodular function and a coverage function in the value oracle
model