166 research outputs found
Scheduling over Scenarios on Two Machines
We consider scheduling problems over scenarios where the goal is to find a
single assignment of the jobs to the machines which performs well over all
possible scenarios. Each scenario is a subset of jobs that must be executed in
that scenario and all scenarios are given explicitly. The two objectives that
we consider are minimizing the maximum makespan over all scenarios and
minimizing the sum of the makespans of all scenarios. For both versions, we
give several approximation algorithms and lower bounds on their
approximability. With this research into optimization problems over scenarios,
we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at
link.springer.co
From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz
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
An electromagnetism-like method for the maximum set splitting problem
In this paper, an electromagnetism-like approach (EM) for solving the maximum set splitting problem (MSSP) is applied. Hybrid approach consisting of the movement based on the attraction-repulsion mechanisms combined with the proposed scaling technique directs EM to promising search regions. Fast implementation of the local search procedure additionally improves the efficiency of overall EM system. The performance of the proposed EM approach is evaluated on two classes of instances from the literature: minimum hitting set and Steiner triple systems. The results show, except in one case, that EM reaches optimal solutions up to 500 elements and 50000 subsets on minimum hitting set instances. It also reaches all optimal/best-known solutions for Steiner triple systems
A Variable Neighborhood Search Approach for Solving the Maximum Set Splitting Problem
This paper presents a Variable neighbourhood search (VNS)
approach for solving the Maximum Set Splitting Problem (MSSP). The algorithm forms a system of neighborhoods based on changing the component for an increasing number of elements. An efficient local search procedure swaps the components of pairs of elements and yields a relatively short running time. Numerical experiments are performed on the instances known in the literature: minimum hitting set and Steiner triple systems. Computational results show that the proposed VNS achieves all optimal or best known solutions in short times. The experiments indicate that the VNS compares favorably with other methods previously used for solving the MSSP. ACM Computing Classification System (1998): I.2.8
Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails
The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves)
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