Qubits Europe 2019 DLR Knowledge for Tomorrow Flight Gate Assignment with a Quantum Annealer Elisabeth Lobe, Tobias Stollenwerk High Performance Computing Simulation and Software Technology DLR German Aerospace Center 26th March 2019
Qubits Europe 2019
DLR
Knowledge for Tomorrow
Flight Gate Assignment with a Quantum Annealer
Elisabeth Lobe, Tobias Stollenwerk
High Performance ComputingSimulation and Software TechnologyDLR German Aerospace Center
26th March 2019
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Algorithmic Quantum Computing Research at DLR
Quantum Optimization Algorithms
Quantum Compiling
Embedding strategies for Quantum Annealing
Complete graph in broken Chimera
Weight distribution problem
DLR
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Aerospace Applications at DLR
for Quantum Annealing
Air Traffic ManagementSatellite Telemetry VerificationEarth Observation Mission PlanningFlight Gate Assignment
for Gate-Based Quantum Computing
QAOA for scheduling problemsHHL for Radar Cross SectionQuantum Simulation for Battery Research
DLR
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Flight Gate Assignment
A day at Frankfurt Airport
about 1300 aircraft movements (arrival and departure)
more than 90% are passenger flights
more than 170000 passengers
about 60% transfer passengers
278 gates
DLR
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Passenger Flows
�i
gate γ
Tαγ
gate α
Tαβ
gate β
securitypoint
baggageclaim
�j �k
tdepα tarr
α
Nji Nik
tini touti
�`
tin`tbuff
narrindep
i
F , G sets of flights and gates
ndep/arri passengers which depart/ arrive with flight i
Nij transfer passengers from flight i to j
tin/outi arrival/departure time of flight i
Tαβ average time to get from gate α to β
tdep/arrα average time to arrive at/ leave from gate α
tbuff buffer time between two flights at the same gate
Which flight should be assigned towhich gate, such that the total transit
time of the passengers is minimal?
A : F → G
DLR
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FGA Binary Program �i
γ
Nik·Tαγ
α
Tαβ ·Nji
β
� �
�j �k
tdepα ·n
depi
tdepα ·n
depi
narri ·t
arrα
narri ·t
arrα
Variables x ∈ {0, 1}F×G with
xiα =
{1, if flight i takes gate α,
0, otherwise
Minimizing the total transfer time with objective function
T (x) = Tarr(x) + Tdep(x) + Ttransfer(x)
=∑iα
narri t
arrα xiα +
∑iα
ndepi tdep
α xiα +∑ijαβ
NijTαβ xiα xjβ
=∑iα
narri t
arrα xiα +
∑iα
ndepi tdep
α xiα︸ ︷︷ ︸linear
+∑ijαβ
NijTαβ xiα xjβ︸ ︷︷ ︸quadratic
⇒ Quadratic Assignment Problemfundamental problem in combinatorial optimization, NP-hard
seems to exploit possible advantages of the D-Wave machine
DLR
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Constraints and Penalty Terms
1. One gate per flight∑α
xiα = 1 ∀i ∈ F
2. Different gates if standing times of two flights overlap forbidden pairs
P ={
(i, j) ∈ F 2 : tini < tinj < touti + tbuff
}xiα + xjα ≤ 1 ⇔ xiα · xjα = 0 ∀(i, j) ∈ P ∀α ∈ G
⇒ Penalty terms Cone(x) =∑i
(∑α
xiα − 1
)2
,
Cnot(x) =∑α
∑(i,j)∈P
xiαxjα where Cone/not
{> 0, if constraint is violated
= 0, if constraint is fulfilled
DLR
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QUBO with Penalty Weights
Q(x) = T (x) + λoneCone(x) + λnotCnot(x)
Need to ensure that a solution always fulfills constraints, hence ∆C > ∆T
⇒ Comparing coefficients in worst cases for
not assigning a flight to any gate
λone > maxi,α
(n
depi tdep
α + narri t
arrα +max
βTαβ
∑j
Nij
)assigning a pair of forbidden flights to the same gate
λnot > maxi,α,γ
(n
depi tdep
α − ndepi tdep
γ + narri t
arrα − narr
i tarrγ +max
β
(Tαβ − Tγβ
)∑j
Nij
)
⇒ Refinement by bisection of weights yielding valid or invalid solutions
DLR
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Airport Data
M. Jung et al. (DLR-FW)
Flight schedule for one day from a mid-sized European airport
Passenger flow from agent-based simulation of Martin Jung
Extracted total instance: 293 flights and 97 gates
⇒ Over 28000 binary variables with about 400 Mio. couplings
DLR
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Instance Preprocessing
0 20000 40000 60000time in seconds
0
50
100
150
200
250
Flig
hts
flights with transfer passengers
Splitting too long on-block timesReducing to only flights with transfersExtracting connected subgraphsFurther slicing of largest
subgraph randomly
⇒163 instances:3 to 16 flights
2 to 16 gates
DLR
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Bin Packing
2 4 6 8 10N
1.0
1.5
2.0
2.5
3.0
3.5
R
50 %90 %99 %100 %
Maximum coefficient ratio of QUBO CQ =maxij |Qij |minij |Qij |
Reducing maximum coefficient ratio to overcome precisionissues
Tαβ , tarrα , t
depα → {0, 1, ...,T } with T ∈ {2, 3, 6, 10}
Nij , narri , n
depi → {0, 1, ...,N} with N ∈ {2, 3, 6, 10}
⇒ CQ � CQ
Approximation ratio (solved with SCIP)
R =Q(argminxQ(x))
minxQ(x)
⇒ Little effect on solution quality
DLR
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Annealing Setup
10 8 6 4 2 0JF
0.0
0.2
0.4
0.6
p
25 %50 %75 %
EmbeddingQuadratic overhead
Up to 84 logical qubits
(#Variables = #Flights ·#Gates)
Intra-logical coupling (JF)Influences success probability p
Best option by scanning: -1 in units of largest coefficient
(Standard) Run parametersAnnealing time 20µs with 1000 runs
Majority voting
DLR
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Annealing Results
0 200 400 600 800 1000Maximum Ising coefficient ratio CI
0.0
0.2
0.4
0.6
0.8
1.0
Succ
ess
Pro
babili
ty p
3 4 5 6 7Number of flights |F|
100
200
300
400
500
T99 (
)
s
25 %50 %75 %
3 4 5 6 7Number of flights |F|
50
100
150
200
CI
25 %50 %75 %
QUBO to Ising transformationincreases maximum coefficient ratio significantly
⇒ large Ising coefficients suppress success probability
Time to solution with 99% certainty T99 = log1−p(1− 0.99)Tanneal
grows with problem size→ because of larger coefficients?
due to small problem sizes asymptotic behaviour unclear
DLR
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Summary
Flight gate assignment is amenable to QA
Precision issues due to large coefficients
Mitigate limited precision by bin packing
Open questions:
How to recombine partial solutions?
How would larger instance perform?
Are these instances hard for classical solvers?
DLR
DLR
Knowledge for Tomorrow
Questions? Related Article:
Elisabeth Lobe Flight Gate Assignment with a Quantum AnnealerT. Stollenwerk, E. Lobe and M. Jung, QTOP, Springer, 2019High Performance Computing
Simulation and Software TechnologyDLR German Aerospace Center