A1258150189yjgFk

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From discrete tomography to maintenance
scheduling
D. de Werra Ecole Polytechnique Fédérale de
Lausanne EPFL
Joint work with C. Bentz, M.C. Costa, C.
Picouleau (CNAM, Paris) B. Ries (EPFL)
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Table of contents

1. Discrete tomography
2. An extension to scheduling
3. Solvable cases
4. Less solvable cases
5. Variations and extensions
6. No conclusion

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1. Discrete tomography
Image (mxn) array A m.n pixels
n columns
color of pixel (i,j) aij 1,2,,or k line row
or column hij pixels with color j
in line i Problem how to
reconstruct the image?




mrows
A
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Graph model
Pi
h(Pl) (3 1 1)
Pl
G (V,E) k colorsP collection of chains Pi
(sets of vertices) H collection of vectors hi
(hi1,hik) hij vertices in Pi
with color j
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Problem ?(G,k, P, H )Find partition V1,, Vk
of vertex set Vsuch that Vj nPi hij
?i,j
NB V1,,Vk not a proper k-coloring ! If V1,Vk
required to be proper
then ?(G,k,P,H)
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2. Extension to scheduling
Stations to be renovated(one month for each)
k months available
hij stations on line i to be closed in month
j for maintenance (renovation)
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Problem Find a schedule of maintenance
in k months
Schedule V1,,Vk Vj stations renovated

in month j
N.B. If no two adjacent stations closed the
same month
?(G,k,P,H)
proper coloring!
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3. Some solvable cases
special family P (Pi i 1,,p) P nested
? Pr, Ps in P Pr ? Ps or Ps ? Pr

or Pr n Ps ØNesticity Nest (P) min number
of nested families
covering P
? polynomial algorithm to find whether Nest (P)
2
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?(G, k 2, P, H) is easy if Nest (P) 2
V 1 2 3 4 5 6 7P (12, 345, 67,
13567, 136, 24, 57)
Compatible flow from a to bgives V1 V2 V V1
N.B. NP-complete if Nest (P) 3
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Cover index of Pc(P) max nb of Pis covering
a vertex
?(G,k 2, P, H) easy if c (P) 2
b-matching in GPi has degree hi1
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If G is a tree with oriented arcs Pi ?
P oriented paththen ?(G, k 2, P, H) is
polynomially solvable
N.B. G need not be anarborescence (rooted tree)
Solution LP with totally unimodular matrix
If G is any graph and Pi ? 2 ? Pi ? P
then ?(G, k, P, H) is polynomially solvable
N.B. ?(G, k 3, P, H) NP-complete if Pi
3 ? Pi ? P, c(P) 2, hij 1 ? i,j
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V1 x x true 2 SAT
V2 x x false
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If G is a tree and Pi n Pf ? 1 ? i
?fthen ?(G, k, P, H) is polynomially solvable
Propagation algorithm
For proper colorings
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?(G, k 3, P, H) is NP-complete if G
graph with vertex v2 G v2 is bipartite
and Pi ? 3 ? i Pi n Pf ? 1 ? i ?f
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5. Variations and extensions
?(G, k, P, H) find E1, E2,,Ek (partition of
E) with Ej n Pi hij ? i,
j
?(G, k, P, H) same with proper edge k-coloring
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?(G, k, P, H) with Pi ? 2 ?i is polynomially
solvable
?(G, k 2, P, H) NP-complete with G tree with
?(T) 3 and Pi chain or bundle of edges
?(G, k 2, P, H) NP-complete if G triangulated
cactus with ?(G) 3 and Pi chain ? i
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6. No conclusion
Further research is needed
special classes of graphs grid graphs
close to tomography!