Royal Holloway
University of London
Gregory Gutin
Department of Computer Science
Joint work with A. Rafiey, A. Yeo
(RHUL) and M. Tso (Man. U.)
www.cs.rhul.ac.uk/home/gutin/
Level of Repair Analysis
and Minimum Cost
Homomorphisms of Graphs
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA
• Level of Repair Analysis (LORA): procedure for
defence logistics
• Complex system with thousands of assemblies,
sub-assemblies, components, etc.
• Has λ ≥2 levels of indenture and with r ≥ 2
repair decisions (λ=2,r=3: UK and USA military,
λ=2,r=5: French military)
• LORA: optimal provision of repair and
maintenance facilities to minimize overall lifecycle costs
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-BR
• Introduced and studied by Barros (1998)
and Barros and Riley (2001) who designed
branch-and-bound heuristics for LORA-BR
• We showed that LORA-BR is polynomialtime solvable
• We proved it by reducing LORA-M via
graph homomorphisms to the max weight
independent set problem on bipartite
graphs (see the paper)
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-BR Formulation-1
• λ=2: Subsystems (S) and Modules (M)
• A bipartite graph G=(S,M;E): sm ε E iff module m
is in subsystem s
• r=3 available repair decisions (for each s and
m): “discard”, “local repair”, “central repair”:
D,L,C (subsystems) and d,l,c (modules).
• Costs (over life-cycle) ci(s), ci(m) of prescribing
repair decision i for subsystem s, module m,
resp.
• The use of any repair decision i incurs a cost ci
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-BR Formulation-2
• We wish to minimize the total cost by
choosing a subset of the six repair
decisions and assigning available repair
options to the subsystems and modules
subject to: R1: Ds → dm, R2: lm → Ls
• For a pair of graphs B and H, a mapping k:
V(B) → V(H) is called a homomorphism of
B to H if xy ε E(B) implies k(x)k(y) ε E(H).
Gregory Gutin, Royal Holloway University of London
Example
Royal Holloway
University of London
u, x → 1
v, y → 2
w, z → 3
Homomorphism
:
1
u
v
B
z
y
w
x
H
3
Gregory Gutin, Royal Holloway University of London
2
Royal Holloway
University of London
LORA-BR Formulation-3
• Let FBR=(Z1,Z2;T) be a bipartite graph with
partite sets Z1={D,C,L} (subsystem repair
options) and Z2 = {d,c,l} (module repair
options) and with T={Dd,Cd,Cc,Ld,Lc,Ll}.
L
d
C
c
D
l
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-BR Formulation-4
• Any homomorphism k of G to FBR such
that k(V1) is a subset of Z1 and k(V2) is a
subset of Z2 satisfies the rules R1 and R2 .
• Let Li is a subset of Zi, i=1,2. A
homomorphism k of G to FBR is an (L1,L2)homomorphism if k(u) ε Li for each u ε Vi.
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-BR Formulation-5
• LORA-BR can be formulated as follows: We are given a
bipartite graph G=(V1,V2;E) and we consider
homomorphisms k of G to FBR.
• Mapping of u ε V(G) to z ε V(FBR) incurs a real cost cz(u).
The use of a vertex z ε V(FBR) in a homomorphism k
incurs a real cost cz.
• We wish to choose subsets Li of Zi, i=1,2, and find an (L1,
L2)-homomorphism k of G to FBR that minimize
ΣuεV ck(u)(u) + ΣzεL cz, where L=L1U L2 .
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
General LORA problem
• General LORA problem: An arbitrary bipartite
graph F instead of FBR
• The list homomorphism problem (LHP) to a fixed
graph F : For an input graph G and a list L(v) (a
subset of V(F)) for each v ε V(G) verify whether
there is homomorphism f from G to H s.t. f(v) ε
L(v) for each v ε V(G).
• LHP is NP-complete unless F is bipartite and its
complement is a circular arc graph (Feder, Hell,
Huang, 1999)
• General LORA problem is NP-hard
Gregory Gutin, Royal Holloway University of London
Royal Holloway
University of London
LORA-M
• A bipartite graph H=(U,W;E) is monotone
if there are orderings u1,…,up and w1,…,wq of U
and W s.t. uiwj ε E implies unwm ε E for each n ≥ i,
m ≥ j.
• The bipartite graph FBR is monotone
• LORA-M is the general LORA problem with
monotone bipartite graphs F.
• LORA-M is polynomial time solvable (using max
weight indep. set problem on bipartite graphs)
Gregory Gutin, Royal Holloway University of London