Level of Repair Analysis and Minimum Cost Homomorphisms of Graphs

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Level of Repair Analysis and Minimum Cost
Homomorphisms of Graphs
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/
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LORA

• Level of Repair Analysis (LORA) procedure for
  defence logistics
• Complex system with thousands of assemblies,
  sub-assemblies, components, etc.
• Has ? 2 levels of indenture and with r 2
  repair decisions (?2,r3 UK and USA military,
  ?2,r5 French military)
• LORA optimal provision of repair and maintenance
  facilities to minimize overall life-cycle costs

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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 polynomial-time
  solvable
• We proved it by reducing LORA-M via graph
  homomorphisms to the max weight independent set
  problem on bipartite graphs (see the paper)

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LORA-BR Formulation-1

• ?2 Subsystems (S) and Modules (M)
• A bipartite graph G(S,ME) sm e E iff module m
  is in subsystem s
• r3 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

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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 e
  E(B) implies k(x)k(y) e E(H).

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Example

• u, x ? 1
• v, y ? 2
• w, z ? 3

Homomorphism
B
z
H
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LORA-BR Formulation-3

• Let FBR(Z1,Z2T) be a bipartite graph with
  partite sets Z1D,C,L (subsystem repair
  options) and Z2 d,c,l (module repair options)
  and with TDd,Cd,Cc,Ld,Lc,Ll.

L
d
c
C
D
l
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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, i1,2. A homomorphism k
  of G to FBR is an (L1,L2)-homomorphism if k(u) e
  Li for each u e Vi.

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LORA-BR Formulation-5

• LORA-BR can be formulated as follows We are
  given a bipartite graph G(V1,V2E) and we
  consider homomorphisms k of G to FBR.
• Mapping of u e V(G) to z e V(FBR) incurs a real
  cost cz(u). The use of a vertex z e V(FBR) in a
  homomorphism k incurs a real cost cz.
• We wish to choose subsets Li of Zi, i1,2, and
  find an (L1, L2)-homomorphism k of G to FBR that
  minimize
• SueV ck(u)(u) SzeL cz, where
  LL1U L2 .

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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 e V(G) verify whether
  there is homomorphism f from G to H s.t. f(v) e
  L(v) for each v e 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

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LORA-M

• A bipartite graph H(U,WE) is monotone
• if there are orderings u1,,up and w1,,wq of
  U and W s.t. uiwj e E implies unwm e 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)