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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. LORA. Level of Repair Analysis (LORA): procedure for defence logistics

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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/

Level of Repair Analysis and Minimum Cost Homomorphisms of Graphs

Gregory Gutin, Royal Holloway University of London

slide2
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 life-cycle costs

Gregory Gutin, Royal Holloway University of London

lora br
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)

Gregory Gutin, Royal Holloway University of London

lora br formulation 1
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

lora br formulation 2
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

1

u

v

w

y

x

2

3

Example

u, x → 1

v, y → 2

w, z → 3

Homomorphism:

B

z

H

Gregory Gutin, Royal Holloway University of London

lora br formulation 3
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

lora br formulation 4
LORA-BR Formulation-4
  • Any homomorphism k of G to FBRsuch that k(V1) is a subset of Z1and k(V2) is a subset of Z2satisfies the rules R1 and R2 .
  • Let Liis a subset of Zi, i=1,2. A homomorphism k of G to FBR is an (L1,L2)-homomorphism if k(u) ε Lifor each u ε Vi.

Gregory Gutin, Royal Holloway University of London

lora br formulation 5
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

general lora problem
General LORA problem
  • General LORA problem: An arbitrary bipartite graph F instead of FBR
  • The list homomorphism problem (LHP) to a fixed graphF : 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

lora m
LORA-M
  • A bipartite graph H=(U,W;E) is monotone

if there are orderings u1,…,upand w1,…,wq of U and W s.t. uiwj εE 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