Level of Repair Analysis and Minimum Cost Homomorphisms of Graphs

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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