1. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London 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/

2. Royal Holloway University of London Gregory Gutin, 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 life- cycle costs

3. Royal Holloway University of London Gregory Gutin, 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 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)

4. Royal Holloway University of London Gregory Gutin, 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) c i (s), c i (m) of prescribing repair decision i for subsystem s, module m, resp. The use of any repair decision i incurs a cost c i

5. Royal Holloway University of London Gregory Gutin, 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: R 1 : D s → d m, R 2 : l m → L s 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).

6. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Example u, x → 1 v, y → 2 w, z → 3 B H z 1 2 3 uv w xy Homomorphism :

7. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-3 Let F BR =(Z 1,Z 2 ;T) be a bipartite graph with partite sets Z 1 ={D,C,L} (subsystem repair options) and Z 2 = {d,c,l} (module repair options) and with T={Dd,Cd,Cc,Ld,Lc,Ll}. Ld c C D l

8. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-4 Any homomorphism k of G to F BR such that k(V 1 ) is a subset of Z 1 and k(V 2 ) is a subset of Z 2 satisfies the rules R 1 and R 2. Let L i is a subset of Z i, i=1,2. A homomorphism k of G to F BR is an (L 1,L 2 )- homomorphism if k(u) ε L i for each u ε V i.

9. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-5 LORA-BR can be formulated as follows: We are given a bipartite graph G=(V 1,V 2 ;E) and we consider homomorphisms k of G to F BR. Mapping of u ε V(G) to z ε V(F BR ) incurs a real cost c z (u). The use of a vertex z ε V(F BR ) in a homomorphism k incurs a real cost c z. We wish to choose subsets L i of Z i, i=1,2, and find an (L 1, L 2 )-homomorphism k of G to F BR that minimize Σ uεV c k(u) (u) + Σ zεL c z, where L=L 1 U L 2.

10. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London General LORA problem General LORA problem: An arbitrary bipartite graph F instead of F BR 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

11. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-M A bipartite graph H=(U,W;E) is monotone if there are orderings u 1,…,u p and w 1,…,w q of U and W s.t. u i w j ε E implies u n w m ε E for each n ≥ i, m ≥ j. The bipartite graph F BR 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)