1. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Some applications of graph theory, combinatorics and number theory Gregory Gutin Department of Computer Science

2. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Two Parts of the Talk Graph-Theoretical Approach to Level of Repair Analysis (joint work with A. Rafiey, A. Yeo and M. Tso, Man. U.) Mediated Digraphs and Quantum Non- Locality (joint work with N. Jones, Bristol U., A. Rafiey, S. Severini, York U., and A. Yeo) www.cs.rhul.ac.uk/~gutin/pppublications.html

3. 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 mil.) LORA: optimal provision of repair and maintenance facilities to minimize overall life- cycle costs

4. 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 solved LORA-BR using branch-and-bound heuristics We show that LORA-BR is polynomial-time solvable Proved by reducing LORA-M to the max weight independent set problem on a bipartite graph

5. 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: "discard", "local repair" central repair“: D,L,C (subsystems) and d,l,c (modules). Costs (over life-cycle) c 1,i (s), c 2,i (m) of prescribing repair decision i for subsystem s, module m, resp.

6. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-2 We wish to minimize the total cost of 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

7. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-3 Assign colors 1,2,3 to vertices of G instead of the repair options Define the color correspondence D → 1, C → 2, L → 3; d → 3, c → 2, l → 1 R 1 (R 2 ) means that if u in V 1 (V 2 ) is assigned color 1, all its neighbors must be assigned color 3 An assignment of colors to vertices of G satisfying R 1 and R 2 is called an R 1 &R 2 - acceptable coloring

8. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-4 We may replace R 1 and R 2 by a bipartite graph F BR with partite sets {1',2',3'} and {1'',2'',3''} and with edges {1'3'',2'3'',2'2'',3'3'',3'2'',3'1''} Indeed, in an R 1 &R 2 -acceptable coloring, we may assign color j to a vertex u in V 1 and color k to a vertex v in V 2 iff j'k'' in E(F BR )

9. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-BR Formulation-5 LORA-BR as a purely graph-theoretical problem: Given: bipartite graph G=(V 1,V 2 ;E), real costs c j (u) of assigning color j in {1,2,3} to a vertex u in V=V 1 U V 2. Also, real costs c ij of using color j for vertices of V i, i ε {1,2}, j ε {1,2,3}. Objective: For each i=1,2, we choose a subset L i of {1,2,3} and find an R 1 &R 2 -acceptable coloring of the vertices of G that minimizes Σ uεV c k(u) (u)+ Σ jεL1 c 1j + Σ jεL2 c 2j where c k(u) (u) is the cost of assigning color k(u) in L i to u in V i and c ij is the cost of using color j for vertices of V i

10. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London General LORA problem F a bipartite graph (color-acceptability graph) with partite sets {1',…,r'} and {1'',…,r''}. An assignment of colors from {1,…,r} to V; assigns a vertex u a color k(u) is an acceptable coloring if for each edge uv ε G, u ε V 1, v ε V 2, we have k'(u)k''(v) ε E(F). For each i=1,2, we choose a subset L i of {1,…,r} and find an acceptable coloring of the vertices of G that minimizes Σ uεV c k(u) (u)+ Σ jεL1 c 1j + Σ jεL2 c 2j where c k(u) (u) is the cost of assigning color k(u) in L i to u in V i and c ij is the cost of using color j for vertices of V i NP-Hard

11. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London LORA-M A bipartite graph B with partite sets {1',…,r'} and {1'',…,r''} monotone if p'q'‘ ε E(B) implies that s't'' ε E(B) for each s≥ p and t ≥ q. The bipartite graph F BR corresponding to both rules of LORA-BR is monotone LORA-M is the general LORA problem with a monotone color-acceptability graph F. POLYNOMIAL TIME SOLVABLE

12. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Solving LORA-M. 1 c 1 (u) ≤ c 2 (u) ≤ … ≤ c k (u) for each u ε V w j (u)=M-c j (u), w ij =M-c ij ≥ 0 w 1 (u) ≥ w 2 (u) ≥ … ≥ w k (u) for each u ε V Max Σ uεV w k(u) (u)+ Σ jεL1 w 1j + Σ jεL2 w 2j Fix L 1 and L 2 Max Σ uεV w k(u) (u)

13. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Solving LORA-M. 2 For fixed subsets L 1 and L 2, LORA-M can be solved in time O(n 1 2 m 1/2 +n 1 m). Graph H with vertices u j : u ε V i, j ε L i u j v k be in H if uv ε E(G), u ε V 1, v ε V 2 and j'k'‘ is not in E(F); r(i) = max {p: p ε L i } For i=1,2, u ε V i and j ε L i, let w(u j ) := w r(i) (u)+M, if j=r(i), and w j (u)- w k (u), where k is the smallest number in L i larger than j, otherwise.

14. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Solving LORA-M. 3 H is bipartite For acceptable coloring k, {u k(u) : u ε V(G)} is independent in H By monotonicity of F, S={ u j : u ε V, j ε L i, j ≥ k(u)} is independent in H S contains S' ={ u r(i) : u ε V}

15. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Solving LORA-M. 4 G has an acceptable coloring iff a maximum weight independent set in H contains S' If G has an acceptable coloring, then an optimal acceptable coloring corresponds to a maximum weight independent set S in H (the difference in weights is Mn)

16. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Mediated Digraphs D=(V,A) is mediated if for each pair x,y of vertices either xy ε A or yx ε A or there is a vertex z such that both xz,yz ε A

17. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Mediation Number x ε V: N - (x)={y: yx ε A}, N - [x]={x} U N - (x) A digraph D is mediated iff for each pair x,y ε V there is a vertex z ε V s.t. x,y ε N - [z] For a digraph D, Δ - (D)=max xεV |N - (x)| The nth mediation number μ(n) is the minimum of Δ - (D) over all mediated digraphs on n vertices

18. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Mediated Families Family F={X 1,X 2,…,X m } of subsets of a finite set X (of points); subsets of X are blocks F symmetric if m=|X| F 2-covering if for each pair j,k ε X there exists a block containing both j and k F mediated if symmetric, 2-covering and has an SDR mcard(F) max cardinality of a block in F μ - (n) the minimum mcard(F) over all mediated families on {1,2,…,n}; we have μ(n)= μ - (n)-1

19. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Proj. Planes and Bounds Projective plane is a (q 2 +q+1,q+1,1)- design; exists when q is prime power Theorem: Let n=q 2 +q+1+m(q+1)-r, where q is a prime power, 1 ≤ m ≤ q+1 and 0 ≤ r ≤ q. Then μ(n) ≤ q+m. Theorem [Baker, Harman and Pintz] For all x>x 0 the interval [x-x 0.525,x] contains prime numbers.

20. Royal Holloway University of London Gregory Gutin, Royal Holloway University of London Bounds and Questions Let f(n)= ┌ ((4n-3) 1/2 -1)/2 ┐ We have μ(n) ≥ f(n) We have μ(n) = f(n) (1+o(1)) Is there a constant c s.t. μ(n) ≤ f(n) + c ? Is μ(n) monotonically increasing ?