Some applications of graph theory, combinatorics and number theory

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

2. 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 Gregory Gutin, Royal Holloway University of London

3. 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 Gregory Gutin, Royal Holloway University of London

4. 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 Gregory Gutin, Royal Holloway University of London

5. 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) c1,i(s), c2,i(m) of prescribing repair decision i for subsystem s, module m, resp. Gregory Gutin, Royal Holloway University of London

6. 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: • R1: Ds → dm, R2: lm → Ls Gregory Gutin, Royal Holloway University of London

7. 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 • R1 (R2) means that if u in V1 (V2) is assigned color 1, all its neighbors must be assigned color 3 • An assignment of colors to vertices of G satisfying R1 and R2 is called an R1&R2- acceptable coloring Gregory Gutin, Royal Holloway University of London

8. LORA-BR Formulation-4 • We may replace R1 and R2 by a bipartite graph FBR 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 R1&R2-acceptable coloring, we may assign color j to a vertex u in V1 and color k to a vertex v in V2 iff j'k'' in E(FBR) Gregory Gutin, Royal Holloway University of London

9. LORA-BR Formulation-5 • LORA-BR as a purely graph-theoretical problem: • Given: bipartite graph G=(V1,V2;E), real costs cj(u) of assigning color j in {1,2,3} to a vertex u in V=V1 U V2. Also, real costs cij of using color j for vertices of Vi, iε {1,2}, j ε {1,2,3}. • Objective: For each i=1,2, we choose a subset Li of {1,2,3} and find an R1&R2-acceptable coloring of the vertices of G that minimizes ΣuεVck(u)(u)+ΣjεL1c1j+ ΣjεL2c2j where ck(u)(u) is the cost of assigning color k(u) in Li to u in Vi and cijis the cost of using color j for vertices of Vi Gregory Gutin, Royal Holloway University of London

10. 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 V1, v ε V2, we have k'(u)k''(v) ε E(F). • For each i=1,2, we choose a subset Li of {1,…,r} and find an acceptable coloring of the vertices of G that minimizes ΣuεVck(u)(u)+ΣjεL1c1j+ ΣjεL2c2j where ck(u)(u) is the cost of assigning color k(u) in Li to u in Vi and cijis the cost of using color j for vertices of Vi NP-Hard Gregory Gutin, Royal Holloway University of London

11. LORA-M • A bipartite graph B with partite sets {1',…,r'} and {1'',…,r''}monotone if p'q'‘ εE(B) implies that s't'' ε E(B) for each s≥ p and t ≥ q. • The bipartite graph FBR 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 Gregory Gutin, Royal Holloway University of London

12. Solving LORA-M. 1 • c1(u) ≤ c2(u) ≤ … ≤ ck(u) for each u ε V • wj(u)=M-cj(u), wij=M-cij ≥ 0 • w1(u) ≥ w2(u) ≥ … ≥ wk(u) for each u ε V • Max ΣuεV wk(u)(u)+ΣjεL1w1j+ ΣjεL2w2j • Fix L1 and L2 • Max ΣuεV wk(u)(u) Gregory Gutin, Royal Holloway University of London

13. Solving LORA-M. 2 • For fixed subsets L1 and L2, LORA-M can be solved in time O(n12m1/2+n1m). • Graph H with vertices uj: u ε Vi, j ε Li • ujvk be in H if uv ε E(G), u ε V1, v ε V2 and j'k'‘ isnot in E(F); r(i) = max {p: p ε Li } • For i=1,2, u ε Vi and j ε Li, let w(uj) := wr(i)(u)+M, if j=r(i), and wj(u)- wk(u), where k is the smallest number in Lilarger than j, otherwise. Gregory Gutin, Royal Holloway University of London

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

15. 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) Gregory Gutin, Royal Holloway University of London

16. 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 Gregory Gutin, Royal Holloway University of London

17. 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)=maxxεV|N-(x)| • The nth mediation number μ(n) is the minimum of Δ-(D) over all mediated digraphs on n vertices Gregory Gutin, Royal Holloway University of London

18. Mediated Families • Family F={X1,X2,…,Xm} of subsets of a finite set X (of points); subsets of X are blocks • Fsymmetric if m=|X| • F2-covering if for each pair j,k ε X there exists a block containing both j and k • Fmediated 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 Gregory Gutin, Royal Holloway University of London

19. Proj. Planes and Bounds • Projective plane is a (q2+q+1,q+1,1)-design; exists when q is prime power • Theorem: Let n=q2+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>x0the interval [x-x0.525,x] containsprime numbers. Gregory Gutin, Royal Holloway University of London

20. 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 ? Gregory Gutin, Royal Holloway University of London