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Gregory Gutin Department of Computer Science. Some applications of graph theory, combinatorics and number theory. 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 .)
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Gregory Gutin Department of Computer Science Some applications of graph theory, combinatorics and number theory 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 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 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 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) c1,i(s), c2,i(m) of prescribing repair decision i for subsystem s, module m, resp. 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: • R1: Ds → dm, R2: lm → Ls 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 • 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
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
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
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
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
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
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
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
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
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
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
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
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
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