The Graph Realization Problem
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1 The Graph Realization Problem via Semi-Definite Programming A. Y. Alfakih Mathematics and Statistics University of Windsor The Graph Realization Problem p.1/21
2 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E. The Graph Realization Problem p.2/21
3 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that C 4 p i p j 2 = ω ij (i, j) E. The Graph Realization Problem p.2/21
4 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.2/21
5 The Graph Realization Problem Given an edge-weighted graph G = (V, E, ω), G is said to be realizable in R r iff p 1, p 2,..., p n R r such that p i p j 2 = ω ij (i, j) E C 4 p 1 p 4 p 2 p 3 The Graph Realization Problem p.2/21
6 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. The Graph Realization Problem p.3/21
7 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. Given G = (V, E, ω), the complexity of the problem: Does G have a realization in some Euclidean space, is still an open problem. The Graph Realization Problem p.3/21
8 GRP Complexity Given G = (V, E, ω) and integer k, the problem: Does G have a realization in R k, is NP-hard. Given G = (V, E, ω), the complexity of the problem: Does G have a realization in some Euclidean space, is still an open problem. Given G = (V, E, ω) and ǫ > 0, the problem: Does G have a realization, with tolerance ǫ, in some Euclidean space is polynomial. The Graph Realization Problem p.3/21
9 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. The Graph Realization Problem p.4/21
10 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: The Graph Realization Problem p.4/21
11 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. The Graph Realization Problem p.4/21
12 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. A characterization of the set of all realizations of graph G. The Graph Realization Problem p.4/21
13 Among the many applications of GRP are: molecular conformation problems, multidimensional scaling, wireless sensor networks. In this talk we will present: A Semi-definite Programming model for the GRP. A characterization of the set of all realizations of graph G. A randomized algorithm for obtaining low-dimensional realizations of graph G. The Graph Realization Problem p.4/21
14 Euclidean distance matrices (EDMs) An n n matrix D = (d ij ) is said to be an EDM iff p 1, p 2,..., p n R r such that d ij = p i p j 2 i, j = 1,..., n. The Graph Realization Problem p.5/21
15 Euclidean distance matrices (EDMs) An n n matrix D = (d ij ) is said to be an EDM iff p 1, p 2,..., p n R r such that d ij = p i p j 2 i, j = 1,..., n. The dimension of the affine span of p 1,..., p n is called the embedding dim of D. The Graph Realization Problem p.5/21
16 Example D = is an EDM. The Graph Realization Problem p.6/21
17 D = Example is an EDM. The points that generate D are: p 1 p 2 p 4 p 3 The Graph Realization Problem p.6/21
18 D = Example is an EDM. The points that generate D are: p 1 p 2 p 4 p 3 Embedding dim of D is 2. The Graph Realization Problem p.6/21
19 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. The Graph Realization Problem p.7/21
20 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. Let G = C 4 then A G = The Graph Realization Problem p.7/21
21 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. A = Embedding dim = 2. The Graph Realization Problem p.7/21
22 GRP = EDM Completion Problem Given a partial matrix A where only some of its elements are specified, The EDMCP is the problem of determining whether or not A can be completed into an EDM. A = Embedding dim = 1. The Graph Realization Problem p.7/21
23 EDM Characterization (Schoenberg 35, Young and Householder 38) An n n symmetric D with diag(d) = 0 is EDM iff B := 1 2 JDJ 0 where J = I eet /n, e = (1,..., 1), and embedding dim of D = rank B. The Graph Realization Problem p.8/21
24 EDM Characterization (Schoenberg 35, Young and Householder 38) An n n symmetric D with diag(d) = 0 is EDM iff B := 1 2 JDJ 0 where J = I eet /n, e = (1,..., 1), and embedding dim of D = rank B. The points p 1,..., p n that generate D are given by the rows of P where B = PP T. The Graph Realization Problem p.8/21
25 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. The Graph Realization Problem p.9/21
26 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. The Graph Realization Problem p.9/21
27 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. rank X = r H is the adjacency matrix of G and denotes Hadamard product. The Graph Realization Problem p.9/21
28 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. This SDP can be solved efficiently using a primal-dual interior point algorithm. The Graph Realization Problem p.9/21
29 SDP formulation of EDMCP (Alfakih,Khandani,Wolkowicz) min H (A K V (X)) 2 F subject to X 0. H is the adjacency matrix of G and denotes Hadamard product. This SDP can be solved efficiently using a primal-dual interior point algorithm. The realization obtained will have the largest possible embedding dimension. The Graph Realization Problem p.9/21
30 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). The Graph Realization Problem p.10/21
31 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The Graph Realization Problem p.10/21
32 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The set of all realizations of G in R r is : {X(y) : y Ω, and rank X(y) = r}. The Graph Realization Problem p.10/21
33 Ω, Set of all Realizations of G Let X 1 be a given realization of G = (V, E, ω). Let Ω := {y : X(y) := X 1 + (i,j) E y ijm ij 0}. The set of all realizations of G in R r is : {X(y) : y Ω, and rank X(y) = r}. The set of all realizations of G in all spaces is : {X(y) : y Ω}. The Graph Realization Problem p.10/21
34 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. The Graph Realization Problem p.11/21
35 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. Ω always contains the origin. The Graph Realization Problem p.11/21
36 Properties of Ω Ω is a convex, closed and in general a non-polyhedral set. Ω always contains the origin. Ω is bounded whenever graph G is connected. The Graph Realization Problem p.11/21
37 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 The Graph Realization Problem p.12/21
38 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 y 2 y 24 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
39 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 y 2 y 24 3 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
40 Ω, An Example Given the following realization of C 4 : p 1 p 2 y 2 y 24 y 3 13 y 1 p 4 p X(0) 3 p 4 p 1 p 3 p 2 X(y 2 ) Set Ω for C 4 The Graph Realization Problem p.12/21
41 Ω, An Example Given the following realization of C 4 : p 1 p 2 p 4 p X(0) 3 p 1 p 2 p 4 p 3 X(y 1 ) y 2 y 24 3 y 13 y 1 Set Ω for C 4 The Graph Realization Problem p.12/21
42 Faces of Ω The facial structure of Ω is closely related to that of the positive semidefinite cone. The Graph Realization Problem p.13/21
43 Faces of Ω The facial structure of Ω is closely related to that of the positive semidefinite cone. The relative interior of faces of Ω are characterized by Gale matrices. The Graph Realization Problem p.13/21
44 Gale Matrices [ ] Given p 1,..., p n R r Λ 1, let Λ = be the Λ 2 matrix whose[ columns form a] basis for p 1 p 2 p n nullspace of The Graph Realization Problem p.14/21
45 Gale Matrices [ ] Given p 1,..., p n R r Λ 1, let Λ = be the Λ 2 matrix whose[ columns form a] basis for p 1 p 2 p n nullspace of [ Z := ΛΛ 1 I n 1 r 1 = Λ 2 Λ 1 is called the Gale 1 matrix corresponding to the p i s. ] The Graph Realization Problem p.14/21
46 For the following configuration: p 1 p 2 p 4 p 3 The Graph Realization Problem p.15/21
47 For the following configuration: p 1 p 2 p 4 p 3 Z = [1, 1, 1, 1] T The Graph Realization Problem p.15/21
48 For the following configuration: p 1 p 2 p 4 p 3 Z = [1, 1, 1, 1] T Theorem (Alfakih): Let y 1, y 2 Ω and let F(y 1 ) denote the smallest face of Ω containing y 1. Then F(y 1 ) = F(y 2 ) iff the realizations X(y 1 ) and X(y 2 ) have the same Gale matrix. The Graph Realization Problem p.15/21
49 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 The Graph Realization Problem p.16/21
50 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
51 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
52 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
53 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
54 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
55 Faces of Ω, An Example y 2 y 1 Set Ω for C 4 p 1 p 2 p 4 p 3 The Graph Realization Problem p.16/21
56 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E The Graph Realization Problem p.17/21
57 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E Theorem: if ŷ is an extreme point of Ω then 8 E +1 1 rank X(ŷ) 2. The Graph Realization Problem p.17/21
58 Extreme Points of Ω Theorem(Barvinok, Pataki, Deza and Laurent): If a G = (V, E, ω) has a realization then it has a realization in R r where r 8 E Theorem: if ŷ is an extreme point of Ω then 8 E +1 1 rank X(ŷ) 2. Theorem(Barvinok): If G is not a union of a complete graph with r + 2 nodes and zero or more isolated nodes, then it has a realization in R r where r 8( E 1) The Graph Realization Problem p.17/21
59 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
60 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
61 Low-Dimensional Realizations Lemma (Alfakih): Assume Ω is full dimensional and let y Ω. If dim N Ω (y) s(s+1) 2 for some positive integer s, then rank X(y) n 1 s. The Graph Realization Problem p.18/21
62 Randomized Algorithm Input: Realization X 1 of G, positive integer t. The Graph Realization Problem p.19/21
63 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do The Graph Realization Problem p.19/21
64 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. The Graph Realization Problem p.19/21
65 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. 2- Solve the SDP problem: max {c kt y : y Ω} and let y k be the corresponding optimal solution. The Graph Realization Problem p.19/21
66 Randomized Algorithm Input: Realization X 1 of G, positive integer t. for k = 1,..., t do 1- Let c k be random vector selected randomly from the unit sphere in R m. 2- Solve the SDP problem: max {c kt y : y Ω} and let y k be the corresponding optimal solution. Pick y min = arg min {X(y k ) : k = 1,..., t}. The Graph Realization Problem p.19/21
67 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The Graph Realization Problem p.20/21
68 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The SDP is solved using SeDuMi 1.05R5 of Sturm. The Graph Realization Problem p.20/21
69 c ij be m numbers randomly drawn from N(0, 1). Then c = 1 c c is uniform over the unit sphere in R m. The SDP is solved using SeDuMi 1.05R5 of Sturm. Numerical tests on 3.06 GHz Pentium-4 PC with 496 MB of RAM. The Graph Realization Problem p.20/21
70 Numerical Tests n = 20 m = Barvinok bound = rank X(y ) The CPU time for optimizing c T y over Ω is in order of 1 sec. The Graph Realization Problem p.21/21
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