Shortest s - t Paths Using Min-Sum Nicholas Ruozzi and Sekhar - PowerPoint PPT Presentation

Background The Shortest s - t Path Problem Min-Sum Conclusion Shortest s - t Paths Using Min-Sum Nicholas Ruozzi and Sekhar Tatikonda Yale University September 25, 2008 Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Previous Work Min-Sum Total Unimodularity Conclusion Previous Work ◮ Max-weight matching (Sanghavi et al., 2007) ◮ Max-weight matching (Bayati et al., 2008) ◮ Max-weight independent set (Sanghavi et al., 2007) Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Previous Work Min-Sum Total Unimodularity Conclusion Previous Work ◮ Take an integer program: max c T x subject to Ax ≤ b and x ∈ { 0 , 1 } n ◮ Factorize it as a product of self-potentials and terms that enforce the constraints ◮ Run the max-product algorithm ◮ Behavior of max-product is related to solutions of the relaxed LP Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Previous Work Min-Sum Total Unimodularity Conclusion Total Unimodularity Definition A matrix A is totally unimodular if every subdeterminant of A is 0 , 1 , or − 1. Theorem Let A be a totally unimodular m × n matrix and b an integral n vector. The polyhedron P = { x | Ax ≤ b } is integral. Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Min-Sum Conclusion The Shortest Path Problem Definition Given a directed graph G = ( V , E ), vertices s and t ∈ V , and for each e ∈ E a weight w e > 0, the shortest s - t path problem is then to find the path of minimum weight in G starting at s and ending at t . If no such path exists the shortest s - t path is infinite. Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Min-Sum Conclusion SP as an Integer Program minimize: � w e X e e ∈ E subject to: X e ∈ { 0 , 1 } � � for v ∈ V − { s , t } , X ( u , v ) = X ( v , u ) ( u , v ) ∈ E ( v , u ) ∈ E � � X ( s , u ) = 1 + X ( u , s ) ( s , u ) ∈ E ( u , s ) ∈ E � � X ( u , t ) = 1 + X ( t , u ) ( u , t ) ∈ E ( t , u ) ∈ E Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Min-Sum Conclusion The Shortest Path Problem We can reformulate the shortest path problem so that it can be solved using min-sum as follows: ◮ For each e ∈ E , let X e ∈ { 0 , 1 } represent whether or not edge e is in the shortest path. ◮ For each e ∈ E , define φ e ( X e ) = e w e X e ◮ For each v ∈ V , define ψ v ( E v ) = 1 if the linear constraints are satisfied at v and zero otherwise Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Min-Sum Conclusion The Shortest Path Problem ◮ Construct an objective function f as follows: � � f ( X E ) = log φ e ( X e ) − log ψ v ( X ∂ v ) e ∈ E v ∈ V ◮ f is infinite for any assignment to the edges that violates the constraints ◮ f is minimized exactly when the integer program is minimized Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion The Min-Sum Algorithm ◮ Initialize all messages to 0. ◮ For e incident to v , � m n − 1 e ′ → v ( x ′ m n v → e ( x ) = x ∂ v : x e = x − log ψ v ( x ∂ v ) + min e ) e ′ : e ′ ∈ ∂ v −{ e } ◮ For e incident to v and u , m n e → v ( x ) = log φ e ( x ) + m n − 1 u → e ( x ) Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion The Min-Sum Algorithm ◮ Compute the beliefs at step n: b n � m n e ( x ) = φ e ( x ) + v → e ( x ) v ∈ ∂ e ◮ Estimate membership of e = ( u , v ) in the min path as if b n e (1) < b n  1 e (0) x en =  if b n e (0) < b n 0 e (1) ? otherwise  Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion The Shortest Path Problem (a) The graph G (b) Factorgraph for G Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Computation Tree Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion ”Reduced” Computation Tree Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Main Theorem Theorem If P ∗ is the unique minimum s-t path in G then an edge e ∈ E is in P ∗ iff every minimal solution on T e ( n ) contains the root for n > w ( P ∗ ) 2 ǫ w min + w ( P ∗ ) w min . Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Proof ◮ Suppose that e is in the optimal s - t path but every minimum solution on T e ( n ) excludes the root. ◮ Fix one such minimum solution M . ◮ Construct a subset of T e ( n ) which consists of an alternating combination of copies of the optimal path and subsets of the paths chosen in M . Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Proof Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Proof Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Proof Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Proof Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Convergence of Min-Sum Min-Sum Non-uniqueness of the Shortest Path Conclusion Other Results If the shortest s - t path in G is not unique, min-sum may not converge or may converge to the wrong answer: Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum

Background The Shortest s - t Path Problem Min-Sum Conclusion Future Work ◮ Extensions to other totally unimodular problems? ◮ Handling non-uniqueness ◮ Approximations for general integer programming problems Nicholas Ruozzi and Sekhar Tatikonda Shortest s - t Paths Using Min-Sum