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

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

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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

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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

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Background The Shortest s-t Path Problem Min-Sum Conclusion Previous Work Total Unimodularity

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

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Background The Shortest s-t Path Problem Min-Sum Conclusion Previous Work Total Unimodularity

Previous Work

◮ Take an integer program: max cTx 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

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Background The Shortest s-t Path Problem Min-Sum Conclusion Previous Work Total Unimodularity

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

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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 we > 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

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Background The Shortest s-t Path Problem Min-Sum Conclusion

SP as an Integer Program

minimize:

  • e∈E

weXe subject to: Xe ∈ {0, 1} for v ∈ V − {s, t},

  • (u,v)∈E

X(u,v) =

  • (v,u)∈E

X(v,u)

  • (s,u)∈E

X(s,u) = 1 +

  • (u,s)∈E

X(u,s)

  • (u,t)∈E

X(u,t) = 1 +

  • (t,u)∈E

X(t,u)

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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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 Xe ∈ {0, 1} represent whether or not edge

e is in the shortest path.

◮ For each e ∈ E, define φe(Xe) = eweXe ◮ For each v ∈ V , define ψv(Ev) = 1 if the linear constraints

are satisfied at v and zero otherwise

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion

The Shortest Path Problem

◮ Construct an objective function f as follows:

f (XE) =

  • e∈E

log φe(Xe) −

  • v∈V

log ψv(X∂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

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

The Min-Sum Algorithm

◮ Initialize all messages to 0. ◮ For e incident to v,

mn

v→e(x) =

min

x∂v:xe=x − log ψv(x∂v) +

  • e′:e′∈∂v−{e}

mn−1

e′→v(x′ e) ◮ For e incident to v and u,

mn

e→v(x) = log φe(x) + mn−1 u→e(x)

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

The Min-Sum Algorithm

◮ Compute the beliefs at step n:

bn

e(x) = φe(x) +

  • v∈∂e

mn

v→e(x) ◮ Estimate membership of e = (u, v) in the min path as

xen =    1

if bn

e(1) < bn e(0)

if bn

e(0) < bn e(1)

?

  • therwise

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

The Shortest Path Problem

(a) The graph G (b) Factorgraph for G

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Computation Tree

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

”Reduced” Computation Tree

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

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 Te(n) contains the root for n > w(P∗)2

ǫwmin + w(P∗) wmin .

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Proof

◮ Suppose that e is in the optimal s-t path but every minimum

solution on Te(n) excludes the root.

◮ Fix one such minimum solution M. ◮ Construct a subset of Te(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

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Proof

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Proof

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Proof

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

Proof

Nicholas Ruozzi and Sekhar Tatikonda Shortest s-t Paths Using Min-Sum

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Background The Shortest s-t Path Problem Min-Sum Conclusion Convergence of Min-Sum Non-uniqueness of the Shortest Path

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

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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