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A Linear Programming Approach to Max-sum Problem: A Review
Tomáš Werner
- Dept. of Cybernetics, Czech Technical
Max-Sum Problem
e.g. the MAP problem on MRFs
A Linear Programming Approach to Max-sum Problem: A Review Tom - - PowerPoint PPT Presentation
A Linear Programming Approach to Max-sum Problem: A Review Tom - - PowerPoint PPT Presentation
A Linear Programming Approach to Max-sum Problem: A Review Tom Werner Dept. of Cybernetics, Czech Technical University Max-Sum Problem e.g. the MAP problem on MRFs Formulation of the Problem G = (T, E) T is a set of objects, x t X is a
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Formulation of the Problem
G = (T, E) T is a set of objects, is a labeling on t G’ = (T × X,EX) gt = (t, x) gtt’ = {(t, x), (t’, x’}
2 T E X x t
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Commutative Semirings
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Rings
) , (S, ) , (Z,
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Semirings
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Semirings CSP
Denote a problem by (G,X, ) – Graph, Domain, Constraints Let t(x), tt’(x,x’) = {0,1} say if an assignment is allowed or forbidden
g g g
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Arc consistency in CSP
The kernel can be obtained by iteratively applying the following relations until no more 0 assignments are made (arc consistency algorithm)
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Semirings Max-sum
Denote a problem by (G,X,g) – Graph, Assignments, Weights
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Equivalent Transformations
Also known as ERs (Wainwright) A problem is called equivalent if (G,X,g) and (G,X,g’) produce the same problem, denoted as g~g’ The simplest such transformation adds a number φtt’(x) to gt(x) while removing from gtt’(x,x’) This formulation corresponds to potentials or messages from message passing
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Schlesinger’s Upper Bound
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Triviality
(t,x) is a maximal node if gt(x) = ut {(t,x), (t’,x’)} is a maximal edge if gtt’(x,x’) = utt’
t(x) = [[gt(x) = ut]] tt’(x) = [[gtt’(x,x’) = utt’]]
A max-sum problem is trivial if a labeling can be formed of a subset of its maximal nodes and edges
g g
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Triviality
Testing for triviality of a max-sum problem is correspondent to solving the CSP generated by its maximal nodes and edges A CSP is a tight solution to all max-sum problems it can be equivalently transformed into
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Equivalent Transformations
(CSPs)
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Linear Programming Relaxation
This gives the polytope which has a set of
- ptimal vertices given by
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Duality of the Relaxations
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More theorems fall out
Finding the kernel does not guarantee finding a solution for the minimal upper bound Obvious by approach from CSPs For problems of boolean variables |X| = 2 finding the kernel is necessary and sufficient for finding the upper bound
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(Super) Submodularity
Known that the (super) submodularity property produces max- sum problems with tractable solutions by conversion to max- flow/min-cut problems Has been suggested that supermodularity is the discrete counterpart of convexity. Lots of work shows that the LP relaxation for a supermodular max-sum problem is tight Supermodular max-sum problems will always form a lattice CSP with a tractable solution
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