Decomposition-based Constraint Optimization and Distributed Execution

1
Decomposition-based Constraint Optimization and
Distributed Execution

• John Stedl, Tsoline Mikaelian, Martin
  Sachenbacher
• September 2003

2
Distributed Execution

• What is Distributed Execution?
• Each agent maintains a subset of the plan
• Each agent operates autonomously using
  communication to coordinate activities
• Two phases
• reformulation phase to create dispatchable plan (
  all agents can communicate )
• temporally flexible execution of the dispatchable
  plan (communication may be limited )
• Motivation
• Reduced communication vs. a leader-follower
  architecture
• If agent-agent communication is limited , we need
  to perform distributed execution
• Reduces computation and memory requirements for
  individual agent
• Increase robustness ( no single point of failure
  )
• Approach
• Use intelligent distributed algorithm selection
  to map centralized algorithm for STNs and STNUs
  in to distributed plan running algorithms
• sub-divide the problem into smaller pieces when
  necessary

distributed STN
3
Distributed Execution Domains
distributed hardware components

• 1. Full Communication and No-Uncertainty
• Use distributed versions of classical STN
  reformulation and dispatching algorithms
• 2. Full Communication with Uncertainty
• Use distributed versions of the Dynamic
  Controllability algorithms for STNUs and
  associated dispatching algorithm
• 3. Limited Communication with Uncertainty
• sacrifice some temporal flexibility in plan to
  make problem easier
• Break up the plan into a natural two-level
  Hierarchy where based on communication
  constraints such that
• agents within a sub-plan are free to communicate
• no communication allowed between different
  sub-plans
• Leverage work on strong and dynamic
  controllability

satellite constellations
rover teams on Mars
group one
group two
Tsamardinos, Muscettola, Morris, Fast
Transformations of Temporal Plans for Efficient
Execution Morris, Muscettola, Vidal,
Dynamic Control of Plans with Temporal
Uncertainty
4
Distributed Mode Estimation

• Based on centralized N-Step mode estimation,
  framed as an OCSP.
• Solve the OCSP using hypertree decomposition
• Exploit tree decomposition for distributed
  problem solving
• Distributed ME process

5
Enabling Distributed Mode Estimation Through
Decomposition-based Constraint Optimization

• Martin Sachenbacher
• September 2003

6
Constraint Satisfaction Problems

• Domains dom(xi)
• Variables X x1, x2, , xn
• Constraints C c1, c2, , cm
• Constraint cj ? dom(xj1) ? ? dom(xjk)

y u
u
x y z
y
1 10 0
0 0 00 1 11 0 11 1 1
x
z
v
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CSP Decomposition

• Transform CSP into equivalent acyclic instance
• By combining constraints responsible for
  cyclicity

Compilation
8
Hypertree Decomposition

• See Gottlob et al., Artificial Intelligence
  124(2000)
• Tree T (N,E) with labeling functions ?, ? such
  that
• For each cj ? C, there is at least one n ? N such
  that scope(cj) ? ?(n) (covering)
• For each variable xi ? X, the set n ? N xi ?
  ?(n) induces a connected subtree of T
  (connectedness)
• For each n ? N, ?(n) ? scope(?(n))
• For each n ? N, scope(?(n)) ? ?(Tn) ? ?(n), where
  Tn is the subtree of T rooted at n
• HT-width of a hypertree decomposition is defined
  as max(?(n)), n?N

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Example

• Boolean Polycell (see Williams, Ragno 2003)

x
a 1
Or1
And1
b 1
f 0
y
Or2
c 1
And2
g 1
d 1
z
Or3
e 0
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Example

• Variables
• a, b, c, d, e, f, g, x, y, z with domain 0,1
• O1, O2, O3, A1, A2 with domain ok,fty
• Constraints
• Or1(O1,a,c,x), Or2(O2,b,d,y), Or3(O3,c,e,z) model
  or-gates
• And1(A1,x,y,f), And2(A2,y,z,g) model and-gates

O1 a c x
A2 y z g
ok 1 1 1fty 1 1 0fty 1 1 1
ok 1 1 1fty 0 0 1fty 0 1 1fty 1 0
1fty 1 1 1
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Example

• Hypertree Decomposition (width 2)

?-label
?-label
O3,A1,c,e,f,x,y,z Or3,And1
Shared variables
y,z
y
c,x
O2,b,d,y Or2
A2,g,y,z And2
O1,a,c,x Or1
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Example

• Resulting acyclic CSP

ok ok 1 0 0 0 1 1ok ok 1 0 0 0 0 1ok ok 1 0 0 1
0 1
O3,A1,c,e,f,x,y,z Or3,And1
y,z
y
c,x
O2,b,d,y Or2
A2,g,y,z And2
O1,a,c,x Or1
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
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Semiring-CSPs and Optimization

• Domains dom(xi)
• Variables X x1, x2, , xn
• Constraints C c1, c2, , cm
• Set S, Constraint ci dom(xi1) ? ? dom(xik) ?
  S
• Operators ? (defines projection) and ? (defines
  join) on S with neutral elements 0 and 1
• (S, ?, ?, 0, 1) forms semiring structure
• Type T ? V specifies variables appearing in
  solutions

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Example

• Boolean Polycell with probabilities
• Or-gates p(ok).99, p(fty).01
• And-gates p(ok).995, p(fty).005
• Semiring (0,1, max, , 0, 1)
• T O1,O2,O3,A1,A2

p
O1 a c x
p
A2 y z g
.99.01.01
.995.005.005.005.005
ok 1 1 1fty 1 1 0fty 1 1 1
ok 1 1 1fty 0 0 1fty 0 1 1fty 1 0
1fty 1 1 1
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Example

• Resulting acyclic SCSP

ok ok 1 0 0 0 1 1ok ok 1 0 0 0 0 1ok ok 1 0 0 1
0 1
.985.985.985
O3,A1,c,e,f,x,y,z Or3,And1
y,z
y
c,x
b,d,y Or2
g,y,z And2
a,c,x Or1
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
.99.01.01.01.01
.99.01.01
.99.01.01
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Solving Tree-Structured SCSPs

• Bottom-up phase for computing values
• Top-down phase for extracting solutions
• Polynomial in width, highly parallelizable

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Example

• Bottom-Up Phase

ok ok 1 0 0 0 1 1ok ok 1 0 0 0 0 1ok ok 1 0 0 1
0 1
.0097
.985.985
O3,A1,c,e,f,x,y,z Or3,And1
y,z
y
c,x
b,d,y Or2
g,y,z And2
a,c,x Or1
ok 1 1 1fty 1 1 1fty 1 0 1fty 1 1 0fty 1 0 0
ok 1 1 1fty 1 1 1fty 1 1 0
ok 1 1 1fty 1 1 1fty 1 1 0
.99.01.01.01.01
.99.01.01
.99.01.01
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Bottom-Up Phase

• Function solve(node)
• For Each tuple ? node.relation
• For Each child ? node.children
• childTuple ? findBestConsistentTuple(c(child),t
  uple)
• If childTuple ? Then
• c(node) ? c(node) \ tuple
• Else value(tuple) ? value(tuple) ?
  value(childTuple)
• End If
• Next child
• Next tuple

Dynamic Programming
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Top-Down Solution Expansion
(True, 0)

(O3A1cxyz ok ok 1011, 0.0097)
Or3,And1

And2
(O3A1A2cxy ok ok ok 111, 0.0097)

Or2
(O2O3A1A2cx ok ok ok ok 11, 0.0097)

Or1
(O1O2O3A1A2 fty ok ok ok ok, 0.0097)
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Discussion

• Conclusion
• Search-free
• Highly parallelizable
• Tractable, if HT-width bounded
• Current Work
• Extension to best-first enumeration
• Extension to symbolic encoding

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Material

• CSP Decomposition Methods

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CSP Decomposition Methods

• Biconnected Components Freuder 85
• Treewidth Robertson and Seymour 86
• Tree Clustering Dechter Pearl 89
• Cycle Cutset Dechter 92
• Bucket Elimination Dechter 97
• Tree Clustering with Minimization Faltings 99
• Hinge Decomposition Gyssens and Paredaens 84
• Hypertree Decomposition Gottlob et al. 99