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Identifying Conflicts in Overconstrained
Temporal Problems
Mark H. Liffiton, Michael D. Moffitt, Martha E. Pollack, and Karem A. Sakallah
University of Michigan
August 5, 2005 (IJCAI-05)
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An Overconstrained Temporal Problem
CC11: (x-y ≤ 1): (x-y ≤ 1)CC22: (y-x ≤ -2): (y-x ≤ -2)CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)CC44: (y-z ≤ -2): (y-z ≤ -2)
CC11: (x-y ≤ 1): (x-y ≤ 1)CC22: (y-x ≤ -2): (y-x ≤ -2)CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)CC44: (y-z ≤ -2): (y-z ≤ -2)
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MUSes
Given: Infeasible set of constraints C Goal: All Minimal Unsatisfiable Subsets of C
Minimal Unsat.: All proper subsets satisfiable Compact explanations of infeasibility
C = {C = {CC11,,CC22,,CC33,,CC44,,CC55} } UNSAT UNSAT
{{CC11,,CC33,,CC44} } UNSAT UNSAT
{ { CC33,,CC44} } SATSAT
{{CC11, , CC44} } SATSAT
{{CC11,,CC33 } } SAT SAT
(MUS)(MUS)
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Disjunctive Temporal Problems A constraint system whose constraints are
disjunctions of inequalities between timepoints:
Generalization of a Simple Temporal Problem (STP), which has one inequality per constraint
Used for temporal reasoning
DTPs
(x(x11-y-y11 ≤ b ≤ b11) ) (x (x22-y-y22 ≤ b ≤ b22) ) (x(xnn-y-ynn ≤ b ≤ bnn))(x(x11-y-y11 ≤ b ≤ b11) ) (x (x22-y-y22 ≤ b ≤ b22) ) (x(xnn-y-ynn ≤ b ≤ bnn))...
timepoints real value
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Why All MUSes? Why DTPs?
Knowing all MUSes can be necessary for determining an optimum relaxation
DTPs allow fine-grained relaxations beyond simply removing entire constraints
CC11: (x-y ≤ 1): (x-y ≤ 1)CC22: (y-x ≤ -2): (y-x ≤ -2)CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)CC44: (y-z ≤ -2): (y-z ≤ -2)
CC11: (x-y ≤ 1): (x-y ≤ 1)CC22: (y-x ≤ -2): (y-x ≤ -2)CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)CC44: (y-z ≤ -2): (y-z ≤ -2)
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Outline
Problem Description / Motivation
Algorithm Foundations
Musilitis (Algorithm Details)
Experimental Results
Conclusion
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Max-CSP Maximum cardinality feasible set of constraints
Maximal Satisfiable Subset (MSS) Inaugmentable feasible set of constraints
Algorithm Foundations
C1: (x-y ≤ 1)
C2: (y-x ≤ -2)
C3: (y-x ≤ -3) (z-y ≤ 1)
C4: (y-z ≤ -2)
C1: (x-y ≤ 1)
C2: (y-x ≤ -2)
C3: (y-x ≤ -3) (z-y ≤ 1)
C4: (y-z ≤ -2)
Max-CSPMSS
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A CoMSS is the complement of an MSS
A CoMSS provides an irreducible “fix” for the system: removing those constraints makes the system satisfiable (turns it into an MSS).
CoMSSes
Solution Space
C: {C1,C2,C3,C4,C5,C6,C7}C1
C2
C3
C4
C5C6C7
MSS CoMSS
Remaining Solutions
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The Link: CoMSSes and MUSes
Known:1. A system is SAT iff it contains no MUSes
2. Removing the constraints in a CoMSS from a system makes it SAT
Removing the constraints in a CoMSS removes at least one constraint from every MUS in a system.
Every CoMSS is an irreducible hitting set of the collection of all MUSes.
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CC11: (x-y ≤ 1): (x-y ≤ 1)
CC22: (y-x ≤ -2): (y-x ≤ -2)
CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)
CC44: (y-z ≤ -2): (y-z ≤ -2)
CC11: (x-y ≤ 1): (x-y ≤ 1)
CC22: (y-x ≤ -2): (y-x ≤ -2)
CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)
CC44: (y-z ≤ -2): (y-z ≤ -2)
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MUS1 = {C1 , C2}MUS2 = {C1 , C3 , C4}
Hitting Sets Example Given a collection of sets M, a hitting set of M is a set
that contains at least one element from each set in M.
CoMSS1 = {C1}CoMSS2 = {C2 , C3}CoMSS3 = {C2 , C4}
Every CoMSS is an irreducible hitting set of the collection of all MUSes.
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The Duality: CoMSSes and MUSES
Additionally, every MUS is an irreducible hitting set of the collection of all CoMSSes
Hitting sets provide a transformation from one collection to the other
MUSesCoMSSeshitting sets
hitting sets
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Musilitis
General operation of Musilitis:1. Find all CoMSSes of a DTP
2. Compute minimal hitting sets of the CoMSSes
DTP
Find CoMSSes CoMSSesCompute Minimal
Hitting Sets
All MUSes
Musilitis
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Musilitis: Finding CoMSSes
Built on Maxilitis (Moffitt & Pollack, 2005) Max-CSP solver for DTPs Utilizes several constraint processing
techniques with a branch-and-bound process General Strategy:
Find a maximal satisfiable subset of constraints Prevent that set from appearing in a future
solution Iterate until no further satisfiable subsets
remain
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Musilitis: Finding CoMSSes
CC11: (x-y ≤ 1): (x-y ≤ 1)
CC22: (y-x ≤ -2): (y-x ≤ -2)
CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)
CC44: (y-z ≤ -2): (y-z ≤ -2)
CC11: (x-y ≤ 1): (x-y ≤ 1)
CC22: (y-x ≤ -2): (y-x ≤ -2)
CC33: (y-x ≤ -3) : (y-x ≤ -3) (z-y ≤ 1) (z-y ≤ 1)
CC44: (y-z ≤ -2): (y-z ≤ -2)
C1 ¬C1
C2 ¬C2
XC31 ¬C3C32
XC4 ¬C4
X{C2,C4}
Candidate CoMSS
C4
{C2,C3}
C2
C3
C4
{C1}
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Musilitis: Computing One MUS
Goal: Irreducible Hitting Set Straightforward construction, no search Iteratively choose constraints to add to the set
• Choice can be arbitrary
For each chosen constraint, alter the remaining problem to make that constraint essential
• Hitting set is irreducible iff every element is essential
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Musilitis: Computing One MUS
1. Select a constraint to add to the MUS (C3)
CoMSSCoMSS11 = {C = {C11}}CoMSSCoMSS22 = {C = {C22 , C , C33}}CoMSSCoMSS33 = {C = {C22 , C , C44}}
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Musilitis: Computing One MUS
1. Select a constraint to add to the MUS (C3)
2. Select a CoMSS in which it appears(CoMSS2)
CoMSSCoMSS11 = {C = {C11}}CoMSSCoMSS22 = {C = {C22 , C , C33}}CoMSSCoMSS33 = {C = {C22 , C , C44}}
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Musilitis: Computing One MUS
1. Select a constraint to add to the MUS (C3)
2. Select a CoMSS in which it appears(CoMSS2)
3. Remove any other constraints in that CoMSS from the problem – makes the chosen constraint essential
CoMSSCoMSS11 = {C = {C11}}CoMSSCoMSS22 = {C = {C22 , C , C33}}CoMSSCoMSS33 = {C = {C22 , C , C44}}
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Musilitis: Computing One MUS
1. Select a constraint to add to the MUS (C3)
2. Select a CoMSS in which it appears(CoMSS2)
3. Remove any other constraints in that CoMSS from the problem – makes the chosen constraint essential
4. Remove any CoMSSes in which the constraint appears, because they are now hit by the MUS
CoMSSCoMSS11 = {C = {C11}}CoMSSCoMSS22 = {C = {C33}}CoMSSCoMSS33 = {C = {C44}}
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Musilitis: Computing One MUS
1. Select a constraint to add to the MUS (C3)
2. Select a CoMSS in which it appears (CoMSS2)
3. Remove any other constraints in that CoMSS from the problem – makes the chosen constraint essential
4. Remove any CoMSSes in which the constraint appears, because they are now hit by the MUS
5. Iterate until no CoMSSes remain
CoMSSCoMSS11 = {C = {C11}}
CoMSSCoMSS33 = {C = {C44}}
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Musilitis: Computing All MUSes
Use general form of single MUS method Branch on choice of constraint and CoMSS to
make all possible MUSes Tree is not irredundant, so ordering heuristics
and memoization are used to reduce and prune the tree
Very fast in practice: Millions of MUSes in minutes
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Experimental Results
Paramaterized Random Test Case Generatork = # disjuncts per constraint
L = max magnitude of inequality bound (x-y≤b)
m = # constraints
N = # timepoints (variables)
R = constraints / timepoints (m/N) Experiments Run
k = 2
L = 100
N = 5, 6, and 7
R = 6 through 11
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Experimental Results
0.01
0.1
1
10
100
1000
10000
100000
6 7 8 9 10 11R
Seco
nds
N=7, Incremental
N=7, Backtracking
N=6, Incremental
N=6, Backtracking
N=5, Incremental
N=5, Backtracking
Median Runtimes for Finding CoMSSes(C)
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0.001
0.01
0.1
1
10
100
1000
10000
10 100 1000 10000 100000 1000000Number of MUSes
Sec
on
ds
Experimental Results
Runtimes of Extracting MUSes(C) vs |MUSes(C)|
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Related Work on MUSes
Many algorithms for single MUSes Mostly for Boolean Satisfiability, some for
Linear Programs Often approximate, i.e., they return a small US
Other work on finding all MUSes Birnbaum & Lozinskii, 2003 (Knowledge Systems) Liffiton & Sakallah, 2005 (Boolean Satisfiability) Bailey & Stuckey, 2005 (Herbrand Constraints)
Finding the Smallest MUS Mneimneh, et al., 2005 (Boolean Satisfiability)
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Future Work
Relaxations / approximations Trade off completeness/correctness for speed Find fewer than all MUSes Find non-minimal USes
Develop techniques for presenting MUSes to users and enabling informed decisions about weakening constraints to obtain feasibility