Solving the MinimumCost Satisfiability Problem Using SAT Based BranchandBound Search

1
Solving the MinimumCost Satisfiability Problem
Using SAT Based BranchandBound Search

• Zhaohui Fu, Sharad Malik
• Princeton University

2
Problem Definition

• MinCostSAT Given a Boolean formula f with
• n variables
• each costs
• Find a variable assignment
• satisfies f
• minimizes

3
Motivation Previous Work

• Various applications, e.g. Automatic Test Pattern
  Generation, FPGA Routing, AI Planning, etc.
• Best branch-and-bound solvers are old
• Bsolo (Manquinho and Marques-Silva, 2002) is
  based on old SAT solver GRASP.
• scherzo (Coudert, 1996) does not scale to large
  problems.
• SAT techniques have advanced dramatically since
  then
• Two literal watching based fast BCP.
• Better decision heuristic, e.g. VSIDS, Berkmin.

4
Classic Covering Algorithms

• The earliest important forms of MinCostSAT are
  the Unate/Binate Covering Problems.

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BranchandBound Search
CC 0 LB 6
CC 0 LB 6
CC 2 LB 5
CC 2 LB 5
CC 2 LB 4
CC 4 LB 3
CC 4 LB 3
CC 4 LB 3
CC 3 LB 5
Solution Found! Cost 7
CC 5 LB 3
CC 5 LB 3
Solution Found! Cost 8
Optimal!
Solution 8
Solution 8
Solution 7
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Maximum Independent Set (MIS) Based Lower
Bounding Functions

• Clauses in MIS do not share any common variable.

Size of the MIS 3
Size of the MIS 4
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Non-MIS Based Lower Bounding Functions

• Convert to Linear Programming
• Minimize Vwxy Vwxz
• Subject to
• Vwxy Vwxz gt 0
• Vwxy Vwxz gt 0
• 0 Vwxy 1, 0 Vwxz 1,
• Integer solution provides better lower bound.
• Cutting Plane techniques accelerate ILP.

8
SAT Based Algorithms

• SAT Based BranchandBound Search
• bsolo (Manquinho and Marques-Silva, 2002)
  implemented on top of GRASP
• MIS Based Lower Bounding Functions
• (x2x4)(x1x2x3)(x4x5)(x5x6x7)
• (x1x2x3)(x4x5) are independent.
• At least 2 variables must be 1
• MIS is computed dynamically, i.e. every time a
  decision is made.

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MinCostChaff

• MIS Based Lower Bounding Function
• Precomputed Static MIS.
• Dynamically maintained.
• Compact Blocking Clause
• Several effective techniques adopted from bsolo.
• SAT Optimization Techniques
• Branch Variable Selection.
• No Expensive Simplifications.

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MIS Based Lower Bound

• Why not LP?
• MIS is simple
• Large SAT instances more constrained than general
  LP ones.
• Challenges
• Two literal watching BCP does not track
  unresolved clauses.
• Efficiency LB is computed each time a decision
  is made.

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Precomputed Static MIS

• An MIS of clauses is selected before the search.
• LB computation only considers the clauses in this
  MIS.
• The status of each clause in the MIS is
  dynamically checked against the current variable
  assignment.

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MIS Construction
Cost c1 1 c2 2 c3 1 c4 6 c5 3 c6
4 c7 9 c8 7 c9 5
Clauses
MIS
Expected Cost
(x1 x2 x3)
4/3
(x1 x2 x6 x8)
12/4
(x5 x7 x8)
(x3 x9)
6/2
(x3 x9)
6/2
(x3 x4 x5 x6)
13/4
(x3 x4 x5 x6)
13/4
(x3 x4 x5 x6)
(x5 x7 x8)
16/3
(x5 x7 x8)
16/3
(x6 x7)
4/2
(x3 x9)
(x7 x9)
9/2
(x8 x5)
3/2
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Computing Lower Bound using Pre-computed MIS
Total Cost
Assignment
Cost c1 1 c2 2 c3 1 c4 6 c5 3 c6
4 c7 9 c8 7 c9 5
MIS
x1 1 x3 1 x5 0
x2 1 x4 0 x5 1 x8 0
S
S
4
10
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Experiments (1)

• 2 literal watching with adaptive (static) MIS
  vs. counter based BCP using dynamic MIS.

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Experiments (2)

• 2 literal watching with adaptive (static) MIS
  vs. 2 literal watching with dynamic MIS.

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Experiments (3)

• 2 literal watching with adaptive (static) MIS
  vs. MiniSat (Eén and Sörensson, 2005) (encoding
  based)

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Experiments (4)
2 literal watching with adaptive (static) MIS
vs. bsolo (Manquinho and Marques-Silva, 2002).
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Experiments (5)
2 literal watching with adaptive (static) MIS
vs. scherzo (Coudert, 1996)
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Experiments (6)
2 literal watching with adaptive (static) MIS
vs. cplex.
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Conclusion

• Presented an efficient solver MinCostChaff that
  applies advanced techniques in SAT to MinCostSAT
  using Branch-and-Bound search.
• Pre-computed, dynamically maintained MIS takes
  full advantage of two literal watching BCP.
• The adaptive lower bounding is effective.
• However, MinCostChaff does not work well on
  classic covering benchmarks.
• The performance of MinCostChaff is comparable to
  MiniSat on benchmarks with small number of
  non-zero cost variables.