Cutting to the Core of Pseudo-Boolean Optimization: Combining Core-Guided Search with Cutting Planes Reasoning

Jo Devriendt ¹²³ Stephan Gocht ¹² Emir Demirović ⁴ Jakob Nordström ²¹ Peter J Stuckey ⁵

 

¹ Lund University, Sweden

² University of Copenhagen, Denmark

³ KU Leuven, Belgium

⁴ TU Delft, The Netherlands

⁵ Monash University, Australia


jodevriendt.com
slides.com/jod/coreguidedpb-full

Objective
(to minimize)

Solutions

Optimization

Solutions

Core-guided Optimization (CGO)

Objective
(to minimize)

Solutions

Objective
(to minimize)

Core-guided Optimization (CGO)

Solutions

Objective
(to minimize)

Core-guided Optimization (CGO)

Solutions

Objective
(to minimize)

Core-guided Optimization (CGO)

Solutions

Objective
(to minimize)

Core-guided Optimization (CGO)

  • All variables          are Boolean (0, 1)
  • Constraints C are linear inequalities
  • Formula φ is a conjunction of constraints
  • Negation:
    • A literal    is a variable or its negation
  • Linear objective function O to be minimized
    • WLOG, objective has only positive coefficients

Prelims

Basically, we are doing 0-1 integer linear programming in this talk

x, y
\overline{x}=1-x
l
  • All variables          are Boolean (0, 1)
  • Constraints C are linear inequalities
  • Formula φ is a conjunction of constraints
  • Negation:
    • A literal    is a variable or its negation
  • Linear objective function O to be minimized
    • WLOG, objective has only positive coefficients
  • Solver input
    • Formula
    • Assumptions
  • Solver output, either
    • Solution satisfying      under
    • Core constraint implied by     and violated by

Prelims: 0-1 linear programming

x, y
\overline{x}=1-x
l
\alpha \colon \{l, \ldots\}
\varphi
\varphi
\alpha
\alpha
\varphi

Prelims

\alpha \colon \{l, \ldots\}
\varphi

Solver abstraction:

  • Input
    • Formula
    • Assumptions
       
  • Output, either
    • Solution satisfying      under
    • Core constraint implied by     and violated by
\varphi
\alpha
\alpha
\varphi

I

Classic

Core-guided Optimization

Solve under assumptions

Extract core

Reformulate problem

Increase lower bound

  • Original core-guided idea [FM06]
  • CGO also succesfully applied in ASP [AKMS12] and CP [GBDS20]
  • Described technique: OLL [AKMS12, MDM14] and ONE [ADR15]

CGO loop

Solve under assumptions

Extract core

Reformulate problem

Increase lower bound

  • Original core-guided idea [FM06]
  • CGO also succesfully applied in ASP [AKMS12] and CP [GBDS20]
  • Described technique: OLL [AKMS12, MDM14] and ONE [ADR15]

CGO loop

Solve under assumptions

  • Assume all variables in objective to be false
\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\

Extract core

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • Classic case: conflict-driven clause learning solver
  • Core is a clause
\mathit{Core}\colon x_2+x_3+x_4 \geq 1

Reformulate problem

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • Determine     :
    smallest coefficient in objective with variable in core
  • Enlarge core to extension constraint Ext with one fewer auxiliary variables
  • Add Ext to     with symmetry breakers
    • Sound as     implies Core
  • Reformulate objective:
    Multiply Ext by    ,
    subtract lhs and add rhs to O
\mathit{Core}\colon x_2+x_3+x_4 \geq 1
\mathbf{c^*}
c^*=2
\varphi
y_i \geq y_{i+1}
\mathit{Ext}\colon x_2+x_3+x_4 = 1+y_1+y_2
\varphi
O'\colon x_1+x_3+2x_4+2+2y_1+2y_2\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
c^*
\varphi' = \varphi \wedge \mathit{Ext} \wedge y_1 \geq y_2

Increase lower bound

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • Lower bound increase is simply
\mathit{Core}\colon x_2+x_3+x_4 \geq 1
c^*=2
\mathit{Ext}\colon x_2+x_3+x_4 = 1+y_1+y_2
O'\colon x_1+x_3+2x_4+2+2y_1+2y_2\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
c^*
\varphi' = \varphi \wedge \mathit{Ext} \wedge y_1 \geq y_2

Rinse and repeat

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • Repeat procedure with reformulated problem
\mathit{Core}\colon x_2+x_3+x_4 \geq 1
c^*=2
\mathit{Ext}\colon x_2+x_3+x_4 = 1+y_1+y_2
O'\colon x_1+x_3+2x_4+2+2y_1+2y_2\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
\varphi' = \varphi \wedge \mathit{Ext} \wedge y_1 \geq y_2
\varphi', O', \alpha'=\{\overline{x_1},\overline{x_3},\overline{x_4},\overline{y_1},\overline{y_2}\}
\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
\mathit{Core}\colon x_2+x_3+x_4 \geq 1
c^*=2
\mathit{Ext}\colon x_2+x_3+x_4 = 1+y_1+y_2
O'\colon x_1+x_3+2x_4+2+2y_1+2y_2\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
\varphi' = \varphi \wedge \mathit{Ext} \wedge y_1 \geq y_2
  • Assume all variables in objective to be false
  • Conflict-driven clause learning solver returns clausal core
  • Determine     :
    smallest coefficient in objective with variable in core
  • Enlarge core to extension constraint Ext with one fewer auxiliary variables
  • Add Ext to     with symmetry breakers
    • Sound as     implies Core
  • Reformulate objective:
    Multiply Ext by    ,
    subtract lhs and add rhs to O
  • Lower bound increase is simply
  • Repeat procedure with reformulated problem
\mathbf{c^*}
\varphi
y_i \geq y_{i+1}
\varphi
c^*
c^*
\varphi', O', \alpha'=\{\overline{x_1},\overline{x_3},\overline{x_4},\overline{y_1},\overline{y_2}\}

II

Cutting-planes

Core-guided Optimization

(our contribution)

Solve under assumptions

  • Assume all variables in objective to be false
\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\

Extract cardinality core

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • With a cutting-planes reasoning
    solver, core is a linear inequality
  • Can contain non-assumptions or negative coefficients
    • Eliminate by adding Boolean axioms
  • Can have arbitrarily large coefficients
    • Reduce to cardinality core, e.g., by division with largest coefficient
\mathit{Core}\colon x_0-x_1+3x_2+3x_3+x_4 \geq 5
1 \geq x, x \geq 0
\mathit{Core'}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card}\colon x_2+x_3+x_4 \geq 2

Reformulate problem

\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
  • Determine    
  • Determine cardinality degree d
  • Enlarge core to extension constraint Ext with d fewer auxiliary variables
  • Add Ext to     with symmetry breakers
  • Reformulate objective
c^*
c^*=2
y_i \geq y_{i+1}
\mathit{Ext}\colon x_2+x_3+x_4 = 2+y_1
\varphi
O'\colon x_1+x_3+2x_4+4+2y_1\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
\varphi' = \varphi \wedge \mathit{Ext}
\mathit{Core}\colon x_0-x_1+3x_2+3x_3+x_4 \geq 5
\mathit{Core'}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card}\colon x_2+x_3+x_4 \geq 2
d=2

Increase lower bound

  • Lower bound increase is
  • Rinse and repeat
c^*d
\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
c^*=2
\mathit{Ext}\colon x_2+x_3+x_4 = 2+y_1
O'\colon x_1+x_3+2x_4+4+2y_1\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
\varphi' = \varphi \wedge \mathit{Ext}
\mathit{Core}\colon x_0-x_1+3x_2+3x_3+x_4 \geq 5
\mathit{Core'}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card}\colon x_2+x_3+x_4 \geq 2
d=2

Advantage of cutting planes vs clausal

  • Stronger (cardinality) core
  • Stronger lower bound per core
  • No encoding of Ext to clauses

Cutting Planes CGO Example

c^*d
\alpha=\{\overline{x_1},\overline{x_2},\overline{x_3},\overline{x_4}\}
\varphi, O\colon x_1+2x_2+3x_3+4x_4\\
c^*=2
\mathit{Ext}\colon x_2+x_3+x_4 = 2+y_1
O'\colon x_1+x_3+2x_4+4+2y_1\\
O' = O - c^*\mathit{lhs_{Ext}} + c^*\mathit{rhs_{Ext}}
\varphi' = \varphi \wedge \mathit{Ext}
\mathit{Core}\colon x_0-x_1+3x_2+3x_3+x_4 \geq 5
\mathit{Core'}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card}\colon x_2+x_3+x_4 \geq 2
d=2
  • Assume all variables in objective to be false
  • With a cutting-planes reasoning
    solver, core is a linear inequality
  • Can contain non-assumptions or negative coefficients
    • Eliminate by adding Boolean axioms
  • Can have arbitrarily large coefficients
    • Reduce to cardinality core, e.g., by division with largest coefficient
  • Determine    
  • Determine cardinality degree d
  • Enlarge core to extension constraint Ext with d fewer auxiliary variables
  • Add Ext to     with symmetry breakers
  • Reformulate objective
  • Lower bound increase is
1 \geq x, x \geq 0
c^*
y_i \geq y_{i+1}
\varphi

III

Cutting-planes

Core-guided Optimization
Optimizations

Cardinality reduction optimization

O\colon x_1+2x_2+3x_3+4x_4\\ \mathit{Core}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card_1}\colon x_2+x_3+x_4 \geq 2\\ \mathit{Card_2}\colon x_3+x_4 \geq 1
  • Multiple implied cardinalities
    • Pick one that leads to best
      lower bound       increase
2*2\\ 1*3
c_1^* = 2\\ c_2^*=3

Lower bounds:

Cardinality reduction optimization

O\colon x_1+2x_2+3x_3+4x_4\\ \mathit{Core}\colon 3x_2+3x_3+x_4 \geq 4
\mathit{Card_1}\colon x_2+x_3+x_4 \geq 2\\ \mathit{Card_2}\colon x_3+x_4 \geq 1
  • Multiple implied cardinalities
  • Pick one that leads to best
    lower bound     increase
2*2\\ 1*3
c_1^* = 2\\ c_2^*=3

Lower bounds:

Lazy extension encoding

\mathit{Card}\colon x_1+x_2+x_3+x_4 \geq 2\\ \mathit{Ext}\colon x_1+x_2+x_3+x_4 = 2+y_1+y_2\\ y_1 \geq y_2
  • Auxiliary variables represent bounds on lhs
  • Extension constraint can be split along the auxiliary variables
  • Setting      to false implies         must be false
  • As long as      is false, the constraint for         is not needed
  • Lazily introduce variable         and corresponding constraint
  • Similar in spirit to [MJML14]
y_1 \Leftrightarrow x_1+x_2+x_3+x_4 \geq 3\\ y_2 \Leftrightarrow x_1+x_2+x_3+x_4 \geq 4\\
y_i
y_{i+1}
y_i
y_{i+1}
y_{i+1}

Lazy extension encoding

\mathit{Card}\colon x_1+x_2+x_3+x_4 \geq 2\\ \mathit{Ext}\colon x_1+x_2+x_3+x_4 = 2+y_1+y_2\\ y_1 \geq y_2
  • Auxiliary variables represent bounds on lhs
  • Extension constraint can be split along the auxiliary variables
  • Setting      to false implies         must be false
  • As long as      is false, the constraint for         is not needed
  • Lazily introduce variable         and corresponding constraint
  • Similar in spirit to [MJML14]
y_1 \Leftrightarrow x_1+x_2+x_3+x_4 \geq 3\\ y_2 \Leftrightarrow x_1+x_2+x_3+x_4 \geq 4\\
y_i
y_{i+1}
y_i
y_{i+1}
y_{i+1}

Interleaving core-guided and linear optimization

Solutions

Objective
(to minimize)

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

Solutions

Objective
(to minimize)

Interleaving core-guided and linear optimization

Related ideas in [ADMR15]

IV
Experiments

Overall performance

[PB16]

Overall performance

[PB16]

Overall performance

[PB16]

Overall performance

[PB16]

Stronger cores reach optimality faster

Fewer # variables due to lazy encoding

Take-away

Our contribution

  • Core-guided optimization with cutting-planes reasoning instead of clausal reasoning
  • Investigated different ways to optimize process

Cutting-planes reasoning advantages

  • More efficient search for cores
  • Can find stronger cores
  • No re-encoding to clauses needed
  • Proposed optimizations are easy to implement

Conclusion

  • Huge performance improvement in RoundingSat
  • Still behind MIP solvers on many instances
  • Future work: incorporate MIP preprocessing techniques?

References

  • [FM06] On Solving the Partial MAX-SAT Problem; Fu, Malik; 2006
  • [AKMS12] Unsatisfiability-Based Optimization in clasp; Andres, Kaufmann, Matheis, Schaub; 2012
  • [MDM14] Core-Guided MaxSAT with Soft Cardinality Constraints; Morgado, Dodaro, Marques-Silva; 2014
  • [MJML14] Martins, Joshi, Manquinho, Lynce; Incremental Cardinality Constraints for MaxSAT; 2014
  • [ADMR15] Alviano, Dodaro, Marques-Silva, Ricca; Optimum stable model search: algorithms and implementation; 2015
  • [ADR15] Alviano, Dodaro, Ricca; A MaxSAT Algorithm Using Cardinality Constraints of Bounded Size; 2015
  • [PB16] Latest PB competition - www.cril.univ-artois.fr/PB16/
  • [GBDS20] Core-Guided and Core-Boosted Search for CP; Gange, Berg, Demirović, Stuckey; 2020

Combining Core-Guided Search with Cutting Planes Reasoning - FULL

By Jo Devriendt

Combining Core-Guided Search with Cutting Planes Reasoning - FULL

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