Watched Propagation for
0-1 Integer Linear Constraints
Jo Devriendt
MIAO research group
Lund University, Sweden
& University of Copenhagen, Denmark
jodevriendt.com
Context: pseudo-Boolean solvers
- Find an (optimal) solution to a 0-1 integer linear program, or prove that none exist
-
CDCL-like: depth-first search, learn 0-1 integer linear (IL) constraints
- huge database of learned constraints
How to efficiently detect conflicts and propagations?
Efficiently detect
conflicts and propagations
For clauses (e.g., SAT solvers)
- Counter approach
- 2-watched propagation [pioneered by Chaff]
2-watched is clearly the most efficient
Efficiently detect
conflicts and propagations
For 0-1 IL constraints (e.g., PB solvers)
- Counter approach [Galena, Sat4J]
-
Watched approach [Galena, Pueblo, Sat4J, RoundingSat]
- lots of variations
- Specialized clause & cardinality propagation [Galena, Sat4J]
- 2-watched & k-watched propagation
Does watched propagation work best?
Contributions
New watched propagation algorithm
- unique optimizations
Experimental results
- New watched algorithm is more efficient than counter propagation ...
- ... but when adding specialized clause & cardinality routines, difference is small
0-1 integer linear constraints
10l_1+5l_2+\sum_{i=3}^{100}{l_i} \geq 10
5x+4y+3\overline{z}+2\overline{u}+\overline{v} \geq 6
x+y+\overline{z}+\overline{u} \geq 2
x+y+\overline{z} \geq 1
\sum_{i=1}^{100}10l_i+2l_{101}+l_{102}+l_{103} \geq 52
Clause
Cardinality constraint
General PB constraint
\sum_i c_i l_i \geq d\\
\text{literals } l_i \text{, } \overline{l_i}=1-l_i \\
\text{coefficients } c_i \in \mathbb{N}^+ \\
\text{degree } d \in \mathbb{N} \\
Pseudo-Boolean (PB) normal form:
eg
Complicating factors
| Clause | Cardinality | General constraint | |
|---|---|---|---|
| coefficients | 1 | 1 | ? |
Complicating factors
| Clause | Cardinality | General constraint | |
|---|---|---|---|
| coefficients | 1 | 1 | ? |
| # watches | O(1), 2 | O(n), d+1 | O(n), variable |
Complicating factors
| Clause | Cardinality | General constraint | |
|---|---|---|---|
| coefficients | 1 | 1 | ? |
| # watches | O(1), 2 | O(n), d+1 | O(n), variable |
| propagation | single | single | multiple |
Complicating factors
| Clause | Cardinality | General constraint | |
|---|---|---|---|
| coefficients | 1 | 1 | ? |
| # watches | O(1), 2 | O(n), d+1 | O(n), variable |
| propagation | single | single | multiple |
| after propagation | satisfied |
satisfied | may still be(come) conflicting |
Core ideas of 0-1 IL
watched propagation
- General:
- slack of a constraint
- conflict condition
- propagation condition
- Algorithmic:
- watch slack invariant
- watch set invariant
Slack under partial assignment
- Partial variable assignment: set of pairwise disjoint literals ρ
- Slack of a constraint C denotes best possible evaluation under ρ
C \equiv 5x+4y+3\overline{z}+2\overline{u}+\overline{v} \geq 6
\mathrm{slack}(C,\rho)\equiv \sum_{i \colon \overline{l_i} \not \in \rho} c_i -d
\mathrm{slack}(C,\{\overline{x},\overline{u},v\})=4+3+2-6=3
eg
Typesetting convention
\textcolor{grey}{5x}+4y+3\overline{z}+\boldsymbol{2\overline{u}}+\textcolor{grey}{\overline{v}} \geq 6 \\
\text{where } \rho = \{\overline{x},\overline{u},v\}
Conflict condition
\mathrm{slack}(C,\rho) < 0
C is conflicting iff
Propagation condition
\mathrm{slack}(C,\rho) < c_i
C \equiv \textcolor{grey}{5x}+4y+3\overline{z}+\boldsymbol{2\overline{u}}+\textcolor{grey}{\overline{v}} \geq 6
\mathrm{slack}(C,\{\overline{x},\overline{u},v\})=3
\text{since } \mathrm{slack}(C,\rho \cup \{\overline{l_i}\})<0
eg
C propagates iff
l_i
\text{where } l_i, \overline{l_i} \not \in \rho
Propagation condition
\mathrm{slack}(C,\rho) < c_i
C \equiv \textcolor{grey}{5x}+4y+3\overline{z}+\boldsymbol{2\overline{u}}+\textcolor{grey}{\overline{v}} \geq 6
\mathrm{slack}(C,\{\overline{x},\overline{u},v\})=3<4 \\
\text{where } y \not \in \{\overline{x},\overline{u},v\}
\text{since } \mathrm{slack}(C,\rho \cup \{\overline{l_i}\})<0
eg
C propagates iff
l_i
\text{where } l_i, \overline{l_i} \not \in \rho
Watch slack invariant
C \equiv \textcolor{grey}{\underline{5x}}+\underline{4y}+\underline{3\overline{z}}+\boldsymbol{2\overline{u}}+\textcolor{grey}{\overline{v}} \geq 6
\mathrm{watches}(C)=\{x,y,\overline{z}\}
eg
\mathrm{watchslack}(C,\rho)\equiv \sum_{i \colon l_i \in \mathrm{watches}(C) \text{ and } \overline{l_i} \not \in \rho} c_i -d
watches(C) is a subset of C's literals
Watch slack invariant:
the algorithm keeps track of the watch slack of a constraint
\mathrm{watchslack}(C,\{\overline{u},v\})=4+3-6=1
Watch slack invariant
\mathrm{watchslack}(C,\rho) \geq \mathrm{maxcoef}(C)
maxcoef(C) is C's largest coefficient
What happens if
if
\mathrm{slack}(C,\rho) \geq \mathrm{watchslack}(C,\rho)
then C is neither conflicting nor propagating
since
\mathrm{watchslack}(C,\rho) < \mathrm{maxcoef}(C)
?
Watch set invariant
hence
\mathrm{watchslack}(C,\rho) < \mathrm{maxcoef}(C) \\ \Rightarrow \\ \forall l \in C \setminus \mathrm{watches}(C) \colon \overline{l} \in \rho
\mathrm{watchslack}(C,\rho) < \mathrm{maxcoef}(C) \\ \Rightarrow \\ \mathrm{watchslack}(C,\rho) = \mathrm{slack(C,\rho)}
Particularities of proposed approach
- Fixed decreasing-coefficient literal order
- Store index of watched literal in watch lists
-
Keep falsified watches if watchslack(C,ρ)<maxcoef(C)
- cheap watchslack update during backjumps
- Important optimizations based on whether a backjump happened since last watch change for a constraint
Example
\underline{4x}+\underline{3y}+\underline{3z}+2u+2v+w \geq 5
\underline{4x}+\underline{3y}+\underline{3z}+2u+2v+\textcolor{grey}{w} \geq 5
\underline{4x}+\underline{3y}+\textcolor{grey}{3z}+\underline{2u}+2v+\textcolor{grey}{w} \geq 5
\rho
C
\textrm{watchslack}(C,\rho)
5
5
4
\{\}
\{\overline{w}\}
\{\overline{w},\overline{z}\}
\textcolor{grey}{\underline{4x}}+\underline{3y}+\textcolor{grey}{3z}+\underline{2u}+\underline{2v}+\textcolor{grey}{w} \geq 5
2
\{\overline{w},\overline{z},\overline{x}\}
\textcolor{grey}{\underline{4x}}+\boldsymbol{\underline{3y}}+\textcolor{grey}{3z}+\underline{2u}+\underline{2v}+\textcolor{grey}{w} \geq 5
2
\{\overline{w},\overline{z},\overline{x},y\}
propagation
\underline{4x}+\underline{3y}+{3z}+\underline{2u}+\underline{2v}+{w} \geq 5
6
\{\}
backjump to root
\underline{4x}+\underline{3y}+{3z}+\textcolor{grey}{2u}+\underline{2v}+{w} \geq 5
4
\{\overline{u}\}
...
Experimental evaluation
- 'watch-opt' implementation in RoundingSat
- Compare to
- old RoundingSat watched propagation
- counter implementation
- with/without specialized clause and cardinality propagation (-cc)
- Measure propagations/second on
PB competition and MIPLIB benchmarks- limited to instances solved within 1s and 5000s by compared configurations
New algorithm improves on
old algorithm
propagations / second
Specialized routines don't change the picture much
propagations / second
Watched propagation is faster than counter ...
propagations / second
... but counter catches up when adding specialized clause & cardinality routines
propagations / second
Future work
- Use a dynamic max coefficient instead of a static max coefficient
- allows to reduce the amount of watched literals
- Handle some constraints with counter propagation, others with watched
- Determine the propagation order of constraints based on their strength
Conclusion
- Investigated watched propagation for 0-1 IL constraints
- Proposed a new watched propagatation variant
- Effective in practice
- Not yet clearly beating counter propagation