This document proposes an approximate encoding method for at-most-k constraints that reduces boolean expressions compared to conventional encodings, though it does not cover all possible solutions. It recursively applies at-most constraints by multiplying the number of variables. For a 2x2 recursive model, it achieves a 44% solution coverage using only 15% of the literals required by a sequential counter encoding. The approximate encoding is intended for use with soft constraints to efficiently search for better solutions, rather than finding all possible solutions.

1. Approximate-At-Most-k Encoding
of SAT for Soft Constraints
Shunji Nishimura
National Institute of Technology, Oita College, Japan
PoS2023
1

2. At-most-k constraints and encodings
 the number of true values ≦ k
 problem: Boolean expressions will explode
 proposed encodings in the past:
binary, sequential counter, commander, product, etc..
2

3.  the number of true values ≦ k
 problem: Boolean expressions will explode
 proposed encodings in the past:
binary, sequential counter, commander, product, etc..
3
are absolutely at-most-k
here is approximately at-most-k
At-most-k constraints and encodings

4.  the number of true values ≦ k
 problem: Boolean expressions will explode
 proposed encodings in the past:
binary, sequential counter, commander, product, etc..
4
are absolutely at-most-k
here is approximately at-most-k
covers all solutions
only covers
a part of solutions
At-most-k constraints and encodings

5. Conventional vs Approximate
5
solution
coverage
purposes
conventional complete
hard and soft
constraints
approximate incomplete
only soft
constraints
 hard constraints: necessities
 solt constraints: to describe optional desires

6. Conventional vs Approximate
6
solution
coverage
purposes
conventional complete
hard and soft
constraints
approximate incomplete
only soft
constraints
 hard constraints: necessities
 soft constraints: to describe optional desires
but drastically reduces
Boolean expressions

7. Soft constraints
Not necessary but preferred
 In common with optimization problems
 Example: university timetabling
 minimize empty time slots in between
 minimize the number of teachers who have continuous classes
 it is preferable a subject is always taught in the same room
7

8. Fundamental idea
8
A
B
at most 2 of 4

9. Fundamental idea
9
2 times
at most
A
A1
B1
B
at most 2 of 4

10. Fundamental idea
10
2 times
at most
A
A1
B1
B
at most 2 of 4
at most 0
0 trues

11. Fundamental idea
11
2 times
at most
A
A1
B1
B
at most 2 of 4
at most 2
1 true

12. Fundamental idea
12
2 times
at most
A
A1
B1
B
at most 2 of 4
at most 4
2 trues

13. Fundamental idea
13
2 times
at most
A
A1
B1
B
at most 2 of 4
0 trues ⇒ at most 0
∧ 1 true ⇒ at most 2
∧ 2 true ⇒ at most 4
=

14. Fundamental idea
14
2 times
at most
at most 2 of 4
at most 4 of 8 (?)
2 times
at most
A
A1
B1
B

15. Fundamental idea
15
2 times
at most
at most 2 of 4
at most 4 of 8 (?)
2 times
at most
A
A1
B1
B
←OK(SAT)
true false

16. Fundamental idea
16
2 times
at most
at most 2 of 4
at most 4 of 8
(approximate)
2 times
at most
A
A1
B1
B
←UNSAT

17. 17
• again, is not a real at-most-k
• should use for only soft constraints

18. 2x2 models
18
x 2 x 2
x2 x2 x2 x2
 two parents and four children
 define recursively

19. 2x2 models
19
x 2 x 2
x2 x2 x2 x2
m
at most k of 4
approximate-at-most-
(k/2 · 2m+1) of 2m+1

20. h x w models
20
h1
w1
h2
w2
h2
w2
hn
wn
hn・m hn・m
hn
wn
 height h and width w

21. h x w models
21
h1
w1
h2
w2
h2
w2
hn
wn
hn・m hn・m
hn
wn
at most k of h1·w1
approximate-at-most-
(k/(h1·w1) · Π hi·wi·m)
of Π hi·wi·m

22. h x w models
22
h1
w1
h2
w2
h2
w2
hn
wn
hn・m hn・m
hn
wn
at most k of h1·w1
approximate-at-most-
(k/(h1·w1) · Π hi·wi·m)
of Π hi·wi·m
notation:
approximate-at-most-(a/b·n) of n

23. Literal number comparison (2x2 models)
23
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
8 16 32 64 128
literals
at-most-1/2 of
approximate literals counter literals
vs sequential counter encoding

24. Literal number comparison (2x2 models)
24
0%
10%
20%
30%
40%
50%
60%
0
5,000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
45,000
8 16 32 64 128
rate
(=approximate
/counter)
literals
at-most-1/2 of
approximate literals counter literals rate

25. Coverages (2x2 models)
25
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
8 16 32 64 128
coverage
/
rate
at-most-1/2 of
solution coverage
= (solutions by approximate) / (all solutions)

26. Coverages (2x2 models)
26
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
8 16 32 64 128
coverage
/
rate
at-most-1/2 of
solution coverage literal rate

27. Coverages (2x2 models)
27
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
8 16 32 64 128
coverage
/
rate
at-most-1/2 of
solution coverage literal rate
covers 44% on 15% literals

28. Coverages and efficiencies (2x2 models)
28
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
8 16 32 64 128
efficiency
(=
coverage
/
literal
rate)
coverage
/
rate
at-most-1/2 of
solution coverage literal rate efficiency
efficiency = coverage / literal rate

29. 8 of 16
Target variables
29
h x w models: adjustment
want to generate arbitrary k of n

30. 8 of 16
fix to false fix to true
30
h x w models: adjustment

31. 8 of 16
fix to false fix to true
4 of 10
31
h x w models: adjustment

32. 32
h x w models: example1
to generate 5 of 10
at most 2
approximate-at-most-
6 of 12

33. 33
h x w models: example1
to generate 5 of 10
fix 1 false and 1 true
at most 2
approximate-at-most-
6 of 12
approximate-at-most-
5 of 10

34. 34
h x w models: example2
to generate 5 of 10
fix 3 falses and 3 trues
at most 2
approximate-at-most-
8 of 16
approximate-at-most-
5 of 10

35. The best efficiencies
35
0
2
4
6
8
10
12
14
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
efficiency
approximate-at-most-
of 10 of 20 of 30

36. The best efficiencies
36
0
2
4
6
8
10
12
14
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
efficiency
approximate-at-most-
of 10 of 20 of 30

37. Low efficiency between highs
37
24 of 30 :
high efficiency
at-most-6
fix 2 falses and 0 trues
26 of 30 :
high efficiency
at-most-7
fix 0 falses and 2 trues
(24 of 32 → 24 of 30) (28 of 32 → 26 of 30)
25 of 30 :
low efficiency
at-most-5
fix 1 false and 5 trues
(30 of 36 → 25 of 30)

38. Low efficiency between highs
38
24 of 30 :
high efficiency
at-most-6
fix 2 falses and 0 trues
26 of 30 :
high efficiency
at-most-7
fix 0 falses and 2 trues
(24 of 32 → 24 of 30) (28 of 32 → 26 of 30)
25 of 30 :
low efficiency
at-most-5
fix 1 false and 5 trues
(30 of 36 → 25 of 30)

39. Discussion1: coverage definition
39
all solutions
∪
at-most-8
∪
at-most-7
∪
:
∪
at-most-1
∪
no trues

40. Discussion1: coverage definition
40
all solutions
∪
at-most-8
∪
at-most-7
∪
:
∪
at-most-1
∪
no trues
this study’s definition

41. Discussion1: coverage definition
41
all solutions
∪
at-most-8
∪
at-most-7
∪
:
∪
at-most-1
∪
no trues
this study’s definition
maybe important

42. Discussion2: probability of finding solutions
When approximate-at-most-k covers 50% of the possible
solutions, every single solution has probability 50% to be found.
42

43. Discussion2: probability of finding solutions
When approximate-at-most-k covers 50% of the possible
solutions, every single solution has probability 50% to be found.
43
For a real-life problem ..
• has 1 solution → 50% to find
• has 2 solutions → 75% to find (whichever)
：
• has 10 solutions → 99.9% to find (whichever)
：

44. Discussion2: probability of finding solutions
When approximate-at-most-k covers 50% of the possible
solutions, every single solution has probability 50% to be found.
44
For a real-life problem ..
• has 1 solution → 50% to find
• has 2 solutions → 75% to find (whichever)
：
• has 10 solutions → 99.9% to find (whichever)
：
coverage ≠ finding probability

45. Conclusion
 at-most-k constraints are recursively applied (with multiplying)
 less Boolean expressions needed than conventional encodings,
but does not cover all solutions
 available for searching better solutions under soft constraints
 Ex. at-most-16 of 32
 only 15% of literal number (vs sequential counter)
 covers 44% of the solution space
45