In [1] a puzzle is mentioned:
I.e. we want to forbid patterns like \(111\),\(1x1x1\), \(1xx1xx1\) and \(000\),\(0x0x0\), \(0xx0xx0\). It is solved as a SAT problem in [1], but we can also write this down as a MIP. Instead of just 0s and 1s we make it a bit more general: any color from a set \(c\) can be used (this allows us to use more colors than just two and we can make nice pictures of the results). A MIP formulation can look like the two-liner:
\[\begin{align}& x_{i,c} + x_{i+k,c} + x_{i+2k,c} \le 2 && \forall i,k,c \text{ such that } i+2k \le n\\ & \sum_c x_{i,c}=1 && \forall i\end{align} \]
where \(x_{i,c} \in \{0,1\}\) indicating whether color \(c\) is selected for position \(i\). For \(n=8\) and two colors we find six solutions:
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For \(n=9\) we can not find any solution. This is sometimes denoted as the Van der Waerden number \(W(3,3)=9\).
For small problems we can enumerate these by adding cuts and resolving. For slightly larger problems the solution pool technique in solvers like Cplex and Gurobi can help.
For three colors \(W(3,3,3)=27\) i.e. for \(n=27\) we cannot find a solution without three equally spaced colors. For \(n=26\) we can enumerate all 48 solutions:
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Only making this problem a little bit bigger is bringing our model to a screeching halt. E.g. \(W(3,3,3,3)=76\). I could not find solutions for \(n=75\) within 1,000 seconds, let alone enumerating them all.
Find a binary sequence \(x_1, \dots , x_8\) that has no three equally spaced \(0\)s and no three equally spaced \(1\)s
I.e. we want to forbid patterns like \(111\),\(1x1x1\), \(1xx1xx1\) and \(000\),\(0x0x0\), \(0xx0xx0\). It is solved as a SAT problem in [1], but we can also write this down as a MIP. Instead of just 0s and 1s we make it a bit more general: any color from a set \(c\) can be used (this allows us to use more colors than just two and we can make nice pictures of the results). A MIP formulation can look like the two-liner:
\[\begin{align}& x_{i,c} + x_{i+k,c} + x_{i+2k,c} \le 2 && \forall i,k,c \text{ such that } i+2k \le n\\ & \sum_c x_{i,c}=1 && \forall i\end{align} \]
where \(x_{i,c} \in \{0,1\}\) indicating whether color \(c\) is selected for position \(i\). For \(n=8\) and two colors we find six solutions:
For \(n=9\) we can not find any solution. This is sometimes denoted as the Van der Waerden number \(W(3,3)=9\).
For small problems we can enumerate these by adding cuts and resolving. For slightly larger problems the solution pool technique in solvers like Cplex and Gurobi can help.
For three colors \(W(3,3,3)=27\) i.e. for \(n=27\) we cannot find a solution without three equally spaced colors. For \(n=26\) we can enumerate all 48 solutions:
Only making this problem a little bit bigger is bringing our model to a screeching halt. E.g. \(W(3,3,3,3)=76\). I could not find solutions for \(n=75\) within 1,000 seconds, let alone enumerating them all.
Python/Z3 version
An SMT or CP version of this model can use integers directly, instead of expanding things into binary variables.
from z3 import *
n = 26
nc = 3
# integer variables X[i]
X = [ Int('x%s' % i ) for i in range(n) ]
# lower bounds X[i] >= 1
Xlo = And([X[i] >= 1for i in range(n)])
# upper bounds X[i] <= nc
Xup = And([X[i] <= nc for i in range(n)])
# forbid equally spaced colors
Xforbid = And( [ Or(X[i] != X[i+k+1], X[i+k+1] != X[i+2*(k+1)] ) for i in range(n) \
for k in range(n) if i+2*(k+1) < n])
# combine all constraints
Cons = [Xlo,Xup,Xforbid]
# find all solutions
s = Solver()
s.add(Cons)
k = 0
res = []
while s.check() == sat:
m = s.model()
k = k + 1
sol = [ m.evaluate(X[i]) for i in range(n)]
res = res + [(k,sol)]
forbid = Or([X[i] != sol[i] for i in range(n) ])
s.add(forbid)
print(res)
Here we use and and or to model our forbid constraint. We could have used the same binary variable approach as done in the integer programming model. I tested that also: it turned out to be slower than using integers directly.
-Portrait-Portr_12193.tif/lossy-page1-330px-ETH-BIB-Waerden%2C_Bartel_Leendert_van_der_(1903-1996)-Portrait-Portr_12193.tif.jpg) |
| Van der Waerden (1903-1996) |
References
- Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 6, Satisfiability, 2015
- Van der Waerden Number, https://en.wikipedia.org/wiki/Van_der_Waerden_number