Number of lattices inside mixed-integer polyhedron
$begingroup$
Given a mixed-integer polyhedron
$P = {(x;z) in mathbb{R}^n times mathbb{Z}^d mid A x + B z leq c }$
with $A in mathbb{Q}^{m times n}$, $B in mathbb{Q}^{m times d}$ and $c in mathbb{Q}^{m}$, is
$textrm{proj}_{mathbb{Z}^d} (P)$ a set with a finite number of elements? That should be equivalent to say that the number of lattices (i.e., "integer points") inside $P$ is finite, right?
If the answer is yes, are there some ways to enumerate them?
combinatorics computational-mathematics computational-geometry mixed-integer-programming
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
$endgroup$
add a comment |
$begingroup$
Given a mixed-integer polyhedron
$P = {(x;z) in mathbb{R}^n times mathbb{Z}^d mid A x + B z leq c }$
with $A in mathbb{Q}^{m times n}$, $B in mathbb{Q}^{m times d}$ and $c in mathbb{Q}^{m}$, is
$textrm{proj}_{mathbb{Z}^d} (P)$ a set with a finite number of elements? That should be equivalent to say that the number of lattices (i.e., "integer points") inside $P$ is finite, right?
If the answer is yes, are there some ways to enumerate them?
combinatorics computational-mathematics computational-geometry mixed-integer-programming
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
$endgroup$
add a comment |
$begingroup$
Given a mixed-integer polyhedron
$P = {(x;z) in mathbb{R}^n times mathbb{Z}^d mid A x + B z leq c }$
with $A in mathbb{Q}^{m times n}$, $B in mathbb{Q}^{m times d}$ and $c in mathbb{Q}^{m}$, is
$textrm{proj}_{mathbb{Z}^d} (P)$ a set with a finite number of elements? That should be equivalent to say that the number of lattices (i.e., "integer points") inside $P$ is finite, right?
If the answer is yes, are there some ways to enumerate them?
combinatorics computational-mathematics computational-geometry mixed-integer-programming
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
$endgroup$
Given a mixed-integer polyhedron
$P = {(x;z) in mathbb{R}^n times mathbb{Z}^d mid A x + B z leq c }$
with $A in mathbb{Q}^{m times n}$, $B in mathbb{Q}^{m times d}$ and $c in mathbb{Q}^{m}$, is
$textrm{proj}_{mathbb{Z}^d} (P)$ a set with a finite number of elements? That should be equivalent to say that the number of lattices (i.e., "integer points") inside $P$ is finite, right?
If the answer is yes, are there some ways to enumerate them?
combinatorics computational-mathematics computational-geometry mixed-integer-programming
combinatorics computational-mathematics computational-geometry mixed-integer-programming
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
edited Jan 15 at 14:25
GuidoLaremi
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
asked Jan 15 at 14:02
GuidoLaremiGuidoLaremi
145
145
add a comment |
add a comment |
1 Answer
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$begingroup$
If $P$ is bounded (hence compact), then the projection has finite cardinality. If $P$ is unbounded, then you cannot really say. It is easy to construct unbounded examples that are empty (no integer-feasible points), and it is easy to construct unbounded examples with an infinite number of integer-feasible points.
If $P$ is bounded, it is possible (given enough time and memory) to enumerate all integer feasible solutions using a MIP solver. CPLEX has a solution pool feature that, if properly configured, will do this. Alternatively, if you recode all the integer variables into linear combinations of binary variables, you can solve for any feasible solution (set the object to minimize or maximize 0), record it, add a "no-good" constraint that cuts it out (and only that one), and iterate ad nauseum. I think you can also do this with a good quality constraint solver.
That said, enumerating all points in the projection can easily overwhelm you in terms of time and memory.
answered Jan 15 at 19:52
prubinprubin
1,555125
$endgroup$
add a comment |
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1 Answer
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$begingroup$
If $P$ is bounded (hence compact), then the projection has finite cardinality. If $P$ is unbounded, then you cannot really say. It is easy to construct unbounded examples that are empty (no integer-feasible points), and it is easy to construct unbounded examples with an infinite number of integer-feasible points.
If $P$ is bounded, it is possible (given enough time and memory) to enumerate all integer feasible solutions using a MIP solver. CPLEX has a solution pool feature that, if properly configured, will do this. Alternatively, if you recode all the integer variables into linear combinations of binary variables, you can solve for any feasible solution (set the object to minimize or maximize 0), record it, add a "no-good" constraint that cuts it out (and only that one), and iterate ad nauseum. I think you can also do this with a good quality constraint solver.
That said, enumerating all points in the projection can easily overwhelm you in terms of time and memory.
answered Jan 15 at 19:52
prubinprubin
1,555125
$endgroup$
add a comment |
$begingroup$
If $P$ is bounded (hence compact), then the projection has finite cardinality. If $P$ is unbounded, then you cannot really say. It is easy to construct unbounded examples that are empty (no integer-feasible points), and it is easy to construct unbounded examples with an infinite number of integer-feasible points.
If $P$ is bounded, it is possible (given enough time and memory) to enumerate all integer feasible solutions using a MIP solver. CPLEX has a solution pool feature that, if properly configured, will do this. Alternatively, if you recode all the integer variables into linear combinations of binary variables, you can solve for any feasible solution (set the object to minimize or maximize 0), record it, add a "no-good" constraint that cuts it out (and only that one), and iterate ad nauseum. I think you can also do this with a good quality constraint solver.
That said, enumerating all points in the projection can easily overwhelm you in terms of time and memory.
answered Jan 15 at 19:52
prubinprubin
1,555125
$endgroup$
add a comment |
$begingroup$
If $P$ is bounded (hence compact), then the projection has finite cardinality. If $P$ is unbounded, then you cannot really say. It is easy to construct unbounded examples that are empty (no integer-feasible points), and it is easy to construct unbounded examples with an infinite number of integer-feasible points.
If $P$ is bounded, it is possible (given enough time and memory) to enumerate all integer feasible solutions using a MIP solver. CPLEX has a solution pool feature that, if properly configured, will do this. Alternatively, if you recode all the integer variables into linear combinations of binary variables, you can solve for any feasible solution (set the object to minimize or maximize 0), record it, add a "no-good" constraint that cuts it out (and only that one), and iterate ad nauseum. I think you can also do this with a good quality constraint solver.
That said, enumerating all points in the projection can easily overwhelm you in terms of time and memory.
answered Jan 15 at 19:52
prubinprubin
1,555125
$endgroup$
If $P$ is bounded (hence compact), then the projection has finite cardinality. If $P$ is unbounded, then you cannot really say. It is easy to construct unbounded examples that are empty (no integer-feasible points), and it is easy to construct unbounded examples with an infinite number of integer-feasible points.
If $P$ is bounded, it is possible (given enough time and memory) to enumerate all integer feasible solutions using a MIP solver. CPLEX has a solution pool feature that, if properly configured, will do this. Alternatively, if you recode all the integer variables into linear combinations of binary variables, you can solve for any feasible solution (set the object to minimize or maximize 0), record it, add a "no-good" constraint that cuts it out (and only that one), and iterate ad nauseum. I think you can also do this with a good quality constraint solver.
That said, enumerating all points in the projection can easily overwhelm you in terms of time and memory.
answered Jan 15 at 19:52
prubinprubin
1,555125
answered Jan 15 at 19:52
prubinprubin
1,555125
answered Jan 15 at 19:52
prubinprubin
1,555125
answered Jan 15 at 19:52
prubinprubin
1,555125
1,555125
add a comment |
add a comment |
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