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Computational Treatment: Relational Isomorphism Problem
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We consider relational systems $(X_1,Y_1,R_1)$ and $(X_2,Y_2,R_2)$ with $R_isubseteq X_itimes Y_i$. An isomorphism is a pair of bijective maps $(alpha,beta)$, with $alpha: X_1rightarrow X_2$ and $beta: Y_1rightarrow Y_2$, such that:
$(x,y)in R_1 Leftrightarrow (alpha x,beta y)in R_2$. This can be seen as a generalization of the graph isomorphism problem. Just take two graphs and put $alpha = beta = gamma$ for some graph isomorphism $gamma$.
Given two relations, I'm looking for a way to decide the existence of an isomorphism computationally. Similarly to graph isomorphisms, the problem can be translated to the existence of two permutation matrices $P,Q$, such that $R_1P=QR_2$ or $R_1=QR_2P^-1$ (here $R_1$ and $R_2$ are identified with there adjacency matrices).
This yields an $0$-$1$ integer linear program:
$$R_1P =QR_2$$
$$ sum_i=1^nP_ij = 1, 1le jle m$$
$$ sum_j=1^nP_ij = 1, 1le ile m$$
$$ sum_i=1^nQ_ij = 1, 1le jle n$$
$$ sum_j=1^nQ_ij = 1, 1le ile n$$
$$P_ij in 0,1, 1 le i,jle m $$
$$Q_ij in 0,1, 1 le i,jle n. $$
What software (with C++ interface) can I use to solve this? I tried Scip, but don't know how to code this problem to solve it with Scip.
I would be glad about another approach too. For example a SAT formulation of this iomorphism problem. Or is there a specialized software (like for graph isomorphisms) which can treat such kind of isomorphism problems?
relations linear-programming graph-isomorphism satisfiability
asked Aug 3 at 9:24
Chris
456
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0
down vote
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We consider relational systems $(X_1,Y_1,R_1)$ and $(X_2,Y_2,R_2)$ with $R_isubseteq X_itimes Y_i$. An isomorphism is a pair of bijective maps $(alpha,beta)$, with $alpha: X_1rightarrow X_2$ and $beta: Y_1rightarrow Y_2$, such that:
$(x,y)in R_1 Leftrightarrow (alpha x,beta y)in R_2$. This can be seen as a generalization of the graph isomorphism problem. Just take two graphs and put $alpha = beta = gamma$ for some graph isomorphism $gamma$.
Given two relations, I'm looking for a way to decide the existence of an isomorphism computationally. Similarly to graph isomorphisms, the problem can be translated to the existence of two permutation matrices $P,Q$, such that $R_1P=QR_2$ or $R_1=QR_2P^-1$ (here $R_1$ and $R_2$ are identified with there adjacency matrices).
This yields an $0$-$1$ integer linear program:
$$R_1P =QR_2$$
$$ sum_i=1^nP_ij = 1, 1le jle m$$
$$ sum_j=1^nP_ij = 1, 1le ile m$$
$$ sum_i=1^nQ_ij = 1, 1le jle n$$
$$ sum_j=1^nQ_ij = 1, 1le ile n$$
$$P_ij in 0,1, 1 le i,jle m $$
$$Q_ij in 0,1, 1 le i,jle n. $$
What software (with C++ interface) can I use to solve this? I tried Scip, but don't know how to code this problem to solve it with Scip.
I would be glad about another approach too. For example a SAT formulation of this iomorphism problem. Or is there a specialized software (like for graph isomorphisms) which can treat such kind of isomorphism problems?
relations linear-programming graph-isomorphism satisfiability
asked Aug 3 at 9:24
Chris
456
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
We consider relational systems $(X_1,Y_1,R_1)$ and $(X_2,Y_2,R_2)$ with $R_isubseteq X_itimes Y_i$. An isomorphism is a pair of bijective maps $(alpha,beta)$, with $alpha: X_1rightarrow X_2$ and $beta: Y_1rightarrow Y_2$, such that:
$(x,y)in R_1 Leftrightarrow (alpha x,beta y)in R_2$. This can be seen as a generalization of the graph isomorphism problem. Just take two graphs and put $alpha = beta = gamma$ for some graph isomorphism $gamma$.
Given two relations, I'm looking for a way to decide the existence of an isomorphism computationally. Similarly to graph isomorphisms, the problem can be translated to the existence of two permutation matrices $P,Q$, such that $R_1P=QR_2$ or $R_1=QR_2P^-1$ (here $R_1$ and $R_2$ are identified with there adjacency matrices).
This yields an $0$-$1$ integer linear program:
$$R_1P =QR_2$$
$$ sum_i=1^nP_ij = 1, 1le jle m$$
$$ sum_j=1^nP_ij = 1, 1le ile m$$
$$ sum_i=1^nQ_ij = 1, 1le jle n$$
$$ sum_j=1^nQ_ij = 1, 1le ile n$$
$$P_ij in 0,1, 1 le i,jle m $$
$$Q_ij in 0,1, 1 le i,jle n. $$
What software (with C++ interface) can I use to solve this? I tried Scip, but don't know how to code this problem to solve it with Scip.
I would be glad about another approach too. For example a SAT formulation of this iomorphism problem. Or is there a specialized software (like for graph isomorphisms) which can treat such kind of isomorphism problems?
relations linear-programming graph-isomorphism satisfiability
asked Aug 3 at 9:24
Chris
456
We consider relational systems $(X_1,Y_1,R_1)$ and $(X_2,Y_2,R_2)$ with $R_isubseteq X_itimes Y_i$. An isomorphism is a pair of bijective maps $(alpha,beta)$, with $alpha: X_1rightarrow X_2$ and $beta: Y_1rightarrow Y_2$, such that:
$(x,y)in R_1 Leftrightarrow (alpha x,beta y)in R_2$. This can be seen as a generalization of the graph isomorphism problem. Just take two graphs and put $alpha = beta = gamma$ for some graph isomorphism $gamma$.
Given two relations, I'm looking for a way to decide the existence of an isomorphism computationally. Similarly to graph isomorphisms, the problem can be translated to the existence of two permutation matrices $P,Q$, such that $R_1P=QR_2$ or $R_1=QR_2P^-1$ (here $R_1$ and $R_2$ are identified with there adjacency matrices).
This yields an $0$-$1$ integer linear program:
$$R_1P =QR_2$$
$$ sum_i=1^nP_ij = 1, 1le jle m$$
$$ sum_j=1^nP_ij = 1, 1le ile m$$
$$ sum_i=1^nQ_ij = 1, 1le jle n$$
$$ sum_j=1^nQ_ij = 1, 1le ile n$$
$$P_ij in 0,1, 1 le i,jle m $$
$$Q_ij in 0,1, 1 le i,jle n. $$
What software (with C++ interface) can I use to solve this? I tried Scip, but don't know how to code this problem to solve it with Scip.
I would be glad about another approach too. For example a SAT formulation of this iomorphism problem. Or is there a specialized software (like for graph isomorphisms) which can treat such kind of isomorphism problems?
relations linear-programming graph-isomorphism satisfiability
asked Aug 3 at 9:24
Chris
456
edited Aug 3 at 9:42
asked Aug 3 at 9:24
Chris
456
asked Aug 3 at 9:24
Chris
456
asked Aug 3 at 9:24
Chris
456
456
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