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Enumeration.hpp File Reference

Plans regarding enumeration of all biclique partitions in (general) graphs. More...

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Plans regarding enumeration of all biclique partitions in (general) graphs.

Todo:
Enumerating biclique partitions
• Given a general graph G, enumerate all biclique partitions.
• As discussed in ComputerAlgebra/Graphs/Lisp/BicliquePartitions/plans/Bicliques.hpp, likely multigraphs are more appropriate, since then we do not need to distinguish between different parallel edges.
• The problem is the same as finding all clause-sets without pure literals with a given conflict multigraph (namely G).
• The simplest approach:
1. Via set_partitions(edge_set) create all possible partitions of the edge-set, and then we compute the subset given by bicliquepartp.
• Alternatively, we translate the biclique partition problem into a SAT problem (see ComputerAlgebra/Graphs/Lisp/BicliquePartitions/plans/Transformations.hpp) and enumerate all solutions.
1. However, this is then a harder problem, since we demand certain properties of the biclique partition.
• Can we write an efficient generator, which creates one biclique partition at a time, with polynomial delay, running through all biclique partitions without repetition?
1. For example, running through the lexicographical ordering?!
2. Most natural seems to consider a backtracking approach.
• Given all biclique partitions, we can then find all isomorphism types:
1. Two biclique partitions are isomorphic if the associated clause-sets are isomorphic.
2. The problem of finding all isomorphism types is then the same as determining all isomorphism types of clause-sets without pure literals and with conflict multigraph G.
• Again the question is whether there is an efficient generator for all isomorphism types?
1. Definitely we do not need to generate first all biclique partitions.
2. Instead we use a backtracking approach, where we investigate new branches only if they are not isomorphic to branches already considered.
• Viewing biclique partitions are solutions of a constraint satisfaction problem:
1. Given a general G = (V,E).
2. Find a function b : E -> {1,...,|E|} which yield biclique partitions.
3. This blows up the search space, but makes the problem more accessible.
4. Making this translation explicit is discussed below in ComputerAlgebra/Graphs/Lisp/BicliquePartitions/plans/Transformations.hpp.
5. But we can explore such a point of view more implicitely.

Definition in file Enumeration.hpp.