16 research outputs found
Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems
An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper