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The vertex set of a \(0/1\)-polytope is strongly \(\mathcal P\)-enumerable. (English) Zbl 0912.68206
Summary: We discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron \(P\) defined by a system of linear inequalities, out put each vertex of \(P\). It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of \({\mathcal P}\)-enumerability introduced by L. G. Valiant [The complexity of enumeration and reliability problems, SIAM J. Comput. 8, No. 3, 410-421 (1979)]. The concept of strong \({\mathcal P}\)-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0/1-polytopes, Our implementation based on the commercial LP solver \(\text{CPLEX}^1\) is superior to general vertex enumeration algorithms. We give an example how simplifications of our algorithm lead to efficient enumeration of combinatorial objects.

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
Full Text: DOI
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