On cycles and the stable multi-set polytope

AbstractStable multi-sets are an integer extension of stable sets in graphs. In this paper, we continue our investigations started by Koster and Zymolka [Stable multi-sets, Math. Methods Oper. Res. 56(1) (2002) 45–65]. We present further results on the stable multi-set polytope and discuss their computational impact.The polyhedral investigations focus on the cycle inequalities. We strengthen their facet characterization and show that chords need not weaken the cycle inequality strength in the multi-set case. This also helps to derive a valid right hand side for clique inequalities.The practical importance of the cycle inequalities is evaluated in a computational study. For this, we revisit existing polynomial time separation algorithms. The results show that the performance of state-of-the-art integer programming solvers can be improved by exploiting this general structure

Stable Intersections of Tropical Varieties

We give several characterizations of stable intersections of tropical cycles and establish their fundamental properties. We prove that the stable intersection of two tropical varieties is the tropicalization of the intersection of the classical varieties after a generic rescaling. A proof of Bernstein's theorem follows from this. We prove that the tropical intersection ring of tropical cycle fans is isomorphic to McMullen's polytope algebra. It follows that every tropical cycle fan is a linear combination of pure powers of tropical hypersurfaces, which are always realizable. We prove that every stable intersection of constant coefficient tropical varieties defined by prime ideals is connected through codimension one. We also give an example of a realizable tropical variety that is connected through codimension one but whose stable intersection with a hyperplane is not.Comment: Revised version, to appear in Journal of Algebraic Combinatoric

Combinatorics in N = 1 Heterotic Vacua

We briefly review an algorithmic strategy to explore the landscape of heterotic E8 \times E8 vacua, in the context of compactifying smooth Calabi-Yau three-folds with vector bundles. The Calabi-Yau three-folds are algebraically realised as hypersurfaces in toric varieties and a large class of vector bundles are constructed thereon as monads. In the spirit of searching for Standard-like heterotic vacua, emphasis is placed on the integer combinatorics of the model-building programme.Comment: 14 pages. An introductory review prepared for the special issue "Computational Algebraic Geometry in String and Gauge Theory" of Advances in High Energy Physic