We provide two algorithms for computing the volume of the convex polytope $\Omega:=\{x\in \R^n_+ \,|\,Ax\leq b\}$, for $A\in\R^{m\times n}, b\in\R^n$. The computational complexity of both algorithms is essentially described by $n^m$, which makes them especially attractive for large $n$ and relatively small $m$, when the other methods with $O(m^n)$ complexity fail. The methodology which differs … Read more
The Cardinality Constrained Circuit Problem (CCCP) is the problem of finding a minimum cost circuit in a graph where the circuit is constrained to have at most $k$ edges. The CCCP is NP-Hard. We present classes of facet-inducing inequalities for the convex hull of feasible circuits. ArticleDownload View PDF
We study the polyhedron associated with a network design problem which consists in determining at minimum cost a two-connected network such that the shortest cycle to which each edge belongs (a “ring”) does not exceed a given length K. We present here a new formulation of the problem and derive facet results for different classes … Read more
In this article we introduce a theory of convexity for the lattices of integer points, which we call a theory of discrete convexity. In particular, we obtain generalizations of Edmonds’ polymatroid intersection theorem and the Hoffman-Kruskal theorem as consequences of our constructions. CitationAdvances in Mathematics (to appear)ArticleDownload View PDF
We investigate the polyhedral structure of a formulation of the k-way equipartition problem and a branch-and-cut algorithm for the problem. The k-way equipartition problem requires dividing the vertices of a weighted graph into k equally sized sets, so as to minimize the total weight of edges that have both endpoints in the same set. Applications … Read more
In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets. For the … Read more