No abstract

We derive a new upper bound on the diameter of a polyhedron P = {x ∈ R n : Ax b}, where A ∈ Z m×n . The bound is polynomial in n and the largest absolute value of a sub-determinant of A, denoted by ∆. More precisely, we show that the diameter of P is bounded by O ∆ 2 n 4 log n∆ . If P is bounded, then we show that the diameter of P is at most O ∆ 2 n 3.5 log n∆ .For the special case in which A is a totally unimodular matrix, the bounds are O n 4 log n and O n 3.5 log n respectively. This improves over the previous best bound of O(m 16 n 3 (log mn) 3 ) due to Dyer and Frieze [DF94]. * An extended abstract of this paper was presented at the 28-th annual ACM symposium on Computational Geometry (SOCG 12) † LIX, École Polytechnique, Palaiseau and IBM,

In this paper we deal with the Critical Node Problem (CNP), i.e., the problem of searching for a given number K of nodes in a graph G, whose removal minimizes the (weighted or unweighted) number of connections between pairs of nodes in the residual graph. In particular, we study the case where the physical network represented by graph G has a hierarchical organization, so that G is a tree. The N P-completeness of this problem for general graphs has been already established (Arulselvan et al.). We study the subclass of CNP over trees, generalizing the objective function and constraints to take into account general nonnegative "costs" of node connections and "weights" for the nodes that are to be removed. We prove that CNP over trees is still N Pcomplete when general connection costs are specified, while the cases where all connections have unit cost are solvable in polynomial time by dynamic programming approaches. For the case with nonnegative connection costs and unit node weights we propose an enumeration scheme whose time complexity is within a polynomial factor from O(1.618034 n ), where n is the number of nodes of the tree. Results from computational experiments are reported for all the proposed algorithms.

In this paper we deal with the critical node problem, where a given number of nodes has to be removed from an undirected graph in order to maximize the disconnections between the node pairs of the graph. We propose an integer linear programming model with a non-polynomial number of constraints but whose linear relaxation can be solved in polynomial time. We derive different valid inequalities and some theoretical results about them. We also propose an alternative model based on a quadratic reformulation of the problem. Finally, we perform many computational experiments and analyze the corresponding results.

The cutting-plane approach to integer programming was initiated more that 40 years ago: Gomory introduced the corner polyhedron as a relaxation of a mixed integer set in tableau form and Balas introduced intersection cuts for the corner polyhedron. This line of research was left dormant for several decades until relatively recently, when a paper of Andersen, Louveaux, Weismantel and Wolsey generated renewed interest in the corner polyhedron and intersection cuts. Recent developments rely on tools drawn from convex analysis, geometry and number theory, and constitute an elegant bridge between these areas and integer programming. We survey these results and highlight recent breakthroughs in this area.