Alain Prodon

Double Description Method Revisited

The double description method is a simple and useful algorithm for enumerating all extreme rays of a general polyhedral cone in ℝd, despite the fact that we can hardly state any interesting theorems on its time and space complexities. In this paper, we reinvestigate this method, introduce some new ideas for efficient implementations, and show some empirical results indicating its practicality in solving highly degenerate problems.

Tree polytope on 2-trees

We give a complete polyhedral characterization of the tree polytope (convex hull of the characteristic vectors of trees in the graph) on 2-trees.

Locating leak detecting sensors in a water distribution network by solving prize-collecting Steiner arborescence problems

We consider the problem of optimizing a novel acoustic leakage detection system for urban water distribution networks. The system is composed of a number of detectors and transponders to be placed in a choice of hydrants such as to provide a desired coverage under given budget restrictions. The problem is modeled as a particular Prize-Collecting Steiner Arborescence Problem. We present a branch-and-cut-and-bound approach taking advantage of the special structure at hand which performs well when compared to other approaches. Furthermore, using a suitable stopping criterion, we obtain approximations of provably excellent quality (in most cases actually optimal solutions). The test bed includes the real water distribution network from the Lausanne region, as well as carefully randomly generated realistic instances.

Disjoint paths in the plane

Given n pairs of points in the Euclidean plane, we address the problem of finding paths of minimum length linking the pairs that can be made disjoint by infinitesimal deformations. We present and compare several fast heuristics and their implementation. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Steiner trees with n terminals among n + 1 nodes

Let G = (V, E) be a connected undirected graph and N a subset of distinguishednodes, called terminals. A Steiner tree on [G, N] is a minimal tree connecting all the terminal nodes. Restricting the instances to the case @?N@? = @?V@? -1, we present an algorithm to construct a minimum weight Steiner tree for any weight function on the edges E of G, and a complete minimal description of the polytope defined as the convex hull of all steiner trees on [G, N].

Optimal node disjoint paths on partial 2-trees: A linear algorithm and polyhedral results

We present anO(p · n) algorithm for the problem of finding disjoint simple paths of minimum total length betweenp given pairs of terminals on oriented partial 2-trees withn nodes and positive or negative arc lengths. The algorithm is inO(n) if all terminals are distinct nodes. We characterize the convex hull of the feasible solution set for the casep=2.

A note on the separation problem for the matching matroid

For a given graph G(V,E) and a given vector (......) the problem of finding a hyperplane which separates x form the polyhedron P of the matching matroid on G or proving that x belongs to P is solved by finding a minimum capacity cut on an auxiliary digraph.