Notes on acyclic orientations and the shelling lemma

AbstractIn this paper we study two lemmas on acyclic orientations and totally cyclic orientations of a graph, which can be derived from the shelling lemma in vector subspaces. We give simple graph theoretical proofs as well as a proof by the interpretations of the shelling lemma in the special setting of graphs. Furthermore, we present similar interpretations of closely related theorems in vector subspaces, which do not seem to admit simple graph theoretical proofs

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 instance

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

New Implementations of the Double Description Method

A pair (A; R) of real matrices A and R is said to be a double description pair or simply DD pair if the relationship Ax 0 if and only if x = 3DR for some 0 holds. Clearly, for a pair (A; R) to be a DD pair, it is necessary that the column size of A is equal to the row size of R, say d. The term "double description" was introduced by Motzkin et al. [MRTT53], and it is quite natural in the sense that such a pair contains two different descriptions of the same polyhedral cone. We will call A a representation matrix and R a generating matrix for this cone that we denote by P (A). Minkowski's and Weyl's theorems suggest the two fundamental problems, one to construct a matrix R from a given matrix A and the converse. It is well known that these two problems are computationally equivalent, and it can be shown that (A; R) is a DD pair if and only if (R T ; A T ) is a DD pair. Thus, we concentrate on the first problem, to find a generating matrix R for a given A. Clearly a more appr..

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

We present an O(p*n) algorithm for the problem of finding disjoint simple paths of minimum total length between p given pairs of terminals on oriented partial 2-trees with n nodes and positive or negative arc lengths. The algorithm is in O(n) if all terminals are distinct nodes. We characterize the convex hull of the feasible solution set for the case p = 2

Chasing Leaks 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 an urban water distribution network composed of a number of detectors and transponders to be placed in a choice of hydrants such as to provide a desired covering 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

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