Multicommodity network design with survivability constraints: Some models and algorithms

We survey the main results presented by the author in his PhD thesis, supervised by F. Malucelli, and defended on the 15th March 2003. The thesis is written in English and is available on the Web page http: //www.elet.polimi.it/upload/belotti/thesis.pdf.gz. We investigate three problems, arising in the field of Telecommunication, of networks design with survivability constraints, and solve them through different approaches on a number of real-world network topologies with up to 40 nodes.Received: April 2004MSC classification: 90B10, 90C57     

An algorithm to generate all spanning trees with flow

Spanning tree enumeration in undirected graphs is an important issue and task in many problems encountered in computer network and circuit analysis. This paper discusses the spanning tree with flow for the case that there are flow requirements between each node pair. An algorithm based on minimal paths (MPs) is proposed to generate all spanning trees without flow. The proposed algorithm is a structured approach, which splits the system into structural MPs first, and also all steps in it are easy to follow.     

Multicommodity flow problem

The problem of nonmixing multicommodity flow is investigated. The theorem on the number of nonmixing flows of different commodities passing simultaneously through an oriented network is proved.Translated from Dinamicheskie Sistemy, No. 5, pp. 95–97, 1986.     

Multicommodity long-term hydrogeneration optimization with capacity and energy constraints

The work presented deals with long-term hydrogeneration optimization in integrated systems when there are no limitations on the availability of fuels for thermal units. A multicommondity network model represents hydrovariables. Hydrogeneration and its unavailability distribution is modeled as a multiblock distribution and a procedure is derived to convolve the hydropower unavailability distribution with the load duration curve. A suitable approximation of the expected production cost is minimized subject to multicommodity network constraints and to hydropower capacity and hydroenergy limit nonlinear constraints. It can be applied to systems with hydrogeneration regardless of its proportion of hydrothermal mix. A realistic case example is solved and the results are discussed. Simulation tests performed with many inflow sequences validate the results obtained.     

Linear verification for spanning trees

Given a rooted tree with values associated with then vertices and a setA of directed paths (queries), we describe an algorithm which finds the maximum value of every one of the given paths, and which uses only $$5n + n\log \frac{{\left| A \right| + n}}{n}$$ comparisons. This leads to a spanning tree verification algorithm usingO(n+e) comparisons in a graph withn vertices ande edges. No implementation is offered.     

Minimal spanning trees

The main result is that a recursive weighted graph having a minimal spanning tree has a minimal spanning tree that is . This leads to a proof of the failure of a conjecture of Clote and Hirst (1998) concerning Reverse Mathematics and minimal spanning trees.

     

Generalized spanning trees

In this paper, we propose a definition for the Generalized Minimal Spanning Tree (GMST) of a graph. The GMST requires spanning at least one node out of every set of disjoint nodes (node partition) in a graph. The analysis of the GMST problem is motivated by real life agricultural settings related to construction of irrigation networks in desert environments. We prove that the GMST problem is NP-hard, and examine a number of heuristic solutions for this problem. Computational experiments comparing these heuristics are presented.     

Efficient spanning trees

The definition of a shortest spanning tree of a graph is generalized to that of an efficient spanning tree for graphs with vector weights, where the notion of optimality is of the Pareto type. An algorighm for obtaining all efficient spanning trees is presented.     

Chain-constrained spanning trees

Mathematical Programming - We consider the problem of finding a spanning tree satisfying a family of additional constraints. Several settings have been considered previously, the most famous being...     

Graphs with homeomorphically irreducible spanning trees

It is an NP-complete problem to decide whether a graph contains a spanning tree with no vertex of degree 2. We show that these homeomorphically irreducible spanning trees are contained in all graphs with minimum degree at least c√n and in triangulations of the plane. They are nearly present in all graphs of diameter 2. They do not necessarily occur in r-regular or r-connected graphs.     

Homeomorphically irreducible spanning trees

We show that if G is a graph such that every edge is in at least two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree). This result was originally asked in a question format by Albertson, Berman, Hutchinson, and Thomassen in 1979, and then conjectured to be true by Archdeacon in 2009.     

Degree-bounded minimum spanning trees

Given n points in the Euclidean plane, the degree-δ minimum spanning tree (MST) problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most δ. The problem is NP-hard for 2≤δ≤3, while the NP-hardness of the problem is open for δ=4. The problem is polynomial-time solvable when δ=5. By presenting an improved approximation analysis for Chan’s degree-4 MST algorithm [T. Chan, Euclidean bounded-degree spanning tree ratios, Discrete & Computational Geometry 32 (2004) 177-194], we show that, for any arbitrary collection of points in the Euclidean plane, there always exists a degree-4 spanning tree of weight at most times the weight of an MST.     

Low-degree minimum spanning trees

Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that, under theL _p norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the best-known bounds for the maximum MST degree for arbitraryL _p metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane an MST exists with maximum degree of at most 4, and for three-dimensional rectilinear space the maximum possible degree of a minimum-degree MST is either 13 or 14. Gabriel Robins was partially supported by NSF Young Investigator Award MIP-9457412. Jeffrey Salowe was partially supported by NSF Grants MIP-9107717 and CCR-9224789.     

Graphs with certain families of spanning trees

Sufficient conditions are given in terms of δ(G) and Δ(T), for a graph G with n vertices to contain a tree T with n vertices. One of these sufficient conditions is used to calculate some of the Ramsey numbers for the pair tree-star. Also necessary conditions are given, in terms of δ(G), for a graph G with n vertices to contain all trees with n vertices.     

Graphs with only caterpillars as spanning trees

A connected graph G is caterpillar-pure if each spanning tree of G is a caterpillar. The caterpillar-pure graphs are fully characterized. Loosely speaking they are strings or necklaces of so-called pearls, except for a number of small exceptional cases. An upper bound for the number of edges in terms of the order is given for caterpillar-pure graphs, and those which attain the upper bound are characterized.     

Bijections for Cayley trees,spanning trees,and their q-analogues

We construct a family of extremely simple bijections that yield Cayley's famous formula for counting trees. The weight preserving properties of these bijections furnish a number of multivariate generating functions for weighted Cayley trees. Essentially the same idea is used to derive bijective proofs and q-analogues for the number of spanning trees of other graphs, including the complete bipartite and complete tripartite graphs. These bijections also allow the calculation of explicit formulas for the expected number of various statistics on Cayley trees.     

On encodings of spanning trees

Deo and Micikevicius recently gave a new bijection for spanning trees of complete bipartite graphs. In this paper we devise a generalization of Deo and Micikevicius's method, which is also a modification of Olah's method for encoding the spanning trees of any complete multipartite graph K(n₁,…,n_r). We also give a bijection between the spanning trees of a planar graph and those of any of its planar duals. Finally we discuss the possibility of bijections for spanning trees of DeBriujn graphs, cubes, and regular graphs such as the Petersen graph that have integer eigenvalues.