Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

New formulations of the Hop-Constrained Minimum Spanning Tree problem via Miller–Tucker–Zemlin constraints

Cataloged from PDF version of article.Given an undirected network with positive edge costs and a natural number p, the Hop-Constrained Minimum Spanning Tree problem (HMST) is the problem of finding a spanning tree with minimum total cost such that each path starting from a specified root node has no more than p hops (edges). In this paper, we develop new formulations for HMST. The formulations are based on Miller–Tucker–Zemlin (MTZ) subtour elimination constraints, MTZ-based liftings in the literature offered for HMST, and a new set of topologyenforcing constraints. We also compare the proposed models with the MTZ-based models in the literature with respect to linear programming relaxation bounds and solution times. The results indicate that the new models give considerably better bounds and solution times than their counterparts in the literature and that the new set of constraints is competitive with liftings to MTZ constraints, some of which are based on well-known, strong liftings of Desrochers and Laporte (1991). 2011 Elsevier B.V. All rights reserved

A polyhedral study of the diameter constrained minimum spanning tree problem

This paper provides a first polyhedral study of the diameter constrained minimum spanning tree problem (DMSTP). We introduce a new set of inequalities, the circular-jump inequalities which strengthen the well-known jump inequalities. These inequalities are further generalized in two ways: either by increasing the number of partitions defining a jump, or by combining jumps with cutsets. Most of the proposed new inequalities are shown to define facets of the DMSTP polytope under very mild conditions. Currently best known lower bounds for the DMSTP are obtained from an extended formulation on a layered graph using the concept of central nodes/edges. A subset of the new families of inequalities is shown to be not implied by this layered graph formulation

Analysis and optimization of highly reliable systems

In the field of network design, the survivability property enables the network to maintain a certain level of network connectivity and quality of service under failure conditions. In this thesis, survivability aspects of communication systems are studied. Aspects of reliability and vulnerability of network design are also addressed. The contributions are three-fold. First, a Hop Constrained node Survivable Network Design Problem (HCSNDP) with optional (Steiner) nodes is modelled. This kind of problems are N P-Hard. An exact integer linear model is built, focused on networks represented by graphs without rooted demands, considering costs in arcs and in Steiner nodes. In addition to the exact model, the calculation of lower and upper bounds to the optimal solution is included. Models were tested over several graphs and instances, in order to validate it in cases with known solution. An Approximation Algorithm is also developed in order to address a particular case of SNDP: the Two Node Survivable Star Problem (2NCSP) with optional nodes. This problem belongs to the class of N P-Hard computational problems too. Second, the research is focused on cascading failures and target/random attacks. The Graph Fragmentation Problem (GFP) is the result of a worst case analysis of a random attack. A fixed number of individuals for protection can be chosen, and a non-protected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. This problem belongs to the N P-Hard class. A mathematical programming formulation is introduced and exact resolution for small instances as well as lower and upper bounds to the optimal solution. In addition to exact methods, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances and exact methods in exponential time. Finally, the concept of separability in stochastic binary systems is here introduced. Stochastic Binary Systems (SBS) represent a mathematical model of a multi-component on-off system subject to independent failures. The reliability evaluation of an SBS belongs to the N P-Hard class. Therefore, we fully characterize separable systems using Han-Banach separation theorem for convex sets. Using this new concept of separable systems and Markov inequality, reliability bounds are provided for arbitrary SBS

Distance Transformation for Network Design Problems

International audienceWe propose a new generic way to construct extended formulations for a large class of network design problems with given connectivity requirements. The approach is based on a graph transformation that maps any graph into a layered graph according to a given distance function. The original connectivity requirements are in turn transformed into equivalent connectivity requirements in the layered graph. The mapping is extended to the graphs induced by fractional vectors through an extended linear integer programming formulation. While graphs induced by binary vectors are mapped to isomorphic layered graphs, those induced by fractional vectors are mapped to a set of graphs having worse connectivity properties. Hence, the connectivity requirements in the layered graph may cut off fractional vectors that were feasible for the problem formulated in the original graph. Experiments over instances of the Steiner Forest and Hop-constrained Survivable Network Design problems show that significant gap reductions over the state-of-the art formulations can be obtained

Network Flow Models for Designing Diameter-Constrained Minimum Spanning and Steiner Trees

The Diameter-Constrained Minimum Spanning Tree Problem seeks a least cost spanning tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge problem after one week of computation. We formulate the problem as a directed tree from a selected central node or a selected central edge. Our model simultaneously finds a central node or a central edge and uses it as the source for the commodities in a directed multicommodity flow model with hop constraints. The new model has been able to solve the 20-node, 100-edge instance to optimality after less than four seconds. We also present model enhancements when the diameter bound is odd (these situations are more difficult). We show that the linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially-solvable cases of the problem. We also examine the Diameter Constrained Minimum Steiner Tree problem. We present computational experience in solving problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints

Computing a Minimum-Cost k-Hop Steiner Tree in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a k-hop constraint which requires the depth of the tree to be at most k. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time n^O(k). For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if k is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the k-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension and bounded doubling dimension