Shortest Paths, Network Design and Associated Polyhedra

We study a specialized version of network design problems that arise in telecommunication, transportation and other industries. The problem, a generalization of the shortest path problem, is defined on an undirected network consisting of a set of arcs on which we can install (load), at a cost, a choice of up to three types of capacitated facilities. Our objective is to determine the configuration of facilities to load on each arc that will satisfy the demand of a single commodity at the lowest possible cost. Our results (i) demonstrate that the single-facility loading problem and certain "common breakeven point" versions of the two-facility and three-facility loading problems are polynomially solvable as a shortest path problem; (ii) show that versions of the twofacility loading problem are strongly NP-hard, but that a shortest path solution provides an asymptotically "good" heuristic; and (iii) characterize the optimal solution (that is, specify a linear programming formulation with integer solutions) of the common breakeven point versions of the two-facility and three-facility loading problems. In this development, we introduce two new families of facets, give geometric interpretations of our results, and demonstrate the usefulness of partitioning the space of the problem parameters to establish polyhedral integrality properties. Generalizations of our results apply to (i) multicommodity applications and (ii) situations with more than three facilities

The Convex Hull of Two Core Capacitated Network Design Problems

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP

Modeling and Solving the Capacitated Network Loading Problem

This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-topoint demand between various pairs of nodes of a network must be met by installing (loading) capacitated facilities on the arcs. The facilities are chosen from a small set of alternatives and loading a particular facility incurs an arc specific and facility dependent cost. The problem is to determine the configuration of facilities to be loaded on the arcs of the network that will satisfy the given demand at minimum cost. Since we need to install (load) facilities to carry the required traffic, we refer to the problem as the network loading problem. In this paper, we develop modeling and solution approaches for the problem. We consider two approaches for solving the underlying mixed integer programming model: (i) a Lagrangian relaxation strategy, and (ii) a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. In particular, we show that a linear programming formulation that includes the valid inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound (as measured by the gaps in the objective functions). We also examine the computational effectiveness of these inequalities on a set of prototypical telecommunications data. The computational results show that the addition of these inequalities considerably improves the gap between the integer programming formulation of the problem and its linear programming relaxation: for 6 - 15 node problems from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

A Dual-Based Algorithm for Multi-Level Network Design

Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher type. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem, and has applications in telecommunication, transportation, and electric power distribution network design. In a companion paper we introduced the problem, studied alternative model formulations, and analyzed the worst-case performance of heuristics based on Steiner network and spanning tree solutions. This paper develops and tests a dual-based algorithm for the Multi-level Network Design (MLND) problem. The method first performs problem preprocessing to fix certain design variables, and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value. We report extensive computational results on large, random networks (containing up to 500 nodes, and 5000 edges) with varying cost structures. The integer programming formulation of the largest of these problems has 20,000 integer variables and over 5 million constraints. Our tests indicate that the dualbased algorithm is very effective, producing solutions guaranteed to be within 0 to 0.9% of optimality

Heuristics, LPs, and Generalizations of Trees on Trees

We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a linear programming relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and linear programming relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, the performance guarantees for both the heuristic and linear programming relaxation of the combined overlay optimization model are 33%. We also develop heuristic and linear programming performance guarantees for specialized models, including a dual path connectivity model with a worst-case performance guarantee of 25%, and an uncapacitated multicommodity network design model with a worst-case performance guarantee (approximately) proportional to the square root of the number of commodities

Heuristics, LPs, and Trees on Trees: Network Design Analyses

We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a LP relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and LP relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, both the heuristic and the LP relaxation of the combined overlay optimization model have performance guarantees of 4/3. We extend this analysis to multiple overlays on the same base solution, producing the first known worst-case bounds (approximately proportional to the square root of the number of commodities) for the uncapacitated multicommodity network design problem. In a companion paper, we develop heuristic performance guarantees for various new multi-tier. survivable network design models that incorporate both multiple facility types or technologies and differential node connectivity levels

Designing Hierarchical Survivable Networks

As the computer, communication, and entertainment industries begin to integrate phone, cable, and video services and to invest in new technologies such as fiber optic cables, interruptions in service can cause considerable customer dissatisfaction and even be catastrophic. In this environment, network providers want to offer high levels of servicein both serviceability (e.g., high bandwidth) and survivability (failure protection)-and to segment their markets, providing better technology and more robust configurations to certain key customers. We study core models with three types of customers (critical, primary, and secondary) and two types of services/technologies (primary and secondary). The network must connect primary customers using primary (high bandwidth) services and, additionally, contain a back-up path connecting certain critical primary customers. Secondary customers require only single connectivity to other customers and can use either primary or secondary facilities. We propose a general multi-tier survivable network design model to configure cost effective networks for this type of market segmentation. When costs are triangular, we show how to optimally solve single-tier subproblems with two critical customers as a matroid intersection problem. We also propose and analyze the worst-case performance of tailored heuristics for several special cases of the two-tier model. Depending upon the particular problem setting, the heuristics have worst-case performance ratios ranging between 1.25 and 2.6. We also provide examples to show that the performance ratios for these heuristics are the best possible

Doubling or Splitting: Strategies for Modeling and Analyzing Survivable Network Design Problems

Survivability is becoming an increasingly important criterion in network design. This paper studies formulations, heuristic worst-case performance, and linear programming relaxations for two classes of survivable network design problems: the low connectivity Steiner (LCS) problem for graphs containing nodes with connectivity requirement of 0, 1, or 2, and a more general multi-connected network with branches (MNB) that requires connectivities of two or more for certain (critical) nodes and single connectivity for other secondary nodes. We consider both unitary and nonunitary MNB problems that respectively require a connected design or permit multiple components. Using a doubling argument, we first show how to generalize heuristic bounds of the Steiner tree and traveling salesman problems to LCS problems. We then develop a disaggregate formulation for the MNB problem that uses fractional edge selection variables to split the total connectivity requirement across each critical cutset into two separate requirements. This model, which is tighter than the usual cutset formulation, has three special cases: a "secondary-peeling" version that peels off the lowest connectivity level, a "connectivity-dividing" version that divides the connectivity requirements for all the critical cutsets, and a "secondarycompletion" version that attempts to separate the design decisions for the multi-connected network from those for the branches. We examine the tightness of the linear programming relaxations for these extended formulations, and then use them to analyze heuristics for the LCS and MNB problems. Our analysis strengthens some previously known heuristic-to-IP worst-case performance ratios for LCS and MNB problems by showing that the same bounds apply to the heuristic-to-LP ratios using our stronger formulations

The Fanconi Anemia Core Complex Is Dispensable during Somatic Hypermutation and Class Switch Recombination

To generate high affinity antibodies during an immune response, B cells undergo somatic hypermutation (SHM) of their immunoglobulin genes. Error-prone translesion synthesis (TLS) DNA polymerases have been reported to be responsible for all mutations at template A/T and at least a fraction of G/C transversions. In contrast to A/T mutations which depend on PCNA ubiquitination, it remains unclear how G/C transversions are regulated during SHM. Several lines of evidence indicate a mechanistic link between the Fanconi Anemia (FA) pathway and TLS. To investigate the contribution of the FA pathway in SHM we analyzed FancG-deficient B cells. B cells deficient for FancG, an essential member of the FA core complex, were hypersensitive to treatment with cross-linking agents. However, the frequencies and nucleotide exchange spectra of SHM remained comparable between wild-type and FancG-deficient B cells. These data indicate that the FA pathway is not involved in regulating the outcome of SHM in mammals. In addition, the FA pathway appears dispensable for class switch recombination