The branch and bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations has been used with great success for decades. Over the years, a variety of researchers have studied ways of making the basic algorithm more effective. Breakthroughs in the fields of computer hardware, computer software, and mathematics have led to increasing success at solving larger and larger MIP instances. The goal of this paper is to survey many of the results regarding branch and bound search strategies and evaluate them again in light of the other advances that have taken place over the years. In addition, novel search strategies are presented and shown to often perform better than those currently used in practice. October 1997 The effectiveness of the branch and bound procedure for solving mixed integer programming (MIP) problems using linear programming relaxations is well documented. After the introduction of this procedure in the 1960's [26] [10]...

We present several classes of facet-defining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints. These problems typically involve assigning capacities to a network with multicommodity demands, such that after a vertex- or edge-deletion at least some prescribed fraction of each demand can be routed.

This paper provides an analysis of capacitated network design cut--set polyhedra. We give a complete linear description of the cut--set polyhedron of the single commodity -- single facility capacitated network design problem. Then we extend the analysis to single commodity -- multifacility and multicommodity -- multifacility capacitated network design problems. The valid inequalities described here have coefficients for both inflow and outflow arcs of a cut--set and are applicable to network design problems with an arbitrary number of facility types and arbitrary capacities. We report a computational study to test the effectiveness of the new inequalities. 1 Introduction Given a network and a set of demands on the vertices of the network, the capacitated network design problem is to install integer multiples of capacities on the arcs of the network and route the flow so that the total capacity installation and flow routing costs are minimized. For instance, installing or leasing fiber...

This paper presents algorithms, heuristics and lower bounds for an optimal sharing of network resources among several multicast groups that coexist in the network. Group (i.e., many-to-many) multicasting is a demanding service since any member can become a sender independently from the others. We consider a shared tree as the backbone of a group multicasting session. Considering each multicast session in isolation and independently may cause congestion on some links and reduce network utilization. Thus, we define the multicast packing problem in which network tries to accommodate simultaneously all the multicast groups while trying to avoid bottlenecks on the links for higher throughput (i.e., minimize the maximum link sharing among multicast groups). Minimization of maximum congestion is achieved at the expense of increasing the size of some multicast tree which in turn impacts the delay. This trade off is addressed by adding a penalty term to the objective function of the optimal pac...

Consider the following classical network design problem: a set of terminals T: {t.i} wants to send traffic to a "root" r in an 'x-node graph G: (V, E). Each terminal ti sends di units of traffic, and enough bandwidth has to be allocated on the edges to permit this. However, bandwidth on an edge e can only be allocated in integral multiples of some base capacity ue-- and hence provisioning k x ue bandwidth on edge e incurs a cost of [k] times the cost of that edge. The objective is a minimum-cost feasible solution. This is one of many network design problems widely studied where the bandwidth allocation being governed by side constraints: edges may only allow a subset of cables to be purchased on them, or certain quality-of-service requirements may have to be met. In this work, we show that the above problem, and in fact, several basic problems in this general network design framework, cannot be approximated better than ~(log log n) unless NP c _ OTIME(,r~°(l°gl°gl°gn)). In particular, we show that this inapproximability threshold holds for (i) the Priority-Steiner Tree problem [7], (ii) the Cost-Distance problem [31], and the single-sink version of an even more fundamental problem, (iii) Fixed Charge Network Flow [33]. Our results provide a further breakthrough in the understanding of the level of complexity of network design problems. These are the first non-constant hardness results known for all these problems.

This paper presents algorithms, heuristics and lower bounds addressing optimization issues in many-to-many multicasting. Two main problems are addressed: (1) a precise combinatorial comparison of optimal multicast trees with optimal multicast rings, (2) an optimized sharing of network resources (i.e., nodes and links) among multiple multicast groups that coexist. The former is central to the choice of multicast protocols and their performance, while the latter is crucial for network utilization. The first problem is treated as a comparison of Steiner Tree and Traveling Salesman problems on the same input set. The underlying integer programming problems are solved to optimum by using cutting-plane inequalities and the branchand-cut algorithm. In addition to these exact solutions, fast heuristics are presented for approximate solutions. The second problem is formulated as a packing problem in which the network tries to accommodate all the multicast groups by optimizing the utilization of resources. Precise mathematical programming formulations, lower bounds and a heuristic for the underlying optimization problem are presented. The heuristic aims to accommodate multiple multicast groups while avoiding bottlenecks on the links for higher throughput. The heuristics and exact algorithms are implemented on various networks and multicast groups. The simulations show that multicast trees can be built by using 25 % fewer links than the rings, both for optimal and suboptimal constructions. The packing heuristic is also implemented and its performance is compared to the constructive lower bound.

In this paper we introduce an evolutionary algorithm for the solution of linear integer programs. The strategy is based on the separation of the variables into the integer subset and the continuous subset; the integer variables are fixed by the evolutionary system, and the continuous ones are determined in function of them, by a linear program solver. We report results obtained for some standard benchmark problems, and compare them with those obtained by branch-and-bound. The performance of the evolutionary algorithm is promising. Good feasible solutions were generally obtained, and in some of the difficult benchmark tests it outperformed branch-and-bound. 1 Introduction Integer linear programming problems are widely described in the combinatorial optimisation literature, and include many well-known and important applications. Typical problems of this type include lot sizing, scheduling, facility location, vehicle routing, and more; see for example [6, 1]. The problem consists of opti...

We present a branch-and-cut algorithm to solve the single commodity uncapacitated fixed charge network flow problem, which includes the Steiner tree problem, uncapacitated lot-sizing problems, and the fixed charge transportation problem as special cases. The cuts used are simple dicut inequalities and their variants. A crucial problem when separating these inequalities is to find the right cut set on which to generate the inequalities. The prototype branch-and-cut system, bc-nd includes a separation heuristic for the dicut inequalities, and problem specific primal heuristics, branching and pruning rules. Computational results show that bc-nd is competitive compared to a variety of special purpose algorithms for problems with explicit flow costs. We also examine how general purpose MIP systems perform on such problems when provided with formulations that have been tightened a priori with dicut inequalities.

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent, design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and another based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.

We present a Heuristic Branch&Price&Cut (BPC) approach for a network design problem. This algorithm integrates several original features including: • a multi-commodity multi-capacity network design problem path based model, • a pricing scheme using pools of generators and promising columns, • cutting-planes applied to the path-based formulation to obtain better lower bounds, to make better heuristic decisions, and to prune a larger part of the tree, • several heuristics and search strategies for quicker primal solutions. The effectiveness of the algorithm is assessed on a large set of benchmarks based on real data. The computational results and comparisons with other well known techniques show that the algorithm is able to produce not only tight lower bounds, but also high quality heuristic solutions (upper bounds). The algorithm has been implemented using the Maestro BPC framework.