Network Design with Coverage Costs

We study network design with a cost structure motivated by redundancy in data traffic. We are given a graph, g groups of terminals, and a universe of data packets. Each group of terminals desires a subset of the packets from its respective source. The cost of routing traffic on any edge in the network is proportional to the total size of the distinct packets that the edge carries. Our goal is to find a minimum cost routing. We focus on two settings. In the first, the collection of packet sets desired by source-sink pairs is laminar. For this setting, we present a primal-dual based 2-approximation, improving upon a logarithmic approximation due to Barman and Chawla (2012). In the second setting, packet sets can have non-trivial intersection. We focus on the case where each packet is desired by either a single terminal group or by all of the groups, and the graph is unweighted. For this setting we present an O(log g)-approximation. Our approximation for the second setting is based on a novel spanner-type construction in unweighted graphs that, given a collection of g vertex subsets, finds a subgraph of cost only a constant factor more than the minimum spanning tree of the graph, such that every subset in the collection has a Steiner tree in the subgraph of cost at most O(log g) that of its minimum Steiner tree in the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result

A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints

This article presents an exact algorithm for the multi-depot vehicle routing problem (MDVRP) under capacity and route length constraints. The MDVRP is formulated using a vehicle-flow and a set-partitioning formulation, both of which are exploited at different stages of the algorithm. The lower bound computed with the vehicle-flow formulation is used to eliminate non-promising edges, thus reducing the complexity of the pricing subproblem used to solve the set-partitioning formulation. Several classes of valid inequalities are added to strengthen both formulations, including a new family of valid inequalities used to forbid cycles of an arbitrary length. To validate our approach, we also consider the capacitated vehicle routing problem (CVRP) as a particular case of the MDVRP, and conduct extensive computational experiments on several instances from the literature to show its effectiveness. The computational results show that the proposed algorithm is competitive against stateof-the-art methods for these two classes of vehicle routing problems, and is able to solve to optimality some previously open instances. Moreover, for the instances that cannot be solved by the proposed algorithm, the final lower bounds prove stronger than those obtained by earlier methods

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm for a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints. These constraints are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on the fly within an efficient solution method. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm and then introduce the associated set of pricing subproblems in detail. The proposed methodology is demonstrated on numerical examples for the multi-stage cutting stock and the quadratic set covering problems

The Pyramidal Capacitated Vehicle Routing Problem

This paper introduces the Pyramidal Capacitated Vehicle Routing Problem (PCVRP) as a restricted version of the Capacitated Vehicle Routing Problem (CVRP). In the PCVRP each route is required to be pyramidal in a sense generalized from the Pyramidal Traveling Salesman Problem (PTSP). A pyramidal route is de ned as a route on which the vehicle rst visits customers in increasing order of customer index, and on the remaining part of the route visits customers in decreasing order of customer index. Provided that customers are indexed in nondecreasing order of distance from the depot, the shape of a pyramidal route is such that its traversal can be divided in two parts, where on the rst part of the route, customers are visited in nondecreasing distance from the depot, and on the remaining part of the route, customers are visited in nonincreasing distance from the depot. Such a route shape is indeed found in many optimal solutions to CVRP instances. An optimal solution to the PCVRP may therefore be useful in itself as a heuristic solution to the CVRP. Further, an attempt can be made to nd an even better CVRP solution by solving a TSP, possibly leading to a non-pyramidal route, for each of the routes in the PCVRP solution. This paper develops an exact branch-and-cut-and-price (BCP) algorithm for the PCVRP. At the pricing stage, elementary routes can be computed in pseudo-polynomial time in the PCVRP, unlike in the CVRP. We have therefore implemented pricing algorithms that generate only elementary routes. Computational results suggest that PCVRP solutions are highly useful for obtaining near-optimal solutions to the CVRP. Moreover, pricing of pyramidal routes may due to its eciency prove to be very useful in column generation for the CVRP.vehicle routing; pyramidal traveling salesman; branch-and-cut-and-price

An Optimal Game Theoretical Framework for Mobility Aware Routing in Mobile Ad hoc Networks

Selfish behaviors are common in self-organized Mobile Ad hoc Networks (MANETs) where nodes belong to different authorities. Since cooperation of nodes is essential for routing protocols, various methods have been proposed to stimulate cooperation among selfish nodes. In order to provide sufficient incentives, most of these methods pay nodes a premium over their actual costs of participation. However, they lead to considerably large overpayments. Moreover, existing methods ignore mobility of nodes, for simplicity. However, owing to the mobile nature of MANETs, this assumption seems unrealistic. In this paper, we propose an optimal game theoretical framework to ensure the proper cooperation in mobility aware routing for MANETs. The proposed method is based on the multi-dimensional optimal auctions which allows us to consider path durations, in addition to the route costs. Path duration is a metric that best reflects changes in topology caused by mobility of nodes and, it is widely used in mobility aware routing protocols. Furthermore, the proposed mechanism is optimal in that it minimizes the total expected payments. We provide theoretical analysis to support our claims. In addition, simulation results show significant improvements in terms of payments compared to the most popular existing methods

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems