Planning wireless networks by shortest path

Transmitters and receivers are the basic elements of wireless networks and are characterized by a number of radio-electrical parameters. The generic planning problem consists of establishing suitable values for these parameters so as to optimize some network performance indicator. The version here addressed, namely the Power Assignment Problem (pap), is the problem of assigning transmission powers to the transmitters of a wireless network so as to maximize the satisfied demand. This problem has relevant practical applications both in radio-broadcasting and in mobile telephony. Typical solution approaches make use of mixed integer linear programs with huge coefficients in the constraint matrix yielding numerical inaccuracy and poor bounds, and so cannot be exploited to solve large instances of practical interest. In order to overcome these inconveniences, we developed a two-phase heuristic to solve large instances of pap, namely a constructive heuristic followed by an improving local search. Both phases are based on successive shortest path computations on suitable directed graphs. Computational tests on a number of instances arising in the design of the national Italian Digital Video Broadcasting (DVB) network are presented.     

A local search heuristic for the Multi-Commodity k-splittable Maximum Flow Problem

The Multi-Commodity $k$ -splittable Maximum Flow Problem consists of maximizing the amount of flow routed through a network such that each commodity uses at most $k$ paths and such that edge capacities are satisfied. The problem is $\mathcal NP $ -hard and has application in a.o. telecommunications. In this paper, a local search heuristic for solving the problem is proposed. The heuristic is an iterative shortest path procedure on a reduced graph combined with a local search procedure to modify certain path flows and prioritize the different commodities. The heuristic is tested on benchmark instances from the literature and solves 83 % of the instances to optimality. For the remaining instances, the heuristic finds good solution values which on average are 1.04 % from the optimal. The heuristic solves all instances in less than a second. Compared to other heuristics, the proposed heuristic again shows superior performance with respect to solution quality.     

A 3-flip neighborhood local search for the set covering problem

The set covering problem (SCP) calls for a minimum cost family of subsets from n given subsets, which together covers the entire ground set. In this paper, we propose a local search algorithm for SCP, which has the following three characteristics. (1) The use of 3-flip neighborhood, which is the set of solutions obtainable from the current solution by exchanging at most three subsets. As the size of 3-flip neighborhood is O(n³), the neighborhood search becomes expensive if implemented naively. To overcome this, we propose an efficient implementation that reduces the number of candidates in the neighborhood without sacrificing the solution quality. (2) We allow the search to visit the infeasible region, and incorporate the strategic oscillation technique realized by adaptive control of penalty weights. (3) The size reduction of the problem by using the information from the Lagrangian relaxation is incorporated, which is indispensable for solving very large instances. According to computational comparisons on benchmark instances with other existing heuristic algorithms for SCP, our algorithm performs quite effectively for various types of problems, especially for very large-scale instances.     

On the Generalized Elementary Shortest Path Problem: A heuristic approach

We introduce the generalized elementary shortest path problem (GESPP) where in addition to the features of the shortest path problem, nodes belong to predefined non-disjoint clusters. Each cluster is associated to a profit to the cost function, obtained if at least one element in the cluster appears in the path. Several applications can be considered as school bus routing, pricing problems, or telecommunication network design. Thus, depending on the case, clusters could be interpreted as groups of nodes with linking features as, for example, being easily reachable from each other, or some kind of coverage guarantee. We compare the GESPP to similar problems in the literature and we propose a two-phase heuristic algorithm for graphs including negative cycles. Tests on random instances with up to 100 nodes show an average gap of 0.3% to the best known solutions computed in 2.8s in average.     

An integer linear programming formulation and heuristics for the minmax relative regret robust shortest path problem

The well-known Shortest Path problem (SP) consists in finding a shortest path from a source to a destination such that the total cost is minimized. The SP models practical and theoretical problems. However, several shortest path applications rely on uncertain data. The Robust Shortest Path problem (RSP) is a generalization of SP. In the former, the cost of each arc is defined by an interval of possible values for the arc cost. The objective is to minimize the maximum relative regret of the path from the source to the destination. This problem is known as the minmax relative regret RSP and it is NP-Hard. We propose a mixed integer linear programming formulation for this problem. The CPLEX branch-and-bound algorithm based on this formulation is able to find optimal solutions for all instances with 100 nodes, and has an average gap of 17 % on the instances with up to 1,500 nodes. We also develop heuristics with emphasis on providing efficient and scalable methods for solving large instances for the minmax relative regret RSP, based on Pilot method and random-key genetic algorithms. To the best of our knowledge, this is the first work to propose a linear formulation, an exact algorithm and metaheuristics for the minmax relative regret RSP.     

An Augmented Lagrangian Algorithm for Large Scale Multicommodity Routing

The linear multicommodity network flow (MCNF) problem has many applications in the areas of transportation and telecommunications. It has therefore received much attention, and many algorithms that exploit the problem structure have been suggested and implemented. The practical difficulty of solving MCNF models increases fast with respect to the problem size, and especially with respect to the number of commodities. Applications in telecommunications typically lead to instances with huge numbers of commodities, and tackling such instances computationally is challenging.In this paper, we describe and evaluate a fast and convergent lower-bounding procedure which is based on an augmented Lagrangian reformulation of MCNF, that is, a combined Lagrangian relaxation and penalty approach. The algorithm is specially designed for solving very large scale MCNF instances. Compared to a standard Lagrangian relaxation approach, it has more favorable convergence characteristics. To solve the nonlinear augmented Lagrangian subproblem, we apply a disaggregate simplicial decomposition scheme, which fully exploits the structure of the subproblem and has good reoptimization capabilities. Finally, the augmented Lagrangian algorithm can also be used to provide heuristic upper bounds.The efficiency of the augmented Lagrangian method is demonstrated through computational experiments on large scale instances. In particular, it provides near-optimal solutions to instances with over 3,600 nodes, 14,000 arcs and 80,000 commodities within reasonable computing time.     

A facility neighborhood search heuristic for capacitated facility location with single-source constraints and flexible demand

We consider a generalization of the well-known capacitated facility location problem with single source constraints in which customer demand contains a flexible dimension. This work focuses on providing fast and practically implementable optimization-based heuristic solution methods for very large scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems.     

A discrete dynamic convexized method for the max-cut problem

The max-cut problem is a classical NP-hard problem in graph theory. In this paper, we adopt a local search method, called MCFM, which is a simple modification of the Fiduccia-Mattheyses heuristic method in Fiduccia and Mattheyses (Proc. ACM/IEEE DAC, pp. 175?C181, 1982) for the circuit partitioning problem in very large scale integration of circuits and systems. The method uses much less computational cost than general local search methods. Then, an auxiliary function is presented which has the same global maximizers as the max-cut problem. We show that maximization of the function using MCFM can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. An algorithm is proposed for the max-cut problem, by maximizing the auxiliary function using MCFM from random initial solutions. Computational experiments were conducted on three sets of standard test instances from the literature. Experimental results show that the proposed algorithm is effective for the three sets of standard test instances.     

Creating very large scale neighborhoods out of smaller ones by compounding moves

This paper discusses neighborhood search algorithms where the size of the neighborhood is “very large” with respect to the size of the input data. We concentrate on such a very large scale neighborhood (VLSN) search technique based on compounding independent moves (CIM) such as 2-opts, swaps, and insertions. We present a systematic way of creating and searching CIM neighborhoods for routing problems with side constraints. For such problems, the exact search of the CIM neighborhood becomes NP-hard. We introduce a multi-label shortest path algorithm for searching these neighborhoods heuristically. Results of a computational study on the vehicle routing problem with capacity and distance restrictions shows that CIM algorithms are very competitive approaches for solving vehicle routing problems. Overall, the solutions generated by the CIM algorithm have the best performance among the current solution methodologies in terms of percentage deviation from the best-known solutions for large-scale capacitated VRP instances.     

Equilibrium strategies for multiple interdictors on a common network

In this work, we introduce multi-interdictor games, which model interactions among multiple interdictors with differing objectives operating on a common network. As a starting point, we focus on shortest path multi-interdictor (SPMI) games, where multiple interdictors try to increase the shortest path lengths of their own adversaries attempting to traverse a common network. We first establish results regarding the existence of equilibria for SPMI games under both discrete and continuous interdiction strategies. To compute such an equilibrium, we present a reformulation of the SPMI game, which leads to a generalized Nash equilibrium problem (GNEP) with non-shared constraints. While such a problem is computationally challenging in general, we show that under continuous interdiction actions, an SPMI game can be formulated as a linear complementarity problem and solved by Lemke’s algorithm. In addition, we present decentralized heuristic algorithms based on best response dynamics for games under both continuous and discrete interdiction strategies. Finally, we establish theoretical lower bounds on the worst-case efficiency loss of equilibria in SPMI games, with such loss caused by the lack of coordination among noncooperative interdictors, and use the decentralized algorithms to numerically study the average-case efficiency loss.     

α-Coverage to extend network lifetime on wireless sensor networks

An important problem in the context of wireless sensor networks is the Maximum Network Lifetime Problem (MLP): find a collection of subset of sensors (cover) each covering the whole set of targets and assign them an activation time so that network lifetime is maximized. In this paper we consider a variant of MLP, where we allow each cover to neglect a certain fraction (1 ? α) of the targets. We analyze the problem and show that the total network lifetime can be hugely improved by neglecting a very small portion of the targets. An exact approach, based on a Column Generation scheme, is presented and a heuristic solution algorithm is also provided to initialize the approach. The proposed approaches are tested on a wide set of instances. The experimentation shows the effectiveness of both the proposed problems and solution algorithms in extending network lifetime and improving target coverage time when some regularity conditions are taken into account.     

Survivable network design with shared-protection routing

In this paper we study the problem of designing a survivable telecommunication network with shared-protection routing. We develop a heuristic algorithm to solve this problem. Recent results in the area of global re-routing have been used to obtain very tight lower bounds for the problem. Our results indicate that in a majority of problem instances, the average gap between the heuristic solutions and the lower bounds is within 5%. Computational experience is reported on randomly generated problem instances with up to 35 nodes, 80 edges and 595 demand pairs and also on the instances available in SNDlib database.     

A fuzzy shortest path with the highest reliability

This paper concentrates on a shortest path problem on a network where arc lengths (costs) are not deterministic numbers, but imprecise ones. Here, costs of the shortest path problem are fuzzy intervals with increasing membership functions, whereas the membership function of the total cost of the shortest path is a fuzzy interval with a decreasing linear membership function. By the max–min criterion suggested in [R.E. Bellman, L.A. Zade, Decision-making in a fuzzy environment, Management Science 17B (1970) 141–164], the fuzzy shortest path problem can be treated as a mixed integer nonlinear programming problem. We show that this problem can be simplified into a bi-level programming problem that is very solvable. Here, we propose an efficient algorithm, based on the parametric shortest path problem for solving the bi-level programming problem. An illustrative example is given to demonstrate our proposed algorithm.     

Upper and lower bounding procedures for the multiple knapsack assignment problem

We formulate the multiple knapsack assignment problem (MKAP) as an extension of the multiple knapsack problem (MKP), as well as of the assignment problem. Except for small instances, MKAP is hard to solve to optimality. We present a heuristic algorithm to solve this problem approximately but very quickly. We first discuss three approaches to evaluate its upper bound, and prove that these methods compute an identical upper bound. In this process, reference capacities are derived, which enables us to decompose the problem into mutually independent MKPs. These MKPs are solved euristically, and in total give an approximate solution to MKAP. Through numerical experiments, we evaluate the performance of our algorithm. Although the algorithm is weak for small instances, we find it prospective for large instances. Indeed, for instances with more than a few thousand items we usually obtain solutions with relative errors less than 0.1% within one CPU second.     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.     

A column generation approach for an emission-oriented vehicle routing problem on a multigraph

In this work, an emission-minimizing vehicle routing problem with heterogeneous vehicles and a heterogeneous road and traffic network is considered as it is typical in urban areas. Depending on the load of the vehicle, there exist multiple emission-minimal arcs for traveling between two locations. To solve the vehicle routing problem efficiently, a column generation approach is presented. At the core of the procedure an emission-oriented elementary shortest path problem on a multigraph is solved by a backward labeling algorithm. It is shown that the labeling algorithm can be sped up by adjusting the dual master program and by restricting the number of labels propagated in the sub-problem. The column generation technique is used to setup a fast heuristic as well as a branch-and-price algorithm. Both procedures are evaluated based on test instances with up to 100 customers. It turns out that the heuristic approach is very effective and generates near-optimal solutions with gaps below 0.1% on average while only requiring a fraction of the runtime of the exact approach.     

Dynamic data resolution to improve the tractability of UMTS network planning

Transmission site selection and configuration for cellular networks is in general an NP-hard optimization problem. Consequently efforts to improve tractability are very valuable and meta-heuristic algorithms are now commonly applied in artificial intelligence frameworks and expert systems. The speed of network evaluation is a binding constraint on performance of meta-heuristic techniques. This is particularly challenging for CDMA-based systems because power allocation is required before coverage can be evaluated. The current most efficient heuristic for achieving this requires $O(n^{2}_{\mathit{cell}} \cdot n_{\mathit{stp}})$ time where n _cell is the number of cells and n _stp is the number of active user locations. In the large-scale scenarios that arise in practice, n _stp is large compared to n _cell and consequently tractability is significantly impeded by n _stp . We introduce a new approach to improve tractability for the network planning problem. This concerns changing the resolution of the problem scenario by creating virtual entities which combine spatial traffic requirements, thus reducing the number of n _stp that require evaluation. By solving a linear programming formulation of the planning problem exactly (for small instances) and using a heuristic (for large instances), we examine in detail the change in quality of solution that this method induces. The results show that only a marginal reduction in quality of network evaluation is observed, while computational tractability is improved.     

On the second point-to-point undirected shortest simple path problem

We address the determination of the second point-to-point shortest simple path in undirected networks. The effective reduced cost concept is introduced to compute the second best solution. This concept is used to prove that a path tree containing the second point-to-point shortest simple path is adjacent to any shortest path tree. Therefore, this result immediately implies a method requiring O(m) time once that the shortest path tree is obtained on an undirected network with n nodes and m edges.     

Improvements to a large neighborhood search heuristic for an integrated aircraft and passenger recovery problem

Because most commercial passenger airlines operate on a hub-and-spoke network, small disturbances can cause major disruptions in their planned schedules and have a significant impact on their operational costs and performance. When a disturbance occurs, the airline often applies a recovery policy in order to quickly resume normal operations. We present in this paper a large neighborhood search heuristic to solve an integrated aircraft and passenger recovery problem. The problem consists of creating new aircraft routes and passenger itineraries to produce a feasible schedule during the recovery period. The method is based on an existing heuristic, developed in the context of the 2009 ROADEF Challenge, which alternates between three phases: construction, repair and improvement. We introduce a number of refinements in each phase so as to perform a more thorough search of the solution space. The resulting heuristic performs very well on the instances introduced for the challenge, obtaining the best known solution for 17 out of 22 instances within five minutes of computing time and 21 out of 22 instances within 10 minutes of computing time.