Approximation algorithms for the dynamic k-level facility location problems

In this paper, we introduce the multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and vertex cover with submodular penalties. We present a combinatorial 3-approximation algorithm, based on the primal-dual scheme for the multicut problem in trees.

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V, we want to find a subset R of rejected elements, and assign other elements to one of m sets A1,A2,⋯,Am. The objective is to minimize the sum of the maximum load among A1,A2,⋯,Am and the rejection penalty of R, where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of m+1 under the heterogenous setting and a min{m,⌈nm⌉+1}=O(n)-approximation combinatorial algorithm under the homogenous setting.

This paper introduces a variant of the dynamic Facility Location Problem with modular capacities. Inspired by the challenges faced by cellular telecommunication networks, the problem seeks to optimize the placement and equipment of facilities to meet the fluctuating demand of clients while minimizing installation and operational costs. Selected facilities must provide coverage to the whole considered area. In addition, clients must be served by the active facility that provides the best quality of service. We propose two integer linear programming formulations and analyze their theoretical properties in terms of lower bound. Furthermore, we propose a reinforcement for one formulation and valid inequalities. To evaluate the performance of the proposed formulations, we performed computational experiments with a commercial solver on instances with up to 20 candidate sites and 60 clients. The formulations proved to be very sensitive to the size of the instances. The analysis, both theoretical and numerical, provides meaningful insights on the formulations performance, and it constitutes a key starting point for developing further approaches, either exact methods or heuristics.

This article considers a class of dynamic single-article facility location problems in which the facility must determine order and inventory levels to meet the dynamic demands of the customers over a finite horizon. The motivating application of this class of problems is in military logistics and the decision makers in this area are not only concerned with the logistical costs of the facility but also with centering the facility among the customers in each time period in order to be able to provide other services. Both the location plan and inventory plan of the facility in the problem must be determined while considering these different metrics associated with the performance of these plans. Effective dynamic programming algorithms for this class of problem are provided for both of these metrics. These dynamic programming algorithms are utilized in order to construct the efficient frontier associated with these two metrics in polynomial time. Computational testing indicates that these algorithms can be used in planning activities for military logistics. [Supplemental materials are available for this article. Go to the publisher’s online edition of IIE Transactions for a worst-case example of constructing the efficient frontier.]

A fuzzy programming approach for dynamic virtual hub location problem

In this paper a memetic algorithm integrating genetic procedures and local search, able to solve capacitated and uncapacitated dynamic location problems, is described. These problems are characterized by explicitly considering the possibility of a facility being opened, closed and reopened more than once during the planning horizon. It is also possible to explicitly consider different open and reopen fixed costs. The problems can be of single or multi-level nature. The computational results obtained show that the algorithm is capable of finding good quality solutions, but at the cost of large computational times, when compared with dedicated primal-dual heuristics and even with a general solver. Some changes are proposed to tackle this disadvantage.

A note on the submodular vertex cover problem with submodular penalties

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv+3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv +3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain’s iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

We consider a competitive facility location model where competing parties (Leader and Follower) make decisions considering changes of the set of customers happening during the planing horizon consisting a known number of time periods. It is assumed that the Leader makes a decision on opening their facilities at the beginning of the planning horizon, while the Follower can revise their decision in each time period. In the present paper, we study perspectives to apply a method for finding the best solution that is based on using HP-relaxation of the bilevel problem considered. The key element of this method is construction of additional inequalities strengthening the HP-relaxation and computation of upper bounds for the objective function of the problem. In the paper, we propose new families of additional constraints to strengthen the HP-relaxation that allow computing nontrivial upper bounds.

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held-Karp relaxation, which may be of independent interest.

In the classical k-median problem, we are given a metric space and would like to open k centers so as to minimize the sum (over all the vertices) of the distance of each vertex to its nearest open center. In this paper, we consider the following generalization of the problem: instead of opening at most k centers, what if each center belongs to one of T different types, and we are allowed to open at most ki centers of type i (for each i=1,2,...,T). The case T = 1 is the classical k-median, and the case of T = 2 is the red-blue median problem for which Hajiaghayi et al. [ESA 2010] recently gave a constant-factor approximation algorithm.Even more generally, what if the set of open centers had to form an independent set from a matroid? In this paper, we give a constant factor approximation algorithm for such matroid median problems. Our algorithm is based on rounding a natural LP relaxation in two stages: in the first step, we sparsify the structure of the fractional solution while increasing the objective function value by only a constant factor. This enables us to write another LP in the second phase, for which the sparsified LP solution is feasible. We then show that this second phase LP is in fact integral; the integrality proof is based on a connection to matroid intersection.We also consider the penalty version (alternately, the so-called prize collecting version) of the matroid median problem and obtain a constant factor approximation algorithm for it. Finally, we look at the Knapsack Median problem (in which the facilities have costs and the set of open facilities need to fit into a Knapsack) and get a bicriteria approximation algorithm which violates the Knapsack bound by a small additive amount.

A robust optimization model for dynamic virtual hub location problem under uncertainty using an M/M/C/K queuing model: two metaheuristic algorithms

We consider the problem of makespan minimization: i.e., scheduling jobs on machines to minimize the maximum load. For the deterministic case, good approximations are known even when the machines are unrelated. However, the problem is not well-understood when there is uncertainty in the job sizes. In our setting the job sizes are stochastic, i.e., the size of a job j on machine i is a random variable Xij, whose distribution is known. (Sizes of different jobs are independent of each other.) The goal is to find a fixed assignment of jobs to machines, to minimize the expected makespan---i.e., the expected value of the maximum load over the m machines. For the identical machines special case when the size of a job is the same across all machines, a constant-factor approximation algorithm has long been known. However, the problem has remained open even for the next-harder related machines case. Our main result is a constant-factor approximation for the most general case of unrelated machines. The main technical challenge we overcome is obtaining an efficiently computable lower bound for the optimal solution. We give an exponential-sized LP that we argue gives a strong lower bound. Then we show how to round any fractional solution to satisfy only a small subset of the constraints, which are enough to bound the expected makespan of our solution. We then consider two generalizations. The first is the budgeted makespan minimization problem, where the goal is to minimize the makespan subject to scheduling any subset of jobs whose reward is at least some target reward R. We extend our above result to a constant-factor approximation here using polyhedral properties of the bipartite matching polytope. The second problem is the q-norm minimization problem, where we want to minimize the expected eq-norm of the load vectors. Here we give an O(q/ log q)-approximation algorithm using a reduction to the deterministic q-norm problem with side constraints.

We present the first polynomial-time approximation scheme (PTAS), i.e., (1+e)-approximation algorithm for any constant e> 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn (loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing light-weight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2+)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge.