Approximation algorithms for the MAXSPACE advertisement problem

In the MAXSPACE problem, given a set of ads A, one wants to schedule a subset A' of A into K slots B_1, ..., B_K of size L. Each ad A_i in A has a size s_i and a frequency w_i. A schedule is feasible if the total size of ads in any slot is at most L, and each ad A_i in A' appears in exactly w_i slots. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We introduce a generalization called MAXSPACE-R in which each ad A_i also has a release date r_i >= 1, and may only appear in a slot B_j with j >= r_i. We also introduce a generalization of MAXSPACE-R called MAXSPACE-RD in which each ad A_i also has a deadline d_i <= K, and may only appear in a slot B_j with r_i <= j <= d_i. These parameters model situations where a subset of ads corresponds to a commercial campaign with an announcement date that may expire after some defined period. We present a 1/9-approximation algorithm for MAXSPACE-R and a polynomial-time approximation scheme for MAXSPACE-RD when K is bounded by a constant. This is the best factor one can expect, since MAXSPACE is NP-hard, even if K = 2

Computing the Largest Bond of a Graph

A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S,VS] of G such that G[S] and G[VS] are both connected. Given s,t in V(G), an st-bond of G is a bond whose removal disconnects s and t. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest st-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that Largest Bond remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless P = NP. We also show that Largest Bond and Largest st-Bond on graphs of clique-width w cannot be solved in time f(w) x n^{o(w)} unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) x n^{O(w)}. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP subseteq coNP/poly

Algorithms for the Bin Packing Problem with Scenarios

This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose an absolute approximation algorithm whose ratio is bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential model and a variable neighborhood search heuristic. To speed up the convergence of the exact algorithm, we also consider lower bounds based on dual feasible functions. Results of these algorithms show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered

Approximation algorithms for median hub location problems

In Hub Location Problems, an input is composed of sets of clients, locations and a pairs of clients; a solution is a subset of locations to open hubs and an assignment for each pair of clients to a route starting in the first client, passing through one or more hubs and ending in the second client. The objective is to find a solution that minimizes the length of all routes plus the cost of opening hubs. The currently known approximation algorithms consider only the case in which the set of hubs is given as part of the input and the problem is assigning clients to hubs. For a metric space setting, this work presents the first constant-factor approximation algorithms for the problem of, simultaneously, selecting hubs and allocating clients. A few variants are considered, depending on whether the number of open hubs is given in the input or a client must be assigned to a single hub. In particular, we give an LP-rounding 2.48-approximation algorithm for the Single-Allocation Median Hub Location Problem, using a new formulation and exploiting its symmetries382375401CNPQ - Conselho Nacional de Desenvolvimento Científico e TecnológicoFAPESP – Fundação de Amparo à Pesquisa Do Estado De São Paulosem informação2015/11937-9, 2016/12006-

Clustering through continuous facility location problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We consider the Continuous Facility Location Problem (ConFLP). Given a finite set of clients C subset of R-d and a number f is an element of R+, ConFLP consists in opening a set F' subset of R-d of facilities, each at cost f, and connecting each client to an open facility. The objective is to minimize the costs of opening facilities and connecting clients. We reduce ConFLP to the standard Facility Location Problem (FLP), by using the so-called approximate center sets. This reduction preserves the approximation, except for an error epsilon, and implies 1.488 + epsilon and 2.04 + epsilon-approximations when the connection cost is given by the Euclidean distance and the squared Euclidean distance, respectively. Moreover, we obtain approximate center sets for the case that the connection cost is the ath power of the Euclidean distance, achieving approximations for the corresponding problems, for any alpha >= 1. As a byproduct, we also obtain a polynomial-time approximation scheme for the k-median problem with this cost function, for any fixed k. (C) 2016 Elsevier B.V. All rights reserved.We consider the Continuous Facility Location Problem (ConFLP). Given a finite set of clients C subset of R-d and a number f is an element of R+, ConFLP consists in opening a set F' subset of R-d of facilities, each at cost f, and connecting each client to657B137145FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)2010/20710-4477692/2012-5; 311499/2014-

Approximation algorithms for k-level stochastic facility location problems

In the k-level facility location problem (FLP), we are given a set of facilities, each associated with one of k levels, and a set of clients. We have to connect each client to a chain of opened facilities spanning all levels, minimizing the sum of opening and connection costs. This paper considers the k-level stochastic FLP, with two stages, when the set of clients is only known in the second stage. There is a set of scenarios, each occurring with a given probability. A facility may be opened in any stage, however, the cost of opening a facility in the second stage depends on the realized scenario. The objective is to minimize the expected total cost. For the stage-constrained variant, when clients must be served by facilities opened in the same stage, we present a (Formula presented.)-approximation, improving on the 4-approximation by Wang et al. (Oper Res Lett 39(2):160–161, 2011) for each k. In the case with (Formula presented.), the algorithm achieves factors 2.56 and 2.78, resp., which improves the (Formula presented.)-approximation for (Formula presented.) by Wu et al. (Theor Comput Sci 562:213–226, 2015). For the non-stage-constrained version, we give the first approximation for the problem, achieving a factor of 3.495 for the case with (Formula presented.), and (Formula presented.) in general. © 2016 Springer Science+Business Media New YorkIn the k-level facility location problem (FLP), we are given a set of facilities, each associated with one of k levels, and a set of clients. We have to connect each client to a chain of opened facilities spanning all levels, minimizing the sum of opening341266278FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO2013/21744-8sem informaçã

A systematic approach to bound factor-revealing LPs and its application to the metric and squared metric facility location problems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the metric facility location problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the squared metric facility location problem (SMFLP), and prove that there is no approximation factor better than 2.04, assuming P not equal NP. Then, we analyze the best known algorithms for the MFLP based on primal-dual and LP-rounding techniques when they are applied to the SMFLP. We prove very tight bounds for these algorithms, and show that the LP-rounding algorithm achieves a ratio of 2.04, and therefore has the best possible factor for the SMFLP. We use UPFRPs in the dualfitting analysis of the primal-dual algorithms for both the SMFLP and the MFLP, improving some of the previous analysis for the MFLP.A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor o1532655685CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULONUMEC - NÚCLEO DE APOIO À PESQUISA EM MODELAGEM ESTOCÁSTICA E COMPLEXIDADEConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [309657/2009-1, 306860/2010-4, 473867/2010-9, 477692/2012-5]FAPESP [2010/20710-4]309657/2009-1; 306860/2010-4; 473867/2010-9; 477692/2012-52010/20710-4sem informaçã