Facility Location on Planar Graphs with Unreliable Links
06 Jun 2018-pp 269-281
TL;DR: It is observed that the coverage function of the Max-Exp-Cover-R problem is submodular and the problem admits a \((1-1/e)\)-approximation for any failure model in which the expected coverage of a set by another set can be computed in polynomial time.
Abstract: Hassin et al. [9] consider the Max-Exp-Cover-R problem to study the facility location problem on a graph in the presence of unreliable links when the link failure is according to the Linear Reliability Order (LRO) model. They showed that for unbounded R the problem is polynomial time solvable and for \(R=1\) and planar graphs the problem is NP-Complete. In this paper, we study the Max-Exp-Cover-1 problem under the LRO edge failure model. We obtain a fixed parameter tractable algorithm for Max-Exp-Cover-1 problem for bounded treewidth graphs, parameterized by the treewidth. We extend the Baker’s technique (Baker, J. ACM 1994) to obtain PTAS for Max-Exp-Cover-1 problem under the LRO model on planar graphs. We observe that the coverage function of the Max-Exp-Cover-R problem is submodular and the problem admits a \((1-1/e)\)-approximation for any failure model in which the expected coverage of a set by another set can be computed in polynomial time.
Citations
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TL;DR: An FPT algorithm parameterized by the product of treewidth and max-degree for maximizing expected coverage in an uncertain graph under the RF model and the Linear Reliability Ordering model is presented.
Abstract: We present a detailed survey of results and two new results on graphical models of uncertainty and associated optimization problems. We focus on two well-studied models, namely, the Random Failure (RF) model and the Linear Reliability Ordering (LRO) model. We present an FPT algorithm parameterized by the product of treewidth and max-degree for maximizing expected coverage in an uncertain graph under the RF model. We then consider the problem of finding the maximal core in a graph, which is known to be polynomial time solvable. We show that the Probabilistic-Core problem is polynomial time solvable in uncertain graphs under the LRO model. On the other hand, under the RF model, we show that the Probabilistic-Core problem is W[1]-hard for the parameter d, where d is the minimum degree of the core. We then design an FPT algorithm for the parameter treewidth.
2 citations
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TL;DR: In this paper, the authors introduce and study the problem of facility location along with the notion of social distancing, where the goal is to locate the facilities such that the people can be served and at the same time the total social distance is maximized.
Abstract: In this paper, we introduce and study the problem of facility location along with the notion of \emph{`social distancing'}. The input to the problem is the road network of a city where the nodes are the residential zones, edges are the road segments connecting the zones along with their respective distance. We also have the information about the population at each zone, different types of facilities to be opened and in which number, and their respective demands in each zone. The goal of the problem is to locate the facilities such that the people can be served and at the same time the total social distancing is maximized. We formally call this problem as the \textsc{Social Distancing-Based Facility Location Problem}. We mathematically quantify social distancing for a given allocation of facilities and proposed an optimization model. As the problem is \textsf{NP-Hard}, we propose a simulation-based and heuristic approach for solving this problem. A detailed analysis of both methods has been done. We perform an extensive set of experiments with synthetic datasets. From the results, we observe that the proposed heuristic approach leads to a better allocation compared to the simulation-based approach.
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10 Mar 2022TL;DR: In this paper , the authors studied the expected coverage problem with non-negative demands and showed that it is W[1]-hard to find k potential facility centers in the network such that neighborhood coverage is maximized.
Abstract: Abstract The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k , the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge e of the network, we associate its survival probability 0 ≤ p e ≤ 1, or equivalently, its failure probability 1 − p e . The failure correlation in LRO is the following: If an edge e fails then every edge $e^{\prime }$ e ′ with $p_{e^{\prime }} \leq p_{e}$ p e ′ ≤ p e surely fails. The task is to identify the positions of k facilities that maximize the expected coverage. We refer to this problem as Expected Coverage problem. We study the Expected Coverage problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the Expected Coverage problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that Expected Coverage is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time $ 2^{{\mathcal {O}}({\textbf {tw}} \log {\varDelta })} n^{{\mathcal {O}}(1)}$ 2 O ( tw log Δ ) n O ( 1 ) , where tw is the treewidth, Δ is the maximum vertex degree, and n the number of vertices of the input graph. In particular, since Δ ≤ n , it means the problem is solvable in time $ n^{{\mathcal {O}}({\textbf {tw}})} $ n O ( tw ) , that is, is in XP parameterized by treewidth.
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13 Sep 2021TL;DR: In this paper, the authors introduce and study the problem of facility location along with the notion of social distancing and propose a simulation-based and heuristic approach for solving this problem.
Abstract: In this paper, we introduce and study the problem of facility location along with the notion of ‘social distancing’. The input to the problem is the road network of a city where the nodes are the residential zones, edges are the road segments connecting the zones along with their respective distance. We also have the information about the population at each zone, different types of facilities to be opened and in which number, and their respective demands in each zone. The goal of the problem is to locate the facilities such that the people can be served and at the same time the total social distancing is maximized. We formally call this problem as the Social Distancing-Based Facility Location Problem. We mathematically quantify social distancing for a given allocation of facilities and proposed an optimization model. As the problem is NP-Hard, we propose a simulation-based and heuristic approach for solving this problem. A detailed analysis of both methods has been done. We perform an extensive set of experiments with synthetic datasets. From the results, we observe that the proposed heuristic approach leads to a better allocation compared to the simulation-based approach.
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TL;DR: In this article, it was shown that the PBDS problem is NP-hard even when restricted to uncertain trees of diameter at most four and that it is FPT in the combined parameters of the budget $k$ and the treewidth.
Abstract: We study the {\em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution $p$ associated with the edges, such that an edge $e$ exists in the graph with probability $p(e)$. The input to the problem consists of a vertex-weighted uncertain graph $\G=(V, E, p, \omega)$ and an integer {\em budget} (or {\em solution size}) $k$, and the objective is to compute a vertex set $S$ of size $k$ that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of $S$. We refer to the problem as the {\em Probabilistic Budgeted Dominating Set}~(PBDS) problem and present the following results. \begin{enumerate} \dnsitem We show that the PBDS problem is NP-complete even when restricted to uncertain {\em trees} of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is \wone-hard for the budget parameter $k$, and under the {\em Exponential time hypothesis} it cannot be solved in $n^{o(k)}$ time.
\item We show that if one is willing to settle for $(1-\epsilon)$ approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time.
\item We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is \wone-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget $k$ and the treewidth.
\item Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. \end{enumerate}
References
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TL;DR: It is shown that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1/K]K times the optimal value, which can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.
Abstract: LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)źz(SźT)+z(SźT) for allS, T inN. Such a function is called submodular. We consider the problem maxSźN{a(S):|S|≤K,z(S) submodular}.
Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem.
We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a "greedy" heuristic always produces a solution whose value is at least 1 ź[(K ź 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e ź 1)/e, where e is the base of the natural logarithm.
4,103 citations
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01 Jan 1978
TL;DR: In this article, the authors considered the problem of finding a maximum weight independent set in a matroid, where the elements of the matroid are colored and the items of the independent set can have no more than K colors.
Abstract: LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)gez(SxcupT)+z(SxcapT) for allS, T inN. Such a function is called submodular. We consider the problem maxSsubN{a(S):|S|leK,z(S) submodular}. Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem. We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a ldquogreedyrdquo heuristic always produces a solution whose value is at least 1 –[(K – 1)/K] K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e – 1)/e, where e is the base of the natural logarithm.
3,351 citations
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TL;DR: A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set.
Abstract: This paper describes a general technique that can be used to obtain approximation schemes for various NP-complete problems on planar graphs. The strategy depends on decompos- ing a planar graph into subgraphs of a form we call k-outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least k/(k + 1)optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = (c log log nl or k = (c log nl, where n is the number of nodes and c is some constant, we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominat- ing set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time.
1,047 citations
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TL;DR: In this article, the authors extended the maximum covering location model to account for the chance that when a demand arrives at the system it will not be covered since all facilities capable of covering the demand are engaged serving other demands.
Abstract: The maximum covering location model has been used extensively in analyzing locations for public service facilities. The model is extended to account for the chance that when a demand arrives at the system it will not be covered since all facilities capable of covering the demand are engaged serving other demands. An integer programming formulation of the new problem is presented. Several properties of the formulation are proven. A heuristic solution algorithm is presented and computational results with the algorithm are discussed. Directions for future study are also discussed.
763 citations
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TL;DR: A short overview of recent results in algorithmic graph theory that deal with the notions treewidth and pathwidth can be found in this paper, where the authors discuss algorithms that find tree-decomposition, algorithms that use treedecompositions to solve hard problems efficiently, graph minor theory, and some applications.
Abstract: A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.
755 citations