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Register a new userA constant-ratio approximation algorithm for a class of hub-and-spoke network design problems and metric labeling problems: star metric case
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Added by
Yuko Kuroki
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Transportation networks frequently employ hub-and-spoke network architectures to route flows between many origin and destination pairs. Hub facilities work as switching points for flows in large networks. In this study, we deal with a problem, called the single allocation hub-and-spoke network design problem. In the problem, the goal is to allocate each non-hub node to exactly one of given hub nodes so as to minimize the total transportation cost. The problem is essentially equivalent to another combinatorial optimization problem, called the metric labeling problem. The metric labeling problem was first introduced by Kleinberg and Tardos in 2002, motivated by application to segmentation problems in computer vision and related areas. In this study, we deal with the case where the set of hubs forms a star, which arises especially in telecommunication networks. We propose a polynomial-time randomized approximation algorithm for the problem, whose approximation ratio is less than 5.281. Our algorithms solve a linear relaxation problem and apply dependent rounding procedures.
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Proposed initially from a practical circumstance, the traveling salesman problem caught the attention of numerous economists, computer scientists, and mathematicians. These theorists were instead intrigued by seeking a systemic way to find the optimal route. Many attempts have been made along the way and all concluded the nonexistence of a general algorithm that determines optimal solution to all traveling salesman problems alike. In this study, we present proof for the nonexistence of such an algorithm for both asymmetric (with oriented roads) and symmetric (with unoriented roads) traveling salesman problems in the setup of constructive mathematics.
We propose a new self-organizing algorithm for fixed-charge network flow problems based on ghost image (GI) processes as proposed in Glover (1994) and adapted to fixed-charge transportation problems in Glover, Amini and Kochenberger (2005). Our self-organizing GI algorithm iteratively modifies an idealized representation of the problem embodied in a parametric ghost image, enabling all steps to be performed with a primal network flow algorithm operating on the parametric GI. Computational tests are carried out on an extensive set of benchmark problems which includes the previous largest set in the literature, comparing our algorithm to the best methods previously proposed for fixed-charge transportation problems, though our algorithm is not specialized to this class. We also provide comparisons for additional more general fixed-charge network flow problems against Cplex 12.8 to demonstrate that the new self-organizing GI algorithm is effective on large problem instances, finding solutions with statistically equivalent objective values at least 700 times faster. The attractive outcomes produced by the current GI/TS implementation provide a significant advance in our ability to solve fixed-cost network problems efficiently and invites its use for larger instances from a variety of application domains.
In the metric multi-cover problem (MMC), we are given two point sets $Y$ (servers) and $X$ (clients) in an arbitrary metric space $(X cup Y, d)$, a positive integer $k$ that represents the coverage demand of each client, and a constant $alpha geq 1$. Each server can have a single ball of arbitrary radius centered on it. Each client $x in X$ needs to be covered by at least $k$ such balls centered on servers. The objective function that we wish to minimize is the sum of the $alpha$-th powers of the radii of the balls. In this article, we consider the MMC problem as well as some non-trivial generalizations, such as (a) the non-uniform MMC, where we allow client-specific demands, and (b) the $t$-MMC, where we require the number of open servers to be at most some given integer $t$. For each of these problems, we present an efficient algorithm that reduces the problem to several instances of the corresponding $1$-covering problem, where the coverage demand of each client is $1$. Our reductions preserve optimality up to a multiplicative constant factor. Applying known constant factor approximation algorithms for $1$-covering, we obtain the first constant approximations for the MMC and these generalizations.
In the Metric Capacitated Covering (MCC) problem, given a set of balls $mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $mathcal{B}subseteq mathcal{B}$ and an assignment of the points in $P$ to the balls in $mathcal{B}$ such that each point is assigned to a ball that contains it and each ball is assigned with at most $U$ points. MCC achieves an $O(log |P|)$-approximation using a greedy algorithm. On the other hand, it is hard to approximate within a factor of $o(log |P|)$ even with $beta < 3$ factor expansion of the balls. Bandyapadhyay~{et al.} [SoCG 2018, DCG 2019] showed that one can obtain an $O(1)$-approximation for the problem with $6.47$ factor expansion of the balls. An open question left by their work is to reduce the gap between the lower bound $3$ and the upper bound $6.47$. In this current work, we show that it is possible to obtain an $O(1)$-approximation with only $4.24$ factor expansion of the balls. We also show a similar upper bound of $5$ for a more generalized version of MCC for which the best previously known bound was $9$.
To save energy and alleviate interferences in a wireless sensor network, the usage of virtual backbone was proposed. Because of accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone, which can be modeled as a $k$-connected $m$-fold dominating set (abbreviated as $(k,m)$-CDS) in a graph. A node set $Csubseteq V(G)$ is a $(k,m)$-CDS of graph $G$ if every node in $V(G)backslash C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$-connected. In this paper, we present an approximation algorithm for the minimum $(3,m)$-CDS problem with $mgeq3$. The performance ratio is at most $gamma$, where $gamma=alpha+8+2ln(2alpha-6)$ for $alphageq4$ and $gamma=3alpha+2ln2$ for $alpha<4$, and $alpha$ is the performance ratio for the minimum $(2,m)$-CDS problem. Using currently best known value of $alpha$, the performance ratio is $lndelta+o(lndelta)$, where $delta$ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. This is the first performance-guaranteed algorithm for the minimum $(3,m)$-CDS problem on a general graph. Furthermore, applying our algorithm on a unit disk graph which models a homogeneous wireless sensor network, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the $(3,m)$-CDS problem on a unit disk graph.
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