We develop approximation algorithms for the problem of placing replicated data in arbitrary networks, where the nodes may both issue requests for data objects and have capacity for storing data objects so as to minimize the average data-access cost. We introduce the data placement problem to model this problem. We have a set of caches $\mathcal{F}$, a set of clients $\mathcal{D}$, and a set of data objects $\mathcal{O}$. Each cache $i$ can store at most $u_i$ data objects. Each client $j\in\mathcal{D}$ has demand $d_j$ for a specific data object $o(j)\in\mathcal{O}$ and has to be assigned to a cache that stores that object. Storing an object $o$ in cache $i$ incurs a storage cost of $f_i^o$, and assigning client $j$ to cache $i$ incurs an access cost of $d_jc_{ij}$. The goal is to find a placement of the data objects to caches respecting the capacity constraints, and an assignment of clients to caches so as to minimize the total storage and client access costs. We present a 10-approximation algorithm for this problem. Our algorithm is based on rounding an optimal solution to a natural linear-programming relaxation of the problem. One of the main technical challenges encountered during rounding is to preserve the cache capacities while incurring only a constant-factor increase in the solution cost. We also introduce the connected data placement problem to capture settings where write-requests are also issued for data objects, so that one requires a mechanism to maintain consistency of data. We model this by requiring that all caches containing a given object be connected by a Steiner tree to a root for that object, which issues a multicast message upon a write to (any copy of) that object. The total cost now includes the cost of these Steiner trees. We devise a 14-approximation algorithm for this problem. We show that our algorithms can be adapted to handle two variants of the problem: (a) a $k$-median variant, where there is a specified bound on the number of caches that may contain a given object, and (b) a generalization where objects have lengths and the total length of the objects stored in any cache must not exceed its capacity.

/pdf/approximation-algorithms-for-data-placement-problems-1qb8vj3rfj.pdf

Approximation Algorithms for Data Placement Problems

Mining useful clusters from high dimensional data have received significant attention of the computer vision and pattern recognition community in the recent years. Linear and nonlinear dimensionality reduction has played an important role to overcome the curse of dimensionality. However, often such methods are accompanied with three different problems: high computational complexity (usually associated with the nuclear norm minimization), nonconvexity (for matrix factorization methods), and susceptibility to gross corruptions in the data. In this paper, we propose a principal component analysis (PCA) based solution that overcomes these three issues and approximates a low-rank recovery method for high dimensional datasets. We target the low-rank recovery by enforcing two types of graph smoothness assumptions, one on the data samples and the other on the features by designing a convex optimization problem. The resulting algorithm is fast, efficient, and scalable for huge datasets with O(n log(n)) computational complexity in the number of data samples. It is also robust to gross corruptions in the dataset as well as to the model parameters. Clustering experiments on 7 benchmark datasets with different types of corruptions and background separation experiments on 3 video datasets show that our proposed model outperforms 10 state-of-the-art dimensionality reduction models. Our theoretical analysis proves that the proposed model is able to recover approximate low-rank representations with a bounded error for clusterable data.

Fast Robust PCA on Graphs

In this paper a typical situation arising in the assembly of printed circuit boards is investigated. The planning problem we face is how to assemble boards of different types using a single line of placement machines. From a practical viewpoint, the multiplicity of board types adds significantly to the complexity of the problem, which is already very hard to solve in the case of a single board type. In addition, relatively few studies deal with the multiple board type case. We propose a solution procedure based on a hierarchical decomposition of the planning problem. An important subproblem in this decomposition is the so-called feeder rack assignment problem. By taking into account as much as possible the individual board type characteristics (as well as the machine characteristics) we heuristically solve this problem. The remaining subproblems are solved using constructive heuristics and local search methods. The solution procedure is tested on real-life instances. It turns out that, in terms of the makespan, we can substantially improve the current solutions. Keywords: heuristics, PCB-assembly, feeder rack assignment problem.

The assembly of printed circuit boards: a case with multiple machines and multiple board types

Based on solid theoretical foundations, we present strong evidence that a number of real-world networks, taken from different domains such as Internet measurements, biological data, web graphs, and social and collaboration networks exhibit tree-like structures from a metric point of view. We investigate a few graph parameters, namely, the tree-distortion and the tree-stretch, the tree-length and the tree-breadth, Gromov's hyperbolicity, the cluster-diameter and the cluster-radius in a layering partition of a graph; such parameters capture and quantify this phenomenon of being metrically close to a tree. By bringing all those parameters together, we provide efficient means for detecting such metric tree-like structures in large-scale networks. We also show how such structures can be used. For example, they are helpful in efficient and compact encoding of approximate distance and almost shortest path information and in quick and accurate estimation of diameters and radii of those networks. Estimating the diameter and estimating the radius of a graph or distances between arbitrary vertices are fundamental primitives in many network and graph mining algorithms. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 671, 49-68 2016

Metric tree-like structures in real-world networks: an empirical study

Although information and communications technologies (ICTs) have the potential of enabling powerful social, economic and environmental benefits, ICT systems give a non-negligible contribution to world electricity consumption and carbon dioxide (CO2) footprint. This contribution will sustain since the increased demand for user′s connectivity and an explosion of traffic volumes necessitate continuous expansion of current ICTs services and deployment of new infrastructures and technologies which must ensure the expected user experiences and performance. In this paper, analyses of costs for the global annual energy consumption of telecommunication networks, estimation of ICT sector CO2 footprint contribution and predictions of energy consumption of all connected user-related devices and equipment in the period 2011–2030 are presented. Since presented estimations of network energy consumption trends for main communication sectors by 2030 shows that highest contribution to global energy consumption will come from wireless access networks and data centres (DCs), the rest of the paper analyses technologies and concepts which can contribute to the energy-efficiency improvements of these two sectors. More specifically, different paradigms for wireless access networks such as millimetre-wave communications, Long-Term Evolution in unlicensed spectrum, ultra-dense heterogeneous networks, device-to-device communications and massive multiple-input multiple-output communications have been analysed as possible technologies for improvement of wireless networks energy efficiency. Additionally, approaches related to the DC resource management, DCs power management, green DC monitoring and thermal management in DCs have been discussed as promising approaches to improvement of DC power usage efficiency. For each of analysed technologies, future research challenges and open issues have been summarised and discussed. Lastly, an overview of the accepted papers in the Special Issue dedicated to the green, energy-efficient and sustainable networks is presented.

https://www.mdpi.com/1424-8220/19/22/4864/pdf

Greener, Energy-Efficient and Sustainable Networks: State-Of-The-Art and New Trends.

We consider a generalization of the well-known domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demands of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies the demand of each vertex in V to be met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees from an algorithmic point of view. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete (even for its integer version) and provide a polynomial time approximation scheme (PTAS). We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.

Capacitated Domination Problem

We consider a local service-requirement assignment problem named capacitated domination from an algorithmic point of view. In this problem, we are given a graph with three parameters defined on each vertex, which are the cost, the capacity, and the demand, of a vertex, respectively. A vertex can be chosen multiple times in order to generate sufficient capacity for the demands of the vertices in its closed neighborhood. The objective of this problem is to compute a demand assignment of minimum cost such that the demand of each vertex is fully-served by some of its closed neighbors without exceeding the amount of capacity they provide.

In this paper, we provide complexity results as well as several approximation algorithms to compose a comprehensive study for this problem. First, we provide logarithmic approximations for general graphs which are asymptotically optimal. From the perspective of parameterized complexity, we show that this problem is W[1]-hard with respect to treewidth and solution size. Moreover, we show that this problem is fixed-parameter tractable with respect to treewidth and the maximum capacity of the vertices. The latter result implies a pseudo-polynomial time approximation scheme for planar graphs under a standard framework.

In order to drop the pseudo-polynomial factor, we develop a constant-factor approximation for planar graphs, based on a new perspective which we call general ladders on the hierarchical structure of outer-planar graphs. We believe that the approach we use can be applicable to other capacitated covering problems.

Capacitated Domination: Problem Complexity and Approximation Algorithms

We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an $f$-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al (SODA 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor in (FOCS 2002). We believe that our rounding technique is of independence interests when hard constraints are considered. 
Our main technical tool for establishing the approximation guarantee is a separation lemma that certifies the existence of a strong partition for solutions that are basic feasible in an extended version of the natural LP.

Iterative Partial Rounding for Vertex Cover with Hard Capacities

We explore the machine-minimizing job scheduling problem, which has a rich history in the line of research, under an online setting. We consider systems with arbitrary job arrival times, arbitrary job deadlines, and unit job execution time. For this problem, we present a lower bound 2.09 on the competitive factor of any online algorithms, followed by designing a 5.2-competitive online algorithm. We would also like to point out a false claim made in an existing paper of Shi and Ye regarding a further restricted case of the considered problem. To the best of our knowledge, what we present is the first concrete result concerning online machine-minimizing job scheduling with arbitrary job arrival times and deadlines.

Competitive Design and Analysis for Machine-Minimizing Job Scheduling Problem

We consider a generalization of the well-known domination problem on graphs. The (soft) capacitated domination problem with demand constraints is to find a dominating set D of minimum cardinality satisfying both the capacity and demand constraints. The capacity constraint specifies that each vertex has a capacity that it can use to meet the demand of dominated vertices in its closed neighborhood, and the number of copies of each vertex allowed in D is unbounded. The demand constraint specifies that the demand of each vertex in V is met by the capacities of vertices in D dominating it. In this paper, we study the capacitated domination problem on trees. We present a linear time algorithm for the unsplittable demand model, and a pseudo-polynomial time algorithm for the splittable demand model. In addition, we show that the capacitated domination problem on trees with splittable demand constraints is NP-complete (even for its integer version) and provide a 3/2-approximation algorithm. We also give a primal-dual approximation algorithm for the weighted capacitated domination problem with splittable demand constraints on general graphs.

Mong-Jen Kao

Papers

Capacitated Domination Problem

Capacitated Domination: Problem Complexity and Approximation Algorithms

Iterative Partial Rounding for Vertex Cover with Hard Capacities

Competitive Design and Analysis for Machine-Minimizing Job Scheduling Problem

Capacitated domination problem