Packing problems constitute an important class of optimization problems, both because of their high practical relevance and theoretical appeal. However, despite the large number of variants that have been studied in the literature, most packing problems encompass a single tier of capacity restrictions only. For example, in the Multiple Knapsack Problem, we assign items to multiple knapsacks such that their capacities are not exceeded. But what if these knapsacks are partitioned into clusters, each imposing an additional capacity restriction on the knapsacks contained in that cluster? In this paper, we study the Maximum Coverage Problem with Cluster Constraints (MCPC), which generalizes the Maximum Coverage Problem with Knapsack Constraints (MCPK) by incorporating cluster constraints. Our main contribution is a general LP-based technique to derive approximation algorithms for cluster capacitated problems. Our technique allows us to reduce the cluster capacitated problem to the respective original packing problem. By using an LP-based approximation algorithm for the original problem, we can then obtain an effective rounding scheme for the problem, which only loses a small fraction in the approximation guarantee. We apply our technique to derive approximation algorithms for MCPC. To this aim, we develop an LP-based $\frac12(1-\frac1e)$-approximation algorithm for MCPK by adapting the pipage rounding technique. Combined with our reduction technique, we obtain a $\frac13(1-\frac1e)$-approximation algorithm for MCPC. We also derive improved results for a special case of MCPC, the Multiple Knapsack Problem with Cluster Constraints (MKPC). Based on a simple greedy algorithm, our approach yields a $\frac13$-approximation algorithm. By combining our technique with a more sophisticated iterative rounding approach, we obtain a $\frac12$-approximation algorithm for certain special cases of MKPC.
We consider a generalization of the classical planar maximum coverage location problem (PMCLP) in which partial coverage is allowed, facilities have adjustable quality of service (QoS) or service range, and demand zones and service zone of each facility are represented by two-dimensional spatial objects such as rectangles, circles, polygons, etc. We denote this generalization by PMCLP-PC-QoS. A key challenge in this problem is to simultaneously decide position of the facilities on a continuous two-dimensional plane and their QoS. We present a greedy algorithm and a pseudo-greedy algorithm for …
This paper proposes a greedy heuristic named as Big step greedy heuristic and investigates the application of Big step greedy heuristic for maximum k-coverage problem. Greedy algorithms construct the solution in multiple steps, the classical greedy algorithm for maximum k-coverage problem, in each step selects one set that contains the greatest number of uncovered elements. The Big step greedy heuristic, in each step selects p (1 <= p <= k) sets such that the union of selected p sets contains the greatest number of uncovered elements by evaluating all possible p-combinations of given sets. Whe…
We develop a randomized approximation algorithm for the classical maximum coverage problem, which given a list of sets $A_1,A_2,\cdots, A_m$ and integer parameter $k$, select $k$ sets $A_{i_1}, A_{i_2},\cdots, A_{i_k}$ for maximum union $A_{i_1}\cup A_{i_2}\cup\cdots\cup A_{i_k}$. In our algorithm, each input set $A_i$ is a black box that can provide its size $|A_i|$, generate a random element of $A_i$, and answer the membership query $(x\in A_i?)$ in $O(1)$ time. Our algorithm gives $(1-{1\over e})$-approximation for maximum coverage problem in $O(p(m))$ time, which is independent of the size…
In mobile wireless sensor networks (MWSNs), each sensor has the ability not only to sense and transmit data but also to move to some specific location. Because the movement of sensors consumes much more power than that in sensing and communication, the problem of scheduling mobile sensors to cover all targets and maintain network connectivity such that the total movement distance of mobile sensors is minimized has received a great deal of attention. However, in reality, due to a limited budget or numerous targets, mobile sensors may be not enough to cover all targets or form a connected networ…
In the classic maximum coverage problem, we are given subsets $T_1, \dots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := |\cup_{i \in S} T_i|$. It is well-known that the greedy algorithm for this problem achieves an approximation ratio of $(1-e^{-1})$ and there is a matching inapproximability result. We note that in the maximum coverage problem if an element $e \in [n]$ is covered by several sets, it is still counted only once. By contrast, if we change the problem and count each element $e$ as man…
This paper studies the problem of clustering in metric spaces while preserving the privacy of individual data. Specifically, we examine differentially private variants of the k-medians and Euclidean k-means problems. We present polynomial algorithms with constant multiplicative error and lower additive error than the previous state-of-the-art for each problem. Additionally, our algorithms use a clustering algorithm without differential privacy as a black-box. This allows practitioners to control the trade-off between runtime and approximation factor by choosing a suitable clustering algorithm …
The Set-Union Knapsack Problem (SUKP) and Budgeted Maximum Coverage Problem (BMCP) are two closely related variant problems of the popular knapsack problem. Given a set of weighted elements and a set of items with nonnegative values, where each item covers several distinct elements, these two problems both aim to find a subset of items that maximizes an objective function while satisfying a knapsack capacity (budget) constraint. We propose an efficient and effective local search algorithm called E2LS for these two problems. To our knowledge, this is the first time that an algorithm has been pr…
This study investigated typical performance of approximation algorithms known as belief propagation, greedy algorithm, and linear-programming relaxation for maximum coverage problems on sparse biregular random graphs. After using the cavity method for a corresponding hard-core lattice--gas model, results show that two distinct thresholds of replica-symmetry and its breaking exist in the typical performance threshold of belief propagation. In the low-density region, the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses. Although…
Given a line barrier and a set of mobile sensors distributed in the plane, the Minimizing Maximum Sensor Movement problem (MMSM) for \textcolor{black}{line barrier coverage} is to compute relocation positions for the sensors in the plane such that the barrier is entirely covered by the monitoring area of the sensors while the maximum relocation movement (distance) is minimized. Its weaker version, decision MMSM is to determine whether the barrier can be covered by the sensors within a given relocation distance bound $D\in\mathbb{Z}^{+}$. This paper presents three approximation algorithms for d…
We study the maximum $k$-set coverage problem in the following distributed setting. A collection of sets $S_1,\ldots,S_m$ over a universe $[n]$ is partitioned across $p$ machines and the goal is to find $k$ sets whose union covers the most number of elements. The computation proceeds in synchronous rounds. In each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end, the coordinator outputs the answer. The main measures of efficiency in this setting are the approxi…
Streaming computation plays an important role in large-scale data analysis. The sliding window model is a model of streaming computation which also captures the recency of the data. In this model, data arrives one item at a time, but only the latest $W$ data items are considered for a particular problem. The goal is to output a good solution at the end of the stream by maintaining a small summary during the stream. In this work, we propose a new algorithmic framework for designing efficient sliding window algorithms via bucketing-based sketches. Based on this new framework, we develop space-ef…
We study the classic NP-Hard problem of finding the maximum $k$-set coverage in the data stream model: given a set system of $m$ sets that are subsets of a universe $\{1,\ldots,n \}$, find the $k$ sets that cover the most number of distinct elements. The problem can be approximated up to a factor $1-1/e$ in polynomial time. In the streaming-set model, the sets and their elements are revealed online. The main goal of our work is to design algorithms, with approximation guarantees as close as possible to $1-1/e$, that use sublinear space $o(mn)$. Our main results are: Two $(1-1/e-ε)$ approximati…
In this paper, we study the problem of moving $n$ sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an $O(n^2 \log n)$ time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $O(n^2)$ time. We present an $O(n \log n)$ time algorithm for this case; further,…
We address the Budgeted Maximum Coverage Problem (BMCP), which is a natural and more practical extension of the standard 0-1 knapsack problem and the set cover problem. Given m elements with nonnegative weights, n subsets of elements with nonnegative costs, and a total budget, BMCP aims to select some subsets such that the total cost of selected subsets does not exceed the budget, and the total weight of associated elements is maximized. In this paper, we propose a variable depth local search algorithm (VDLS) for the BMCP. VDLS first generates an initial solution by a greedy algorithm, then it…
Microsoft Research Asia（微软亚洲研究院）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We revisit the optimization from samples (OPS) model, which studies the problem of optimizing objective functions directly from the sample data. Previous results showed that we cannot obtain a constant approximation ratio for the maximum coverage problem using polynomially many independent samples of the form $\{S_i, f(S_i)\}_{i=1}^t$ (Balkanski et al., 2017), even if coverage functions are $(1 - ε)$-PMAC learnable using these samples (Badanidiyuru et al., 2012), which means most of the function values can be approximately learned very well with high probability. In this work, to circumvent th…
We study the complexity of the maximum coverage problem, restricted to set systems of bounded VC-dimension. Our main result is a fixed-parameter tractable approximation scheme: an algorithm that outputs a $(1-\eps)$-approximation to the maximum-cardinality union of $k$ sets, in running time $O(f(\eps,k,d)\cdot poly(n))$ where $n$ is the problem size, $d$ is the VC-dimension of the set system, and $f(\eps,k,d)$ is exponential in $(kd/\eps)^c$ for some constant $c$. We complement this positive result by showing that the function $f(\eps,k,d)$ in the running-time bound cannot be replaced by a fun…
We study the online maximum coverage problem on a line, in which, given an online sequence of sub-intervals (which may intersect among each other) of a target large interval and an integer $k$, we aim to select at most $k$ of the sub-intervals such that the total covered length of the target interval is maximized. The decision to accept or reject each sub-interval is made immediately and irrevocably (no preemption) right at the release timestamp of the sub-interval. We comprehensively study different settings of this problem regarding both the length of a released sub-interval and the total nu…
Maximum coverage and minimum set cover problems --collectively called coverage problems-- have been studied extensively in streaming models. However, previous research not only achieve sub-optimal approximation factors and space complexities, but also study a restricted set arrival model which makes an explicit or implicit assumption on oracle access to the sets, ignoring the complexity of reading and storing the whole set at once. In this paper, we address the above shortcomings, and present algorithms with improved approximation factor and improved space complexity, and prove that our result…
An important application of wireless sensor networks is the deployment of mobile sensors to periodically monitor (cover) a set of points of interest (PoIs). The problem of Point Sweep Coverage is to deploy fewest sensors to periodically cover the set of PoIs. For PoIs in a Eulerian graph, this problem is known NP-Hard even if all sensors are with uniform velocity. In this paper, we study the problem when PoIs are on a line and prove that the decision version of the problem is NP-Complete if the sensors are with different velocities. We first formulate the problem of Max-PoI sweep coverage on p…
In the maximum coverage problem, we are given subsets $T_1, \ldots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := \Big|\bigcup_{i \in S} T_i\Big|$. It is a classic result that the greedy algorithm for this problem achieves an optimal approximation ratio of $1-e^{-1}$. In this work we consider a generalization of this problem wherein an element $a$ can contribute by an amount that depends on the number of times it is covered. Given a concave, nondecreasing function $φ$, we define $C^φ(S) := \sum_{a …
We show, assuming the (randomized) Gap Exponential Time Hypothesis (Gap-ETH), that the following tasks cannot be done in $T(k) \cdot N^{o(k)}$-time for any function $T$ where $N$ denote the input size: - $\left(1 - \frac{1}{e} + ε\right)$-approximation for Max $k$-Coverage for any $ε> 0$, - $\left(1 + \frac{2}{e} - ε\right)$-approximation for $k$-Median (in general metrics) for any constant $ε> 0$. - $\left(1 + \frac{8}{e} - ε\right)$-approximation for $k$-Mean (in general metrics) for any constant $ε> 0$. - Any constant factor approximation for $k$-Unique Set Cover, $k$-Nearest Codeword Probl…
Maximum Coverage $k$-Antichains and Chains: A Greedy Approach
Department of Computer Science, University of Helsinki（赫尔辛基大学计算机科学系）；Aalto University（阿尔托大学）；Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机科学系）；LAMSADE, CNRS UMR7243, Université Paris Dauphine-PSL（巴黎达芬普SLM实验室，法国国家科学研究中心联合研究单位7243，巴黎达芬普-PSL大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
Given a graph $G$, a non-negative integer $k$, and a weight function that maps each vertex in $G$ to a positive real number, the \emph{Maximum Weighted Budgeted Independent Set (MWBIS) problem} is about finding a maximum weighted independent set in $G$ of cardinality at most $k$. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in $G$, is called the \emph{Maximum Independent Vertex Coverage (MIVC)} problem. In other words, the MIVC problem is about finding an independent set of cardinality at most $k$ with maximum coverage. Since it is a generalization of…
