Showing papers on "Set cover problem published in 2022"

Approximation algorithms for the minimum power cover problem with submodular/linear penalties

Abstract: In this paper, we introduce the minimum power cover problem with submodular and linear penalties. Suppose U is a set of users and S is a set of sensors in a d-dimensional space Rd with d≥2. Each sensor can adjust its power and the relationship between the power p(s) and the radius r(s) of the service area of sensor s satisfies p(s)=c⋅r(s)α, where c>0 and α≥1. Let p be the power assignment for each sensor and R be the set of users who are not covered by any sensor supported by p. The objective is to minimize the total power of p plus the rejected penalty of R. For a submodular penalty function that is normalized and nondecreasing, we present a combinatorial primal-dual (3α+1)-approximation algorithm. For the case in which the submodular penalty function is linear, we present a polynomial time approximation scheme based on a plane subdivision technique.

The weighted set covering problem combined to computational fluid dynamics for optimisation of gas detectors

Abstract: We propose a new approach to optimise the number and location of gas detectors employing the weighted set covering problem combined with computational fluid dynamics (CFD). The optimisation mesh that represents the nodes and the links of the Graph V { N , L } ( N is the set of nodes and L is the set of links) of the set covering problem was calculated using computational fluid dynamics. The 0-1 integer programming model is solved using Balas algorithm. The developed model covered 100% of the area. We also tested the model in a typical chemical process module and the same coverage was attained. The optimal distribution of gas detectors was able to detect the ammonia release within 5 s. We also show that the procedure can be combined with the leak frequency for the determination of the weights of the subareas (nodes of the Graph V { N , L } ). Numerical findings show that the weighted approach led to a shorter detection time when compared with the case where the subareas were not assigned a weight.

Clustering Geometrically-Modeled Points in the Aggregated Uncertainty Model

Abstract: The k-center problem is to choose a subset of size k from a set of n points such that the maximum distance from each point to its nearest center is minimized. Let Q = {Q1, . . . , Qn} be a set of polygons or segments in the region-based uncertainty model, in which each Qi is an uncertain point, where the exact locations of the points in Qi are unknown. The geometric objects such as segments and polygons can be models of a point set. We define the uncertain version of the k-center problem as a generalization in which the objective is to find k points from Q to cover the remaining regions of Q with minimum or maximum radius of the cluster to cover at least one or all exact instances of each Qi, respectively. We modify the region-based model to allow multiple points to be chosen from a region, and call the resulting model the aggregated uncertainty model. All these problems contain the point version as a special case, so they are all NP-hard with a lower bound 1.822 for the approximation factor. We give approximation algorithms for uncertain k-center of a set of segments and polygons. We also have implemented some of our algorithms on a data-set to show our theoretical performance guarantees can be achieved in practice.

Network Inspection for Detecting Strategic Attacks

Abstract: This article studies a problem of strategic network inspection, in which a defender (agency) is tasked with detecting the presence of multiple attacks in the network. An inspection strategy entails monitoring the network components, possibly in a randomized manner, using a given number of detectors. We formulate the network inspection problem $(\mathcal{P})$ as a large-scale bilevel optimization problem, in which the defender seeks to determine an inspection strategy with minimum number of detectors that ensures a target expected detection rate under worst-case attacks. We show that optimal solutions of $(\mathcal{P})$ can be obtained from the equilibria of a large-scale zero-sum game. Our equilibrium analysis involves both game-theoretic and combinatorial arguments, and leads to a computationally tractable approach to solve $(\mathcal{P})$. Firstly, we construct an approximate solution by utilizing solutions of minimum set cover (MSC) and maximum set packing (MSP) problems, and evaluate its detection performance. In fact, this construction generalizes some of the known results in network security games. Secondly, we leverage properties of the optimal detection rate to iteratively refine our MSC/MSP-based solution through a column generation procedure. Computational results on benchmark water networks demonstrate the scalability, performance, and operational feasibility of our approach. The results indicate that utilities can achieve a high level of protection in large-scale networks by strategically positioning a small number of detectors.

A Divide and Conquer Strategy for Sweeping Coverage Path Planning

Abstract: One of the main challenges faced by floor treatment service robots is to compute optimal paths that completely cover a set of target areas. Short paths are of noticeable importance because their length is directly proportional to the energy used by the robot. Such a problem is known to be NP-hard; therefore, efficient algorithms are needed. In particular, computation efficiency is important for mobile robots with limited onboard computation capability. The general problem is known as coverage path planning (CPP). The CPP has several variants for single regions and for disjoint regions. In this research, we are investigating the solutions for disjoint target regions (rooms) that have fixed connection points (doors). In particular, we have found effective simplifications for the cases of rooms with one and two doors, while the challenging case of an unbounded number of rooms can be solved by approximation. As a result, this work presents a divide and conquer strategy (DnCS) to address the problem of finding a path for a sweeping robot that needs to sweep a set of disjoint rooms that are connected by fixed doors and corridors. The strategy divides the problem into computing the sweeping paths for the target rooms and then merges those paths into one solution by optimising the room visiting order. In this strategy, a geometrical approach for room coverage and an undirected rural postman problem optimisation are strategically combined to solve the coverage of the entire area of interest. The strategy has been tested in several synthetic maps and a real scenario showing short computation times and complete coverage.

Dynamic Geometric Set Cover, Revisited

Abstract: Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Dynamic Geometric Set Cover, RevisitedTimothy M. Chan, Qizheng He, Subhash Suri, and Jie XueTimothy M. Chan, Qizheng He, Subhash Suri, and Jie Xuepp.3496 - 3528Chapter DOI:https://doi.org/10.1137/1.9781611977073.139PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Geometric set cover is a classical problem in computational geometry, which has been extensively studied in the past. In the dynamic version of the problem, points and ranges may be inserted and deleted, and our goal is to efficiently maintain a set cover solution (satisfying certain quality requirement) for the dynamic problem instance. In this paper, we give a plethora of new dynamic geometric set cover data structures in 1D and 2D, which significantly improve and extend the previous results. Our results include the following: The first data structure for (1 + ∊)-approximate dynamic interval set cover with polylogarithmic amortized update time. Specifically, we achieve an update time of O(log3 n/∊), improving the O(nδ/∊) bound of Agarwal et al. [SoCG'20], where δ > 0 denotes an arbitrarily small constant. A data structure for O(1)-approximate dynamic unit-square set cover with amortized update time, substantially improving the O(n1/2+δ) update time of Agarwal et al. [SoCG'20]. A data structure for O(1)-approximate dynamic square set cover with O(n1/2+δ) randomized amortized update time, improving the O(n2/3+δ) update time of Chan and He [SoCG'21]. A data structure for O(1)-approximate dynamic 2D halfplane set cover with O(n17/23+δ) randomized amortized update time. The previous solution for halfplane set cover by Chan and He [SoCG'21] is slower and can only report the size of the approximate solution. The first sublinear results for the weighted version of dynamic geometric set cover. Specifically, we give a data structure for (3 + o(1))-approximate dynamic weighted interval set cover with amortized update time and a data structure for O(1)-approximate dynamic weighted unit-square set cover with O(nδ) amortized update time. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

An Efficient Algorithm to Enhance Nonoverlapping Coverage Area with Less Energy Consumption in WSN

Abstract: Several chargeable sensor nodes are deployed randomly to cover the target points with an efficient heuristic approach for the mobility of sensor nodes in an area of interest (AoI). The heuristic approach generates the cover set that includes the targets for a prolonged time. The cover sets are the subset of the total sensor node area where each set is capable of representing all the targets. The functionality of the sensor nodes depends upon the network lifetime of the target points covering an AoI. The network lifetime would improve by reducing the consumption of battery power through heuristic process. The proposed heuristic process can do this by generating cover sets and selecting sensor nodes with the highest remaining battery power. These cover sets remove the redundant sensor node in an AoI that causes the overlapping issue and assign the maximum lifetime which is the minimum amount of battery power of the sensor node, participating in the cover set. The results show the improvement in the mobility of sensor nodes by coverage and attain maximum network lifetime as compared to the existing algorithms.

Multi-Agent Coverage Path Planning using a Swarm of Unmanned Aerial Vehicles

Abstract: In recent years, rapid technological advancements in unmanned aerial vehicles (UAVs) have propelled their applications to a wide range of areas such as agriculture, mapping & surveying, surveillance, and many more. A swarm of UAVs can efficiently accomplish the goals rather than a single UAV due to its ability to cover a larger area. Multi-agent coverage path planning (CPP) is the process of determining efficient coverage paths for the swarm of UAVs to completely cover an area of interest. A cellular grid is formed using hexagonal decomposition to ensure complete coverage, with the centroid of each cell acting as a waypoint. The k-means clustering algorithm is implemented to divide the set of waypoints such that the workload is uniformly distributed among the UAVs. Multi-agent CPP is transformed into multiple single-agent CPP, thereby alleviating the exploratory complexity of multi-agent CPP. A mixed-integer linear programming based vehicle routing problem is formed to optimize the path of each UAV considering various constraints.

A General Framework for Approximating Min Sum Ordering Problems

Abstract: We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of min sum set cover, several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the min sum ordering problem (MSOP), which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time α-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time [Formula: see text]-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results. Summary of Contribution: This paper provides a general framework for min sum ordering problems. Within the realm of theoretical computer science, these problems include min sum set cover and its generalizations, as well as problems in Boolean function evaluation. On the operations research side, they include problems in search theory and scheduling. We present and analyze a very general algorithm for these problems, unifying several previous results on various min sum ordering problems and resulting in new constant factor guarantees for others.

Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems

Abstract: In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.

Reducing the Complexity of the Sensor-Target Coverage Problem Through Point and Set Classification

Abstract: The problem of covering random points in a plane with sets of a given shape has several practical applications in communications and operations research. One especially prominent application is the coverage of randomly-located points of interest by randomly-located sensors in a wireless sensor network. In this article we consider the situation of a large area containing randomly placed points (representing points of interest), as well a number of randomly-placed disks of equal radius in the same region (representing individual sensors' coverage areas). The problem of finding the smallest possible set of disks that cover the given points is known to be NP-complete. We show that the computational complexity may be reduced by classifying the disks into several definite classes that can be characterized as necessary, excludable, or indeterminate. The problem may then be reduced to considering only the indeterminate sets and the points that they cover. In addition, indeterminate sets and the points that they cover may be divided into disjoint ``islands'' that can be solved separately. Hence the actual complexity is determined by the number of points and sets in the largest island. We run a number of simulations to show how the proportion of sets and points of various types depend on two basic scale-invariant parameters related to point and set density. We show that enormous reductions in complexity can be achieved even in situations where point and set density is relatively high.

Online Learning for Min Sum Set Cover and Pandora's Box

Abstract: Two central problems in Stochastic Optimization are Min Sum Set Cover and Pandora's Box. In Pandora's Box, we are presented with $n$ boxes, each containing an unknown value and the goal is to open the boxes in some order to minimize the sum of the search cost and the smallest value found. Given a distribution of value vectors, we are asked to identify a near-optimal search order. Min Sum Set Cover corresponds to the case where values are either 0 or infinity. In this work, we study the case where the value vectors are not drawn from a distribution but are presented to a learner in an online fashion. We present a computationally efficient algorithm that is constant-competitive against the cost of the optimal search order. We extend our results to a bandit setting where only the values of the boxes opened are revealed to the learner after every round. We also generalize our results to other commonly studied variants of Pandora's Box and Min Sum Set Cover that involve selecting more than a single value subject to a matroid constraint.

Differentially Private Partial Set Cover with Applications to Facility Location

Abstract: It was observed in \citet{gupta2009differentially} that the Set Cover problem has strong impossibility results under differential privacy. In our work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a $\rho$-fraction of the elements in the universe, for some $\rho\in(0,1)$. We show that this relaxation enables us to avoid the impossibility results: under loose conditions on the input set system, we give differentially private algorithms which output an explicit set cover with non-trivial approximation guarantees. In particular, this is the first differentially private algorithm which outputs an explicit set cover. Using our algorithm for Partial Set Cover as a subroutine, we give a differentially private (bicriteria) approximation algorithm for a facility location problem which generalizes $k$-center/$k$-supplier with outliers. Like with the Set Cover problem, no algorithm has been able to give non-trivial guarantees for $k$-center/$k$-supplier-type facility location problems due to the high sensitivity and impossibility results. Our algorithm shows that relaxing the covering requirement to serving only a $\rho$-fraction of the population, for $\rho\in(0,1)$, enables us to circumvent the inherent hardness. Overall, our work is an important step in tackling and understanding impossibility results in private combinatorial optimization.

Online Set Cover With Rated Subsets

Abstract: : In this paper, we introduce the Online Set Cover With Rated Subsets problem (OSC-RS), a generalization of the well-known Online Set Cover problem, in which we are given a universe of elements and a collection of subsets of the universe, each associated with a subset cost and a rating cost. In each step, the algorithm is given a request containing elements from the universe. The algorithm serves a request by assigning it to a number of purchased subsets that jointly cover the requested elements. The algorithm pays the subset costs associated with the subsets purchased and for each request, it pays the sum of the rating costs associated with the subsets assigned to the request. The aim is to serve all requests as soon as revealed, while minimizing the total subset and rating costs paid. OSC-RS is motivated by intrinsic client-service-providing scenarios in which service providers are rated and their ratings are included in the decision-making process, so as higher-rated service providers are associated with lower rating costs. That is, the decisions about serving clients take into account the quality of the services provided. We propose the first online algorithm for OSC-RS and evaluate it using the standard notion of competitive analysis . The latter compares the performance of the online algorithm to that of an optimal offline algorithm that is assumed to know all the input sequence in advance.

Online learning for min-max discrete problems

Abstract: We study various discrete nonlinear combinatorial optimization problems in an online learning framework. In the first part, we address the computational complexity of designing vanishing regret (and vanishing approximate regret) algorithms. We provide a general reduction showing that many (min-max) polynomial time solvable problems not only do not have a vanishing regret, but also no vanishing approximation α-regret, for some α, unless NP=RP. In particular, for the min-max version of the vertex cover problem, which is solvable in polynomial time in the offline case, our reduction implies that there is no (2−ϵ)-regret online randomized algorithm unless Unique Game is in RP. Besides, we prove that the bound is tight by providing an online efficient algorithm based on the online gradient descent method. In the second part, we turn our attention to online learning algorithms that are based on an offline optimization oracle that, given a set of multiple instances of the problem, is able to compute the optimum static solution that performs best on the set of instances overall. We show that for several min-max (nonlinear) discrete optimization problems, it is strongly NP-hard to solve the offline optimization oracle, even for problems that can be solved in polynomial time in the single-instance static case (e.g. min-max vertex cover, min-max perfect matching, etc.). This also provides a useful insight into the connection between the non-linear nature of some problems and the drastic change of their computational hardness when moved to an online learning setting.

New Classes of Unconstrained Permutation-Based Problems and Their Solutions

Abstract: Three permutation-based matrix combinatorial sets were introduced: a) a matrix set of multipermutations of row entries; b) a matrix set of permutations of rows; c) a matrix set of multipermutations of rows and row entries. A polynomial (on the dimension of matrices) solvability of linear combinatorial optimization problems on these sets was justified. The connection between these problems with Linear Assignment Problems is established. Thus a class of polynomially solvable optimization problems is expanded remarkably. These results can be applied in different areas of Operations Research, such as Geometric Design and optimal management.

A Bi-objective Optimal Scheme for 5G Base Station Deployment Based on BPSO

Abstract: The 5G mobile network is a kind of critical information infrastructure for future Internet of Things. Due to its rapid development, the planning and deployment of 5G network base stations is a more urgent and meaningful problem than ever before from the aspect of optimization. Because the purpose of establishing a 5G base station is to make its wireless signal cover as many areas as possible, a 0–1 programming model with two optimal objectives is proposed for 5G base stations based on the cover set. Then after analyzing the difficulties in solving the problem expressed by above model, employing the idea of divide and conquer and the strategies of parallelization, a scheme based on Binary Particle Swarm Optimization (BPSO) algorithm is proposed to solve this problem. In the first step of the scheme, the partition processing based on coordinate position is considered by the way of dividing the target area into many smaller sub-areas with the algorithm of Balanced Iterative Reducing and Clustering using Hierarchies (BIRCH). The next step is to use BPSO algorithm improved by sparse particles initialization and a suppressor factor to solve each small sub-area with the parallel computing and the memory mapping technology. The proposed approach has been tested in several experiments and the experimental results demonstrate its effectiveness.

Minimum Ply Covering of Points with Unit Squares

Abstract: Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection, called the minimum ply cover number of $P$ with $U$. Biedl et al. [Comput. Geom., 94:101712, 2020] showed that determining the minimum ply cover number for a set of points by a set of axis-parallel unit squares is NP-hard, and gave a polynomial-time 2-approximation algorithm for instances in which the minimum ply cover number is constant. The question of whether there exists a polynomial-time approximation algorithm remained open when the minimum ply cover number is $\omega(1)$. We settle this open question and present a polynomial-time $(8+\varepsilon)$-approximation algorithm for the general problem, for every fixed $\varepsilon>0$.

Local Search Methods for Min-Max Clearing Problem

Abstract: In the ongoing power system reform in Japan, the division of transmission and distribution is separated from the former general electric utility and became a general power transmission and distribution business operator. After the reform, the power transmission and distribution business operators will procure reserve power from the reserve market. We have formulated the clearing problem in the reserve market as a 0-1 integer programming problem. This problem can be viewed as a Min-Max set multi-cover problem. Since the set covering problem belongs NP-hard, it is not easy to solve large-scale problems in practical time. In this paper, we propose a greedy solution and local search solution to the Min-Max set multi-cover problem and evaluate its performance by numerical experiments.

Dynamic Algorithms for Packing-Covering LPs via Multiplicative Weight Updates

Abstract: In the dynamic linear program (LP) problem, we are given an LP undergoing updates and we need to maintain an approximately optimal solution. Recently, significant attention (e.g., [Gupta et al. STOC'17; Arar et al. ICALP'18, Wajc STOC'20]) has been devoted to the study of special cases of dynamic packing and covering LPs, such as the dynamic fractional matching and set cover problems. But until now, there is no non-trivial dynamic algorithm for general packing and covering LPs. In this paper, we settle the complexity of dynamic packing and covering LPs, up to a polylogarithmic factor in update time. More precisely, in the partially dynamic setting (where updates can either only relax or only restrict the feasible region), we give near-optimal deterministic $\epsilon$-approximation algorithms with polylogarithmic amortized update time. Then, we show that both partially dynamic updates and amortized update time are necessary; without any of these conditions, the trivial algorithm that recomputes the solution from scratch after every update is essentially the best possible, assuming SETH. To obtain our results, we initiate a systematic study of the multiplicative weights update (MWU) method in the dynamic setting. As by-products of our techniques, we also obtain the first online $(1+\epsilon)$-competitive algorithms for both covering and packing LPs with polylogarithmic recourse, and the first streaming algorithms for covering and packing LPs with linear space and polylogarithmic passes.

Approximating the Pareto frontier for a challenging real-world bi-objective covering problem

Abstract: Abstract We study a bi-objective covering problem stemming from a real-world application concerning the design of camera surveillance systems for large-scale outdoor areas. It is in this application prohibitively costly to surveil the entire area, and therefore necessary to be able to present a decision-maker with trade-offs between total cost and the portion of the area that is surveilled. The problem can be stated as a set covering problem with two objectives, describing cost and portion of covering constraints that are fulfilled. Finding the Pareto frontier for these objectives is very computationally demanding and we therefore derive a method for finding a good approximate frontier in a practically feasible computing time. The method is based on the ϵ-constraint reformulation, an established heuristic for set covering problems, and subgradient optimization.

Quantum Social Computing Approaches for Influence Maximization

Abstract: Influence Maximization (IM), which seeks a small set of important nodes that spread the influence widely into the network, is a fundamental problem in social networks. It finds applications in viral marketing, epidemic control, and assessing cascading failures within complex systems. Despite the huge amount of effort, finding near-optimal solutions for IM is difficult due to its NP-completeness. In this paper, we propose the first social quantum computing approaches for IM, aiming to retrieve near-optimal solutions. We propose a two-phase algorithm that 1) converts IM into a Max-Cover instance and 2) provides efficient quadratic unconstrained binary optimization formulations to solve the Max-Cover instance on quantum annealers. Our experiments on the state-of-the-art D-Wave annealer indicate better solution quality compared to classical simulated annealing, suggesting the potential of applying quantum annealing to find high-quality solutions for IM.