Set Covering Research Articles

Set Cover Model with Path-Segment Concept to Determine the Optimum Rest Areas on Toll Road as the Location of Electric Vehicle Battery Charging Stations

The rapid adoption of electric vehicles (EVs), especially battery-based electric vehicles (BEVs), requires battery recharging facilities. Battery charging station (BCS) is an important component in EVs ecosystem. EVs requires BCS infrastructure for battery recharging. Limited battery range in one full charge is a concern for EVs users which is often called range anxiety so the number and placement of BCS must be adequate to prevent EVs from running out of battery in the middle of the trip. This study uses the real data of Trans Java toll road network, specifically the Semarang-Ngawi section for running numerical experiment to test the performance of the proposed model. Rest areas on toll roads are considered as candidate points for BCS locations. The proposed model is a set cover model formulated into binary integer programming. The Simplex algorithm in Microsoft Excel Solver is used to find the optimum solution on numerical experiment. Four locations are found as the optimum locations of BCS, i.e rest area r3, r6, r9, and rest area r12. The novelty of this study is combining the path-based approach with node-based approach to get a more compact set cover model.

Statistical-machine-learning-based intelligent relaxation for set-covering location models to identify locations of charging stations for electric vehicles

Europe strengthens its policies on climate change, green transition, and sustainable energy by addressing the high greenhouse-gas emissions in the transportation sector. Europe aims to reduce such emissions and reach a state of carbon neutrality by 2030 and 2050, respectively. This is feasible only if electric vehicles dominate the transportation sector. Paving the way for electric vehicle deployment on roads is subject to the provision of electric-vehicle-charging stations on the roads such that sufficiently good driving experience without any obstacles can be achieved. To address this timely societal challenge, we proposed a novel methodology by using the well-known facility-location-allocation methodology named set-covering location models with statistical machine learning and developed it for the problem settings of identifying electric-vehicle-charging station locations. Statistical machine learning was employed in the proposed model to more precisely identify and determine feasible coverage sets. We demonstrated the efficiency of the proposed model for the Capital Region of Denmark, where the green transition is part of the political agenda and is of severe societal concern, by using the newly collected main road transportation dataset.

Energy Proficient Reduced Coverage Set with Particle Swarm Optimization for Distributed Sensor Network

Retransmission avoidance is an essential need for any type of wireless communication. As retransmissions induce the unnecessary presence of redundant data in every accessible node. As storage capacity is symmetrical to the size of the memory, less storage capacity is experienced due to the restricted size of the respective node. In this proposed work, we have discussed the integration of the Energy Proficient Reduced Coverage Set with Particle Swarm Optimization (PSO). PSO is a metaheuristic global search enhancement technique that promotes the searching of the best nodes in the search space. PSO is integrated with a Reduced Coverage Set, to obtain an optimal path with only high-power transmitting nodes. Energy Proficient Reduced Coverage Set with PSO constructs a set of only best nodes based on the fitness solution, to cover the whole network. The proposed algorithm has experimented with a different number of nodes. Comparison has been made between original and improved algorithm shows that improved algorithm performs better than the existing by reducing the redundant packet transmissions by 18% ~ 40%, thereby increasing the network lifetime.

Pre-Election Political Campaigns Covered by Pakistani TV Channels: A Content Analysis

This research analyzes the political coverage by Pakistani News Channel before the general election 2013. The prime time of the channels for one month was selected for supervising, examining and organizing determinations of coverage of general elections under the code of conduct and procedure of coverage set by regulators. The researcher conducted content analysis of prime time which includes headlines of news bulletin, talk shows and issued addressed in program content broadcast on TV channels. Results of the study showed that all news channels failed to fulfill the fundamental obsessions of impartial balance and reasonable reporting. Because of the indistinct and insufficient necessities of ECP regarding rules of ethics for media, the portion of airtime on electronic media was exceptionally discriminating. In this study as the analysis showed the conduct of the elections, though having eminent place in the state news plan, it devastatingly explained the major political parties.

Set cover model-based optimum location of electric vehicle charging stations

Set cover model-based optimum location of electric vehicle charging stations

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

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.

Proposed Optimization Algorithm for Solving CCTV Camera Placement

The installation of CCTV cameras monitoring the street sections of one of the most visited areas of Manila may serve as deterrence against theft, crime, abduction, and even act of lasciviousness. Furthermore, the redundant orientations of some of the units in the system were recognized as possible inhibitors of the efficiency of the local surveillance system. In line with this, the study proposed a model for CCTV camera placement in Intramuros by representing the community as a graph in a 2-dimensional space. The paper presents a two-phase approach in determining the best placements of CCTV cameras. Phase I took care of the ideal installation spots as a set-covering problem while Phase II identified the optimal CCTV orientation using the Proposed algorithm. In Phase I, a binary integer programming model was formulated and solved using the data solver function of Microsoft Excel. The designed algorithm in Phase II was based on greedy heuristics utilizing the results in Phase I to identify the optimal orientation of the CCTV units. Findings suggest that out of the seventeen candidate locations, nine of them are optimal for CCTV installation. A total of twenty-three CCTV units are required to cover all the entry and exit points of the streets in district 5 of Intramuros. The proposed algorithm produced two optimal solutions A and B. Comparison with the existing CCTV system in the district and discussions on each optimal installation suggested that result B is better than A. Recommendations on the results of the study were addressed to the authorities of district 5 for immediate implementation.

Algorithm for Solving the Generalized Set Cover Problem for a Special Class of Problems

Algorithm for Solving the Generalized Set Cover Problem for a Special Class of Problems

Rechargeable Battery Cabinet Deployment for Public Bike System

Public Bike Systems (PBSs) offer the popular service for the short distance in daily life. The battery powered bike is an interesting and feasible method to extend the bike trip length, which can promote the PBS service but faces the challenges caused by the limited budget for the battery cabinet deployment and user demand. Thus, the realistic problem is how to deploy the cabinets near a part of public bike stations by considering the challenges. This paper is the first to study the novel problem, Cabinet Deployment Problem (CDP) in PBS, based on the features extracted from the real dataset of PBS in Hangzhou China, and proposes our strategies in the case of the Euclidean space and Manhattan model. In the Euclidean space, CDP can be specified as the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${e}$ </tex-math></inline-formula> lectric-bike Set Cover problem (e-SC), and this paper proposes a Greedy Station Coverage algorithm (GSC). Its distributed version, called the Localized Greedy Selection algorithm (LGS), is also presented because of the large amount of bike stations. In many cities, the roads have Manhattan-type directions, i.e., either east-west or south-north. In order to close to the realistic scenario, this paper develops a Genetic Algorithm based Cabinet Search algorithm (GAS) to determine the locations for the cabinet deployment in the Manhattan model. The extensive numerical experiment is conducted for our strategies, which are compared to a straightforward method, the Random Placement Strategy (RPS) under the diverse parameter settings.

Fix-and-optimize metaheuristics for minmax regret binary integer programming problems under interval uncertainty

The Binary Integer Programming problem (BIP) is a mathematical optimization problem, with linear objective function and constraints, on which the domain of all variables is {0, 1}. In BIP, every variable is associated with a determined cost coefficient. The Minmax regret Binary Integer Programming problem under interval uncertainty (M-BIP) is a generalization of BIP in which the cost coefficient associated to the variables is not known in advance, but are assumed to be bounded by an interval. The objective of M-BIP is to find a solution that possesses the minimum maximum regret among all possible solutions for the problem. In this paper, we show that the decision version of M-BIP is Σp2-complete. Furthermore, we tackle M-BIP by both exact and heuristic algorithms. We extend three exact algorithms from the literature to M-BIP and propose two fix-and-optimize heuristic algorithms. Computational experiments, performed on the Minmax regret Weighted Set Covering problem under Interval Uncertainties (M-WSCP) as a test case, indicates that one of the exact algorithms outperforms the others. Furthermore, it shows that the proposed fix-and-optimize heuristics, that can be easily employed to solve any minmax regret optimization problem under interval uncertainty, are competitive with ad-hoc algorithms for the M-WSCP.

Location Optimization of The Fighter Squadron in The National Air Defense System

This study uses the set covering problem (SCP) method to optimize the placement of the F-16 Fighting Falcon fighter in the national air defense system (Sishanudnas). Then the results of the SCP maximize the covering ability by minimizing the average distance between air bases with the P-Median Problem (PMP). The results of this study indicate that the Indonesian Air Force needs nine squadrons of F-16s, to be able to cover all of Indonesia's airspace. However, if the requirements are to include a base that currently operates fighter aircraft as a location for squadron placement, the Indonesian Air Force needs 11 squadrons of F-16 aircraft.

Interval scheduling with economies of scale

Motivated by applications in cloud computing, we study interval scheduling problems exhibiting economies of scale. An instance is given by a set of jobs, each with start time, end time, and a function representing the cost of scheduling a subset of jobs on the same machine. Specifically, we focus on the max-weight function and non-negative, non-decreasing concave functions of total schedule weight. The goal is a partition of the jobs that minimizes the total schedule cost, where overlapping jobs cannot be processed on the same machine. We propose a set cover formulation and a column generation algorithm to solve its linear relaxation. For the max-weight function, which is already NP-hard, we give a polynomial-time pricing algorithm; for the more general case of a function of the total weight, we have a pseudo-polynomial algorithm. To obtain integer solutions, we extend the column generation approach using branch-and-price. We computationally evaluate our methods on two different functions, using both random instances and instances derived from cloud computing data; our algorithm significantly outperforms known integer programming formulations (when these are available) and is able to provably optimize instances with hundreds of jobs.

A fault diagnosis method to defend scapegoating attack in network tomography

The scapegoating attack can cause persistent and inconspicuous performance degradation in network tomography. Defense of scapegoating attack is therefore a critical problem. Theoretically, the ideal defending scheme is to add monitoring paths to make all the links in the network be identifiable. This requires very high monitoring cost, which is unaffordable. To overcome this problem, this paper proposes a diagnosis-based defending scheme for scapegoating attack, which diagnoses scapegoating attack when problematic links are detected by network tomography. The latent fact is that a scapegoating attack can be launched only when the link set manipulated by the attacker cuts the probing paths going through the scapegoat links and is not traversed by any monitoring path. This cut set is called unobserved cut set (UCS). To defense, we propose to find the UCS and add the minimum number of probing paths to traverse the UCS, so that the condition of scapegoating attack is broken and the attacking links can be detected if any scapegoating attack exists. A minimum set cover model is proposed to select the least number of defense links to cover the UCS, and a polynomial time algorithm is proposed to generate the least number of probing paths to go through the selected defense links. Evaluations on various network dataset show the effectiveness of the proposed attack and defense strategies.

A comprehensive multi-topology minimum set cover link-state routing approach for emerging random all-IP access network topologies

Wireless access networks are ever-changing their setup, topologies and provisions of coverage. Forthcoming deployment trends include densification of cells and convergence of the ‘native IP’ with cellular access networks with more than best-effort expectations. Greater meshing in networks due to practical deployment requirements is inevitable and needed. This paper proposes a routing protocol to address the “randomness” of topology interconnections, routing paths and sizes in emerging IP access networks. Termed as Minimum Set Cover (MSC) approach, it is a generalization of the NP and NP complete mathematical problem. The offline component of the MSC multiplies intra-domain routing installations, called Routing Planes (RPs) modelled as graphs, to suitably cover the whole routing topology prior to the traffic injections in the online component. We introduce novel offline optimization features using a dynamic cost function that plans availability of capacities and correlation of routing paths via the chosen set of RPs. Our simulations verify effective path diversity using MSC with its modest protocol overhead and a heuristic used for RPs selection. For randomly constructed access network topologies with 7, 18 and 33 routers and various meshing levels, our results show convincing suitability and key performances gains compared with rival routing solutions.

Dynamic Geometric Set Cover and Hitting Set

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our article are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ 1 and ℝ 2 . The first framework uses bootstrapping and gives a (1 + ε)-approximate data structure for dynamic interval set cover in ℝ 1 with O ( n α / ε) amortized update time for any constant α &gt; 0; in ℝ 2 , this method gives O (1)-approximate data structures for unit-square set cover and hitting set with O ( n 1/2+α ) amortized update time. The second framework uses local modification and leads to a (1 + ε)-approximate data structure for dynamic interval hitting set in ℝ 1 with Õ(1/ε) amortized update time; in ℝ 2 , it gives O (1)-approximate data structures for unit-square set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.

Covering and packing of triangles intersecting a straight line

We study four geometric optimization problems: set cover , hitting set , piercing set , and independent set with right-triangles (a triangle is a right-triangle whose base is parallel to the x -axis, perpendicular is parallel to the y -axis, and the slope of the hypotenuse is − 1 ). The input triangles are constrained to be intersecting a straight line . The straight line can either be a horizontal or an inclined line (a line whose slope is − 1 ). A right-triangle is said to be a λ -right-triangle , if the length of both its base and perpendicular is λ . For 1-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are NP -hard, whereas the piercing set and independent set problems are in P . The same results hold for 1-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are NP -hard. Finally, we give an n O ( ⌈ log c ⌉ + 1 ) time exact algorithm for the independent set problem with λ -right-triangles intersecting a straight line such that λ takes more than one value from [ 1 , c ] , for some integer c . We also present O ( n 2 ) -time dynamic programming algorithms for the independent set problem with 1-right-triangles where the triangles intersect a horizontal line and an inclined line.

Coverings of planar and three-dimensional sets with subsets of smaller diameter

Quantitative estimates related to the classical Borsuk problem of splitting set in Euclidean space into subsets of smaller diameter are considered. For a given $k$ there is a minimal diameter of subsets at which there exists a covering with $k$ subsets of any planar set of unit diameter. In order to find an upper estimate of the minimal diameter we propose an algorithm for finding sub-optimal partitions. In the cases $10 \leqslant k \leqslant 17$ some upper and lower estimates of the minimal diameter are improved. Another result is that any set $M \subset \mathbb{R}^3$ of a unit diameter can be partitioned into four subsets of a diameter not greater than $0.966$.

Rapid quantitative measure of speech intelligibility through personal protective equipment facemasks using two head &amp; torso simulators

A test method has been developed that provides a rapid and quantitative measure of speech intelligibility via speech transmission index (STI) through personal protective facemasks using two Brüel &amp; Kjær Head and Torso Simulators (HATS) under a variety of acoustical environmental conditions. STI measurements were made using this new Test Method (TM) on a collection of 3M prototype and production facemasks and respirators, a competitor facemask, and a “no mask” baseline. This new TM enables rapid differentiation among facemask designs. Comparisons among the set of face coverings shows a clear tradeoff of filtration level for STI. New 3M prototypes with 90% filtration achieved an Excellent STI of 0.85, only 0.03 below a perfect no mask baseline. There is a dramatic step from maximum STI with no filtration at all to Excellent STI with 90% filtration. Most of the 95% filtration masks had STIs clustered near the lower end of the Excellent band at 0.78. P100 filtration significantly compromised speech intelligibility in the 3M 6300 half facepiece reusable respirator with a Fair quality rating on the STI scale at 0.60.

Robust Adaptive Submodular Maximization

The goal of a sequential decision-making problem is to design an interactive policy that adaptively selects a group of items, each selection is based on the feedback from the past, to maximize the expected utility of selected items. It has been shown that the utility functions of many real-world applications are adaptive submodular. However, most of existing studies on adaptive submodular optimization focus on the average-case, that is, their objective is to find a policy that maximizes the expected utility over a known distribution of realizations. Unfortunately, a policy that has a good average-case performance may have very poor performance under the worst-case realization. In this study, we propose to study two variants of adaptive submodular optimization problems, namely, worst-case adaptive submodular maximization and robust submodular maximization. The first problem aims to find a policy that maximizes the worst-case utility and the latter one aims to find a policy, if any, that achieves both near optimal average-case utility and worst-case utility simultaneously. We introduce a new class of stochastic functions, called worst-case submodular function. For the worst-case adaptive submodular maximization problem subject to a p-system constraint, we develop an adaptive worst-case greedy policy that achieves a [Formula: see text] approximation ratio against the optimal worst-case utility if the utility function is worst-case submodular. For the robust adaptive submodular maximization problem subject to cardinality constraints (respectively, partition matroid constraints), if the utility function is both worst-case submodular and adaptive submodular, we develop a hybrid adaptive policy that achieves an approximation close to [Formula: see text] (respectively, 1/3) under both worst- and average-case settings simultaneously. We also describe several applications of our theoretical results, including pool-base active learning, stochastic submodular set cover, and adaptive viral marketing. History: Accepted by Andrea Lodi, Area Editor for Design &amp; Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1239 .

Probability maximization via Minkowski functionals: convex representations and tractable resolution

In this paper, we consider the maximizing of the probability $${\mathbb {P}}\left\{ \, \zeta \, \mid \, \zeta \, \in \, {\mathbf {K}}({\mathbf {x}}) \, \right\} $$ over a closed and convex set $${\mathcal {X}}$$ , a special case of the chance-constrained optimization problem. Suppose $${\mathbf {K}}({\mathbf {x}}) \, \triangleq \, \left\{ \, \zeta \, \in \, {\mathcal {K}}\, \mid \, c({\mathbf {x}},\zeta ) \, \ge \, 0 \right\} $$ , and $$\zeta $$ is uniformly distributed on a convex and compact set $${\mathcal {K}}$$ and $$c({\mathbf {x}},\zeta )$$ is defined as either $$c({\mathbf {x}},\zeta )\, \triangleq \, 1-\left| \zeta ^T{\mathbf {x}}\right| ^m$$ where $$m\ge 0$$ (Setting A) or $$c({\mathbf {x}},\zeta ) \, \triangleq \, T{\mathbf {x}}\, - \, \zeta $$ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions, $${\mathbb {P}}\left\{ \,\zeta \, \mid \, \zeta \, \in \, {\mathbf {K}}({\mathbf {x}}) \, \right\} $$ can be expressed as the expectation of a suitably defined continuous function $$F(\bullet ,\xi )$$ with respect to an appropriately defined Gaussian density (or its variant), i.e. $${\mathbb {E}}_{{{\tilde{p}}}} \left[ \, F({\mathbf {x}},\xi )\, \right] $$ . Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of $$g\left( {\mathbb {E}}\left[ \, F(\bullet ,\xi )\, \right] \right) $$ over $${\mathcal {X}}$$ , where g is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of $$g\left( {\mathbb {E}}\left[ F(\bullet ,\xi )\right] \right) $$ over $$\mathcal X$$ , since conditionally unbiased sampled gradients are unavailable. We then develop a regularized variance-reduced stochastic approximation (r-VRSA) scheme that obviates the need for such unbiasedness by combining iterative regularization with variance-reduction. Notably, (r-VRSA) is characterized by almost-sure convergence guarantees, a convergence rate of $$\mathcal {O}(1/k^{1/2-a})$$ in expected sub-optimality where $$a > 0$$ , and a sample complexity of $$\mathcal {O}(1/\epsilon ^{6+\delta })$$ where $$\delta > 0$$ . To the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. Preliminary numerics on a portfolio selection problem (Setting A) and a set-covering problem (Setting B) suggest that the scheme competes well with naive mini-batch SA schemes as well as integer programming approximation methods.