Set Covering Research Articles

Quantum Annealing with Inequality Constraints: The Set Cover Problem

AbstractQuantum annealing is a promising method for solving hard optimization problems by transforming them into quadratic unconstrained binary optimization (QUBO) problems. However, when constraints are involved, particularly multiple inequality constraints, incorporating them into the objective function poses challenges. In this paper, the authors present two novel approaches for solving problems with multiple inequality constraints on a quantum annealer and apply them to the set cover problem (SCP). The first approach uses the augmented Lagrangian method to represent the constraints, while the second approach employs a higher‐order binary optimization (HUBO) formulation. The experiments show that both approaches outperform the standard approach for solving the SCP on the D‐Wave Advantage quantum annealer. The HUBO formulation performs slightly better than the augmented Lagrangian method in solving the SCP, but its scalability in terms of embeddability in the quantum chip is worse. The results demonstrate that the proposed augmented Lagrangian and HUBO methods can successfully implement a large number of inequality constraints, making them applicable to a broad range of constrained problems beyond the SCP.

Binary Extended Formulations and Sequential Convexification

A binarization of a bounded variable x is obtained via a system of linear inequalities that involve x together with additional variables [Formula: see text] in [Formula: see text] so that the integrality of x is implied by the integrality of [Formula: see text]. A binary extended formulation of a mixed-integer linear set is obtained by adding to its original description binarizations of its integer variables. Binary extended formulations are useful in mixed-integer programming as imposing integrality on 0/1 variables rather than on general integer variables has interesting convergence properties and has been studied from both the theoretical and practical points of view. We study the behavior of binary extended formulations with respect to sequential convexification. In particular, given a binary extended formulation and one of its variables x, we study a parameter that measures the progress made toward enforcing the integrality of x via application of sequential convexification. We formulate this parameter, which we call rank, as the solution of a set-covering problem and express it exactly for the classic binarizations from the literature. Funding: M. Aprile and M. Di Summa are supported by the University of Padova [SID 2019 Grant C94I20000280005].

A Point-of-Interest Recommender System for Tourist Groups Based on Cooperative Location Set Cover Problem

Trip recommendation for groups of tourists (TRGT) is a challenging task in tourism since many tourists travel in groups, inducing social interaction and bringing various social benefits. However, TRGT must address various real-life constraints such as limited time for touring, cost, etc. TRGT aims to design personalized tours that meet the preferences of all group members by addressing a variety of tourists’ requirements that may sometimes result in conflicts and stress for the group members. TRGT should satisfy that both the preferences of group members need to be achieved as much as possible and the preferences of group members need to be achieved as evenly as possible. In this paper, we present a methodology for tackling the TRGT problem by reducing it to the Cooperative Location Set Cover Problem (CLSCP), formulated as an integer linear program. The CLSCP aims to select a group of facilities that can satisfy, in aggregate, all demand points. To tackle the CLSCP, we present a new method based on detecting frequent patterns. We also demonstrate the efficiency of the proposed methodology by presenting extensive experimental tests.

EASC: An exception-aware semantic compression framework for real-world knowledge graphs

Knowledge graphs (KGs) have achieved great success in many real applications, and great efforts have been dedicated to constructing larger knowledge graphs. An obvious trend in KG construction is that the KGs become ever-increasingly bigger. However, we argue that constructing a KG by directly inserting more triples may harm the performance of the KG, and one possible solution is KG compression. In this paper, we propose an exception-aware semantic lossless compression framework EASC to compress a KG. Since many triples can be inferred from other triples with semantic rules, we remove the triples that can be inferred and store the rules and exception cases. Specifically, we formalize the lossless compression problem as a weighted set cover problem, which is NP-hard, and propose a semantic lossless compression algorithm to get an approximation result. We conduct extensive experiments on seven real-world large-scale KGs. The results show that EASC achieves state-of-the-art performance in semantic compression methods. Furthermore, by combining EASC as an independent module with syntactic compression methods, we achieve state-of-the-art performance in lossless compression methods.

A site selection framework for urban power substation at micro‐scale using spatial optimization strategy and geospatial big data

AbstractThe world is facing more energy crises due to extreme weather and the rapidly growing demand for electricity. Siting new substations and optimizing the location of existing ones are necessary to address the energy crisis. The current site selection lacks consideration of spatial and temporal heterogeneity in urban power demand, which results in unreasonable energy transfer and waste, leading to power outages in some areas. Aiming to maximize the grid coverage and transformer utilization, we propose a multi‐scene micro‐scale urban substation siting framework (UrbanPS): (1) The framework uses multi‐source big data and the machine learning model to estimate fine‐scale power consumption for different scenarios; (2) the region growing algorithm is used to divide the power supply area of substations; and the (3) location set coverage problem and genetic algorithm are introduced to optimize the substation location. The UrbanPS was used to perform siting optimization of 110 kV terminal substations in Pingxiang City, Jiangxi Province. Results show that the coverage and utilization rate of the optimization results under different power consumption scenarios are close to 99%. We also found that the power can be saved by dynamic regulation of substation operation.

Cost Allocation and Strategyproof Mechanism for Cover-set Games

In this paper, we discuss several cost-sharing methods for cover-set games, for example, approximately budget-balanced, in the core, and/or strategyproof. Different from the traditional set cover game, in cover-set game, the elements stand for services, and each agent is both a service provider and receiver. In this study, for unselfish agents, we design a cost allocation method that can recover no less than [Formula: see text] fraction of the total cost, where [Formula: see text] and [Formula: see text] is the maximum number of receiving services among all agents. In addition, for agents who are selfish service providers with privately known valuations, we present a strategyproof charging mechanism, further, the total cost is no more than [Formula: see text] times that of an optimal solution.

The Computational Challenges of Means Selection Problems: Network Structure of Goal Systems Predicts Human Performance.

We study human performance in two classical NP-hard optimization problems: Set Cover and Maximum Coverage. We suggest that Set Cover and Max Coverage are related to means selection problems that arise in human problem-solving and in pursuing multiple goals: The relationship between goals and means is expressed as a bipartite graph where edges between means and goals indicate which means can be used to achieve which goals. While these problems are believed to be computationally intractable in general, they become more tractable when the structure of the network resembles a tree. Thus, our main prediction is that people should perform better with goal systems that are more tree-like. We report three behavioral experiments which confirm this prediction. Our results suggest that combinatorial parameters that are instrumental to algorithm design can also be useful for understanding when and why people struggle to choose between multiple means to achieve multiple goals.

Attacking the Core Structure of Complex Network

The concept of <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core in complex networks plays a key role in many applications, e.g., understanding the global structure or identifying central/critical nodes, of a network. A malicious attacker with a jamming ability can exploit the vulnerability of the <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core structure to attack the network and invalidate the network analysis methods, e.g., reducing the <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-shell values of nodes can deceive graph algorithms, leading to the wrong decisions. In this article, we investigate the robustness of the <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core structure under adversarial attacks by deleting edges, for the first time. First, we give the general definition of the targeted <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core attack, map it to the set cover problem, which is NP-hard, and further introduce a series of evaluation metrics to measure the performance of attack methods. Then, we propose the <inline-formula> <tex-math notation="LaTeX">$Q$</tex-math> </inline-formula> index theoretically as the probability that the terminal node of an edge does not belong to the innermost core, which is further used to guide the design of our heuristic attack methods, namely, COREATTACK and GreedyCOREATTACK. The experiments on a variety of real-world networks demonstrate that our methods behave much better than a series of baselines, in terms of much smaller edge change rate (ECR) and false attack rate (FAR), achieving state-of-the-art attack performance. More impressively, for certain real-world networks, only deleting one edge from the <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core may lead to the collapse of the innermost core, even if this core contains dozens of nodes. Such a phenomenon indicates that the <inline-formula> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core structure could be extremely vulnerable under adversarial attacks, and its robustness, thus, should be carefully addressed to ensure the security of many graph algorithms. An open-source implementation is available at https://github.com/Yocenly/COREATTACCK.

Parker: Data fusion through consistent repairs using edit rules under partial keys

Data integration is the problem of consolidating information provided by multiple sources. After schema mapping and duplicate detection have been dealt with, the problem consists in fusing disparate data that comes from the sources. In this work, we treat this problem as a data cleaning problem where consistency within and between sources is modelled by respectively edit rules and a partial key. The combination of these constraints leads to the definition of edit rules under a partial key (EPKs). We show that we can adapt the well-known set cover method for edit rules to the setting of EPKs and this yields an efficient algorithm to find minimal-cost repairs. These repairs then provide solutions for the initial data fusion problem. The methodology is implemented in a repair engine called Parker. Empirical results show that Parker is several orders of magnitude faster than state-of-the-art repair tools. At the same time, the quality of the repairs in terms of F1-score ranges from comparable to better compared to these tools.

Exceptional sets

In this paper, we study sequences of complex numbers of the first order. Multiple terms are allowed for such sequences. We also consider complex sequences with a finite maximum density. We construct special coverings of multiple sets {λk,nk} consisting of circles centered at points λk of special radii. In particular, we construct coverings are with connected components of a relatively small diameter, as well as coverings that are C0-sets. These coverings act as exceptional sets for entire functions of exponential type. Outside these sets, we obtain a representation of the logarithm of the modulus of an entire function. Previously, a similar representation was obtained by B. Ya. Levin outside the exceptional set, with respect to which only its existence is asserted. In contrast to this, in this paper we present a simple effective construction of an exceptional set. We construct bases of the invariant subspace of analytic functions in a convex domain. They consist of linear combinations of eigenfunctions and associated functions (exponential monomials) of the differentiation operator divided into relatively small groups.

The Effects of the Pandemic on the Selected Hotel Industry in Puerto Princesa City, Palawan: Basis for the Development of Recovery Strategies

Puerto Princesa City is one of the most popular tourist sites in the nation. The City encompasses the east coast, which faces the Sulu Sea, and the west coast, which faces the West Philippines Sea and is located in the province of Palawan's center. It occupies a vast tract of land characterized by a high forest cover set in a chain of mountain ranges home to diverse flora and fauna. Its imposing mountains and lush rainforests ensure adequate coordination from the Department of Trade in the industry in helping the hospitality industry watershed, where water overflows into life-giving rivers, waterfalls, and streams. The study enerally aimed to ascertain the Pandemic's effects on the chosen hotel sector in Puerto Princesa City, Palawan, and to develop a basis for developing recovery strategies. It was conducted in twenty hotels on the initiatives and interventions of the government during this covid-19 Pandemic. Henceforth, on the perceived recovery strategies by the hotel owners/managers, the hotel suggested adopting digital solutions, pricing tactics, and safety protocols that fit in this new normal.

An Improved Algorithm for Online Min-Sum Set Cover

We study a fundamental model of online preference aggregation, where an algorithm maintains an ordered list of n elements. An input is a stream of preferred sets R_1, R_2, ..., R_t, ... Upon seeing R_t and without knowledge of any future sets, an algorithm has to rerank elements (change the list ordering), so that at least one element of R_t is found near the list front. The incurred cost is a sum of the list update costs (the number of swaps of neighboring list elements) and access cost (the position of the first element of R_t on the list). This scenario occurs naturally in applications such as ordering items in an online shop using aggregated preferences of shop customers. The theoretical underpinning of this problem is known as Min-Sum Set Cover. Unlike previous work that mostly studied the performance of an online algorithm ALG in comparison to the static optimal solution (a single optimal list ordering), in this paper, we study an arguably harder variant where the benchmark is the provably stronger optimal dynamic solution OPT (that may also modify the list ordering). In terms of an online shop, this means that the aggregated preferences of its user base evolve with time. We construct a computationally efficient randomized algorithm whose competitive ratio (ALG-to-OPT cost ratio) is O(r^2) and prove the existence of a deterministic O(r^4)-competitive algorithm. Here, r is the maximum cardinality of sets R_t. This is the first algorithm whose ratio does not depend on n: the previously best algorithm for this problem was O(r^(3/2) * n^(1/2))-competitive and Ω(r) is a lower bound on the performance of any deterministic online algorithm.

Characterization of complex structural variation in the CYP2D6-CYP2D7-CYP2D8 gene loci using single-molecule long-read sequencing.

Complex regions in the human genome such as repeat motifs, pseudogenes and structural (SVs) and copy number variations (CNVs) present ongoing challenges to accurate genetic analysis, particularly for short-read Next-Generation-Sequencing (NGS) technologies. One such region is the highly polymorphic CYP2D loci, containing CYP2D6, a clinically relevant pharmacogene contributing to the metabolism of >20% of common drugs, and two highly similar pseudogenes, CYP2D7 and CYP2D8. Multiple complex SVs, including CYP2D6/CYP2D7-derived hybrid genes are known to occur in different configurations and frequencies across populations and are difficult to detect and characterize accurately. This can lead to incorrect enzyme activity assignment and impact drug dosing recommendations, often disproportionally affecting underrepresented populations. To improve CYP2D6 genotyping accuracy, we developed a PCR-free CRISPR-Cas9 based enrichment method for targeted long-read sequencing that fully characterizes the entire CYP2D6-CYP2D7-CYP2D8 loci. Clinically relevant sample types, including blood, saliva, and liver tissue were sequenced, generating high coverage sets of continuous single molecule reads spanning the entire targeted region of up to 52kb, regardless of SV present (n = 9). This allowed for fully phased dissection of the entire loci structure, including breakpoints, to accurately resolve complex CYP2D6 diplotypes with a single assay. Additionally, we identified three novel CYP2D6 suballeles, and fully characterized 17 CYP2D7 and 18 CYP2D8 unique haplotypes. This method for CYP2D6 genotyping has the potential to significantly improve accurate clinical phenotyping to inform drug therapy and can be adapted to overcome testing limitations of other clinically challenging genomic regions.

Determination of Distribution Center Location using Analysis of Time-Based Set Covering Model and Maximal Covering Model Analysis

The distribution or delivery process is one factor that affect customer satisfaction as the goal of supply chain. In order for supply chain to be competitive in competition, delivery time is an important factor to manage, so that it could provide high service level value. One factor that could affects distribution or delivery is distribution center location. A strategic distribution center location would facilitate and speed up distribution process. This research discusses model for determining location of distribution center so that delivery could be made at the right amount and in the right time by analyzing of time-based using set covering model and maximal covering model. The numerical example in this research is case study on paint supply chain in Bandung which has a number of delivering clusters. Each cluster consists of a number of building material stores which are located scattered. There is one cluster that has problems in fulfilling demands and delivery times. The analysis made could be used to determine the distribution center location so that it could serve all consumer demands. Based on the calculation results, it was found that there were two distribution center locations. Two distribution centers could fulfill the objective function of cost minimization and satisfy all demands. The results of the sensitivity analysis show that there are several factors that influence the determination of location number.

The Efficiency of SCS-CN, HEC-1, HEC-HMS, TR55, RATIONAL, and SNYDER UNIT Hydrograph Models for Determining Peak Flood Discharge in the Upper Part of Lesser Zab Basin, Kurdistan Region, Iraq

The Unit Hydrograph (UH) and Soil Conservation Service Curve Number (SCS-CN) are prevalent methods for estimating peak flow and peak time within the hydrological river basins. Different types of data, such as gauging data, morphometric analysis, and Land Use-Land Cover (LULC), are used to derive UH for the Kanarwe River Basin (KRB), which is an off-the-Lesser Zab River Basin (LZRB). Different hydrograph models, including HEC-1 (Hydrological Engineering Centre), TR55 (Technical release 55), HEC-HMS (Hydrological Engineering Centre-Hydrological Modeling System), Rational method, and Snyder unit hydrograph, have been applied and correlated with field data. Metrological data, geological setting, and land cover were integrated into the Geographic Information Systems (GIS) and Watershed Modelling System (WMS 11.1). The peak time (Tp) and peak flow (Qp) were estimated based on the five applied models. The results for models are (Qp = 739.93 m3/sec, Tp = 20 hr), (Qp 181.4 m3/sec, Tp = 14 hr), (Qp = 800 m3/sec, Tp = 12 hr, (Qp = 341.13 m3/sec, Tp = 11.65 hr), (Qp = 443 m3/sec, Tp = 19.9 hr), (Qp = 243 m3/sec) for HEC-1, TR55, HEC-HMS, Rational method, Synder unit hydrograph, and observed data respectively. The observed model (field data) peak time and peak flow value best agreed with the peak time and peak flow value of TR55, Snyder, and Rational models. Our finding confirmed that the geomorphoclimatic unit hydrograph, such as (Snyder) is highly efficient and more realistic for estimating peak time and peak flow for large basins than other models because it relates to basin characteristics.

Seagrass meadow stability and composition influence carbon storage

ContextSeagrass ecosystems are lauded for storing organic carbon in underlying sediments, but storage is highly variable, even at relatively small spatial scales. While environmental setting and seagrass cover are known drivers of carbon storage capacity, it is unclear how other seagrass features such as species composition influence carbon storage, and whether historical vs. contemporary features are better predictors of storage.ObjectivesWe examined the influence of historical and contemporary seagrass variables on surface (0–10 cm) sediment organic carbon storage at the meadow-scale (~ 25 km2), in addition to the influence of environmental drivers. Our study area was located within a subtropical mixed-species seagrass meadow along a low-energy coastline in the northeastern Gulf of Mexico (Cedar Key, Florida, USA).MethodsWe derived historical metrics of seagrass cover and composition from 14-year seagrass monitoring datasets and measured surface sediment carbon densities and grain size, contemporary seagrass biomass and species composition, as well as environmental characteristics related to hydrology and physical disturbance (i.e., relative exposure, elevation, and distance to navigation channels). We assessed bivariate relationships between predictor variables and surface carbon densities with linear regression analyses and used path analysis to assess hypothesized relationships between a subset of predictor variables and carbon densities.ResultsWhile low relative to global values, surface carbon densities in Cedar Key seagrass meadows varied by an order of magnitude. Sediment grain size was strongly related to carbon densities, but environmental variables had only indirect effects on carbon densities. Historical seagrass cover, variability in cover, and species diversity were generally better predictors of storage than contemporary variables. Historical and contemporary species identity–specifically the presence of Thalassia testudinum–were also significant drivers of storage.ConclusionsIn Cedar Key, historically diverse and persistent seagrass meadows dominated by late-successional species contained the largest surface carbon stores. Our results highlight the importance of site history in terms of meadow stability (inversely measured as variability in cover) as well as species identity and diversity in enhancing surface carbon storage. The environmental variables we examined had comparatively weak effects on carbon densities, however, relative exposure and elevation may not be the most relevant hydrological drivers of carbon storage at the meadow scale. Together, these findings suggest that drivers of seagrass meadow carbon storage are context and scale dependent.

Open Problems Column

This Issue's Column! This issue's Open Problem Column is by Erika Melder. It is about lower bounds on approximating set cover. The overarching question is: are there easier proofs?

Effective Homology of Universal Covers

We present an algorithm for constructing the effective homology for the universal cover of simplicial sets with effective homology, provided that a nilpotency condition holds. This can be useful for computing higher homotopy groups, since this process can be seen as the first step in the Whitehead tower. Our algorithm can be applied to some spaces satisfying a particular condition. We also present implementations of this method in SageMath and Kenzo.

Minimization Problems with Non-Submodular Cover Constraint

The set cover problem has been studied extensively for many years. Submodular function plays a key role in combinatorial optimization. Extending the set cover problem, we consider three submodular cover problems. The first two problems minimize linear and submodular functions, respectively, subject to the same non-submodular cover constraint. The third problem minimizes a submodular function subject to non-submodular cover and precedence constraints. Based on the concepts of submodular ratio and gap, and Lovász extension, we devise greedy and primal–dual approximation algorithms for these problems.

Approximation Algorithms and Linear Programming Relaxations for Scheduling Problems Related to Min-Sum Set Cover

We consider single-machine scheduling problems that are natural generalizations or variations of the min-sum set-cover problem. For these scheduling problems, we give new approximation algorithms. Some of these algorithms rely on time-indexed linear programming relaxations. We show how a variant of alpha-point scheduling leads to the best known approximation ratios, including a guarantee of four for an interesting special case of the so-called generalized min-sum set-cover problem. We also make explicit the connection between the greedy algorithm for min-sum set cover and the concept of Sidney decomposition for precedence-constrained single-machine scheduling and show how this leads to a 4-approximation algorithm for single-machine scheduling with so-called bipartite OR-precedence constraints. Funding: This work has been supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research.