Minimum Set Cover Research Articles

A succinct and approximate greedy algorithm for the Minimum Set Cover Problem

The Minimum Set Cover Problem (MSCP) is a combinatorial optimization problem belonging to the NP-Hard class in computer science. For this reason, there is no algorithm that in the worst case ensures finding an optimal solution in polynomial-time. For a given universe X, the popular greedy heuristic, called Greedy-SetCover, is the main theoretical contribution to obtain an approximate solution for the MSCP in polynomial-time, offering an optimal approximate ratio of (ln|X|+1). In this article, we propose an approximate algorithm for MSCP within a succinct representation of the input dataset, whose empirical performance improves Greedy-SetCover both in quality and execution time, while offering the same optimal approximation ratio for the problem. Our experiments show that the proposed algorithm is magnitudes of times faster than the aforementioned greedy one, obtaining on average a cardinality much closer to the optimal solution. Furthermore, because we work on a succinct representation that allows us to compute operations between sets using bitwise operators, we can process much larger datasets than state-of-the-art solutions. As a result, our proposal is also a suitable alternative for processing large datasets as required by the current Big Data era.

Hardness and approximation of submodular minimum linear ordering problems

AbstractThe minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $$f(\cdot )$$ f ( · ) due to an ordering $$\sigma $$ σ of the items (say [n]), i.e., $$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ min σ ∑ i ∈ [ n ] f ( E i , σ ) , where $$E_{i,\sigma }$$ E i , σ is the set of items mapped by $$\sigma $$ σ to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a $$(2-\frac{1+\ell _{f}}{1+|E|})$$ ( 2 - 1 + ℓ f 1 + | E | ) -approximation for monotone submodular MLOP where $$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ℓ f = f ( E ) max x ∈ E f ( { x } ) satisfies $$1 \le \ell _f \le |E|$$ 1 ≤ ℓ f ≤ | E | . Our theory provides new approximation bounds for special cases of the problem, in particular a $$(2-\frac{1+r(E)}{1+|E|})$$ ( 2 - 1 + r ( E ) 1 + | E | ) -approximation for the matroid MLOP, where $$f = r$$ f = r is the rank function of a matroid. We further show that minimum latency vertex cover is $$\frac{4}{3}$$ 4 3 -approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.

Observability and Detectability of Stochastic Labeled Graphs

In this paper, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">observable stochastic graphs and detetectable stochastic graphs</i> are, respectively, defined with the detailed implementation for the observability and detectability of stochastic discrete-time and discrete-state dynamic systems. More specifically, they are generally two classes of vertex-colored and edge-labeled graphs rendering a walking agent therein to determine his initial and current positions, respectively, in probability one by measuring the color sequence of his traversed vertices. In the part of analysis, we follow the aforementioned two definitions and establish the corresponding polynomial-time verifying algorithms. Notably, the implicit formulas are also established to calculate the observability and detectability probability for any pairwise vertex pair. In the synthesis part, the minimal number of colors dyeing the vertices to make any considered graph stochastically observable and detectable is investigated, respectively. Our results indicate that the minimal coloring problem is NP-hard for observability in the stochastic situations but is solvable in polynomial time for detectability in the deterministic cases. The observability and detectability of stochastic finite-valued systems, assembling with the finite-cardinal state spaces, are validated as a compelling application of these two types of directed graphs, whereas the minimal sensors placement problems subject to observability and detectability problems are accordingly interpreted by the minimum set cover algorithm.

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.

Can greedy-like heuristics be useful for solving the Weighted Orthogonal Art Gallery Problem under regular grid discretization?

In this paper we deal with the Weighted Orthogonal Art Gallery Problem. The task is to place guards on some vertices of an orthogonal polygon P, such that the total sum of prices assigned to the chosen vertices is minimal and all points from the polygon P are covered. The problem has applications in practice, for example, in installing cameras at the corners of a building such that all interior space is covered by at least one of the cameras but the price of installation is the smallest possible. To solve the problem, the regular grid discretization of the area of polygon is applied. We propose a novel greedy approach which is based on balancing the tradeoff between the total sum of guards' costs and the total number of not yet covered points from the discretization. This new approach and an existing greedy algorithm are further hybridized with the Integer Linear programming, which is originally formulated for the wellknown Minimum Set Cover problem. Our experimental results are conducted on two sets of polygons from the literature: one with small area and the other one with large area. They proved that the proposed greedy methods can achieve the optimal solutions in most cases for the class of large-area polygons, while in case of the small area polygons, they achieve solutions of reasonable quality within lower runtime than the exact algorithms.

Computational design and analysis of modular cells for large libraries of exchangeable product synthesis modules

Microbial metabolism can be harnessed to produce a large library of useful chemicals from renewable resources such as plant biomass. However, it is laborious and expensive to create microbial biocatalysts to produce each new product. To tackle this challenge, we have recently developed modular cell (ModCell) design principles that enable rapid generation of production strains by assembling a modular (chassis) cell with exchangeable production modules to achieve overproduction of target molecules. Previous computational ModCell design methods are limited to analyze small libraries of around 20 products. In this study, we developed a new computational method, named ModCell-HPC, that can design modular cells for large libraries with hundreds of products with a highly-parallel and multi-objective evolutionary algorithm and enable us to elucidate modular design properties. We demonstrated ModCell-HPC to design Escherichia coli modular cells towards a library of 161 endogenous production modules. From these simulations, we identified E. coli modular cells with few genetic manipulations that can produce dozens of molecules in a growth-coupled manner with different types of fermentable sugars. These designs revealed key genetic manipulations at the chassis and module levels to accomplish versatile modular cells, involving not only in the removal of major by-products but also modification of branch points in the central metabolism. We further found that the effect of various sugar degradation on redox metabolism results in lower compatibility between a modular cell and production modules for growth on pentoses than hexoses. To better characterize the degree of compatibility, we developed a method to calculate the minimal set cover, identifying that only three modular cells are all needed to couple with all compatible production modules. By determining the unknown compatibility contribution metric, we further elucidated the design features that allow an existing modular cell to be re-purposed towards production of new molecules. Overall, ModCell-HPC is a useful tool for understanding modularity of biological systems and guiding more efficient and generalizable design of modular cells that help reduce research and development cost in biocatalysis.

On the computational cost of the set cover approach for the optimal camera placement problem and possible set-cover-inspired trade-offs for surveillance infrastructure design

The 𝒩𝒫-hard minimum set cover problem (SCP) is a very typical model to use when attempting to formalise optimal camera placement (OCP) applications. In a generic form, the OCP problem relates to the positioning of individual cameras such that the overall network is able to cover a given area while meeting a set of application-specific requirements (image quality, redundancy, ...) and optimising an objective, typically minimum cost or maximum coverage. In this paper, we focus on an application called global or persistent surveillance: camera networks which ensure full coverage of a given area. As preliminary work, an instance generation pipeline is proposed to create OCP instances from real-world data and solve them using existing literature. The computational cost of both the instance generation process and the solving algorithms however highlights a need for more efficient methods for decision makers to use in real-world settings. In this paper, we therefore propose to review the suitability of the approach, and more specifically to question two key elements: the impact of sampling frequencies and the importance of rigid full-coverage constraints. The results allow us to quickly provide decision makers with an overview of available solutions and trade-offs.

Parallel approximation for partial set cover

In a minimum partial set cover problem (MinPSC), given a ground set E with n elements, a collection S of subsets of E with |S|=m, a cost function c:S→R+, and an integer k≤n, the goal of MinPSC is to find a minimum cost sub-collection of S that covers at least k elements of E. In this paper, we design a parallel algorithm for MinPSC which yields a solution with approximation ratio at most f1−2ε in O(1εlogmnε) rounds, where f is the maximum number of sets containing a common element, and 0<ε<1/2 is a constant. We also design a parallel algorithm for a special MinPSC problem, the minimum power partial cover problem (MinPPC), which achieves approximation ratio at most (3+2ε)α1−2ε in O(1εlogmnεlog2m) rounds, where α≥1 is the attenuation factor of power.

A distributed algorithm for a set cover game

In this paper, we propose a set cover game and show that any Nash equilibrium is a minimal set cover, and also a Pareto optimal solution. Then we present a distributed algorithm to realize the game. The algorithm is self-organizing in the sense that all players can make decisions by themselves simultaneously. It is local in the sense that each player makes his decision based only on information in his neighborhood. It is efficient in the sense that it converges to a minimal set cover in polynomial time when [Formula: see text] is upper bounded by a polynomial of the input size, where [Formula: see text] and [Formula: see text] are the maximum cost and the minimum cost of sets, respectively.

An Exact Method for the Minimum Feedback Arc Set Problem

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G . Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem . Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2 n ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

PE-MSC: partial entailment-based minimum set cover for text summarization

The notion of Textual Entailment (TE) is an established indicator of text connectedness. It captures semantic relationships between texts. Recently, it has been used successfully for determining sentence salience in many text summarization methods. However, it has been reported in previous works that the standard textual entailment is not ideal for measuring sentence salience. This is because textual entailment relationships between sentences are quite rare in real-world texts. Therefore, we suggest using partial TE to accomplish the task of recognizing standard TE. We present the single document summarization problem as an optimization problem which is solved using a weighted Minimum Set Cover (wMSC) algorithm. In this method, sentences are broken into fragments and Partial TE is used to form sets of fragments. Finally, wMSC is applied to the sets to obtain the minimum set cover, which corresponds to the summary of the document. The results achieved on the DUC 2002 dataset using ROUGE and other quality metrics show that the proposed method outperforms the state of the art.

Algorithms and Complexity for a Class of Combinatorial Optimization Problems with Labelling

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling problems into the original problems by polynomial-time reduction. We show that, although the labelling version of these problem seems more universal than their original counterparts, they are actually equivalent to the corresponding original problem from an algorithmic point of view. Moreover, we generalize the greedy approach for solving submodular cover problem to its labelling version, and as simple applications of our new method, we use it to solve the labelling versions of the minimum weighted spanning tree and connected vertex cover problem in a unified way.

Cluster load based content distribution and speculative execution for geographically distributed cloud environment

The scale of big data has shown an explosive growth, which makes the processing of big data put forward higher requirements on data centers, and a single data center can no longer meet the needs of big data processing. To deal with this situation, a geographically distributed cloud system needs to be built. However, in the geographically distributed cloud system, each data center is distributed in different geographic locations, which makes the data placement operations in the geographically distributed cloud system lead to greater overhead. To solve this problem, this paper proposes a data placement strategy. This strategy comprehensively considers the data transmission latency, bandwidth cost, cloud server storage capacity, and load capacity during the data placement process, and formulates a data placement problem that minimizes the energy consumption of data transmission. Then the minimum set cover method based on Lagrangian relaxation is used to solve this problem and obtain the optimal data placement scheme. On the other hand, in a geographically distributed cloud data center, the execution progress of the job submitted by the user will be affected by the straggler task. To solve this problem, this paper proposes a speculative execution strategy for the geographically distributed cloud system. This strategy performs different speculative execution operations according to the state of the cluster load, and then calculates the load capacity of the nodes in the cluster. The node with the strongest load capacity in the cluster is used to perform speculative execution operations. Experimental results show that the proposed data placement strategy can effectively improve the performance of the energy consumption, the data storage cost, the network transmission cost and the data transmission time. The proposed speculative execution strategy can effectively improve the performance of the job completion time, cluster throughput and QoS satisfaction rate.

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Revisiting the minimum set cover, the maximal coverage problems and a maximum benefit area selection problem to make climate‐change‐concerned conservation plans effective

Abstract Informed decisions for the selection of protected areas (PAs) are grounded in two general problems in Operations Research: the minimum set covering problem (minCost), where a set of ecological constraints are established as conservation targets and the minimum cost PAs are found, and the maximal coverage problem (maxCoverage) where the constraint is uniquely economic (i.e. a fixed budget) and the goal is to maximize the number of species having conservation targets adequately covered. We adjust minCost and maxCoverage to accommodate the dynamic effects of climate change on species’ ranges. The selection of sites is replaced by the selection of time‐ordered sequences of sites (climate change corridors), and an estimate of the persistence of each species in corridors is calculated according to the expected suitability of each site in the respective time period and the capacity of species to disperse between consecutive sites along corridors. In these problems, conservation targets are expressed as desired (and attainable) species persistence levels. We also introduce a novel problem (minShortfall) that combines minCost and maxCoverage. Unlike these two problems, minShortfall allows persistence targets to be missed and minimizes the sum of those gaps (i.e. target shortfalls), subject to a limited budget. We illustrate the three problems with a case study using climatic suitability estimates for 10 mammal species in the Iberian Peninsula under a climate change scenario until 2080. We compare solutions of the three problems with respect to species persistence and PA costs, under distinct settings of persistence targets, number of target‐fulfilled species and budgets. The solutions from different problems differed with regard to the areas to prioritize, their timings and the species whose persistence targets were fulfilled. This analysis also allowed identifying groups of species sharing corridors in optimal solutions, thus allowing important financial savings in site protection. We suggest that enhancing species persistence is an adequate approach to cope with habitat shifts due to climate change. We trust the three problems discussed can provide complementary and valuable support for planners to anticipate decisions in order that the negative effects of climate change on species’ persistence are minimized.

3Scover: Identifying Safeguard TF from Cell Type-TF Specificity Network by an Extended Minimum Set Cover Model.

SummaryTranscription factors (TFs) define cellular identity either by activating target cell program or by silencing donor program as demonstrated by intensive cell reprogramming studies. Here, we propose an extended minimum set cover model with stable selection (3Scover) to systematically identify silencing TFs, named safeguard TFs, from omics data. First, a cell type-TF specificity network is constructed to systematically link cell types with their specifically expressed TFs. Then we search the minimum TF set to cover this network with “many but one specificity” characteristic and integrate many subsampling models for a stable solution. 3Scover identified 30 safeguard TFs in human and mouse. These safeguard TFs are significantly enriched in the experimentally discovered reprogramming panel with their protein-protein interactors. In addition, they tend to interact closely with chromatin regulators, negatively regulate transcription, and function earlier in development. Collectively, 3Scover allows us to probe master TFs and combinatorial regulation in controlling cell identity.

Quasi-Polynomial Time Approximation Schemes for Packing and Covering Problems in Planar Graphs

We consider two optimization problems in planar graphs. In Maximum Weight Independent Set of Objects we are given a graph G and a family $${\mathcal {D}}$$ of objects, each being a connected subgraph of G with a prescribed weight, and the task is to find a maximum-weight subfamily of $${\mathcal {D}}$$ consisting of pairwise disjoint objects. In Minimum Weight Distance Set Cover we are given a graph G in which the edges might have different lengths, two sets $${\mathcal {D}},{\mathcal {C}}$$ of vertices of G, where vertices of $${\mathcal {D}}$$ have prescribed weights, and a nonnegative radius r. The task is to find a minimum-weight subset of $${\mathcal {D}}$$ such that every vertex of $${\mathcal {C}}$$ is at distance at most r from some selected vertex. Via simple reductions, these two problems generalize a number of geometric optimization tasks, notably Maximum Weight Independent Set for polygons in the plane and Weighted Geometric Set Cover for unit disks and unit squares. We present quasi-polynomial time approximation schemes (QPTASs) for both of the above problems in planar graphs: given an accuracy parameter $$\epsilon >0$$ we can compute a solution whose weight is within multiplicative factor of $$(1+\epsilon )$$ from the optimum in time $$2^{{\mathrm {poly}}(1/\epsilon ,\log |{\mathcal {D}}|)}\cdot n^{{\mathcal {O}}(1)}$$, where n is the number of vertices of the input graph. We note that a QPTAS for Maximum Weight Independent Set of Objects would follow from existing work. However, our main contribution is to provide a unified framework that works for both problems in both a planar and geometric setting and to transfer the techniques used for recursive approximation schemes for geometric problems due to Adamaszek and Wiese (in Proceedings of the FOCS 2013, IEEE, 2013; in Proceedings of the SODA 2014, SIAM, 2014) and Har-Peled and Sariel (in Proceedings of the SOCG 2014, SIAM, 2014) to the setting of planar graphs. In particular, this yields a purely combinatorial viewpoint on these methods as a phenomenon in planar graphs.

Tight Bounds for Single-Pass Streaming Complexity of the Set Cover Problem

We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $\alpha$-approximate set cover (for any $\alpha= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $\Theta(mn/\alpha)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of $\alpha$ saving in the space is achievable in this case and that this is the best possible. In other words, we show that $\Theta(mn/\alpha^2)$ space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of $\alpha$. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.

Near-Optimal Multicast Tree Construction in Leaf-Spine Data Center Networks

In this paper, we revisit the optimization problem of constructing multicast trees for cloud applications in the context of leaf-spine data center network (DCN). On one hand, we find that, the maximum multicast rate a network can provide equals to the minimum of the maximum unsplit throughput it can provide to each of the receivers. On the other hand, by taking the characteristic of leaf-spine DCN into account, we prove that the optimal construction of multicast tree in leaf-spine DCNs is equivalent to the well-known problem of minimum set cover , distinguished from the conventional wisdom that models the problem as a general-purpose directed minimum Steiner . Based on these findings, we develop simple yet near-optimal algorithms to construct multicast trees and deal with network failures.