Set Multicover Problem Research Articles

Approximation algorithm for the stochastic prize-collecting set multicover problem

We study a scenario-based stochastic and prize-collecting version of the set multicover problem, where there is uncertainty about a demand for each element to be covered and the penalty cost is imposed linearly on the shortfall in each demand. This problem is NP-hard and has an application in shift scheduling in crowdsourced delivery services. For this problem, we present an LP-based O(ln⁡n)-approximation algorithm, where n is the number of elements to be covered.

Approximation algorithm for the multicovering problem

Let \(\mathcal {H}=(V,\mathcal {E})\) be a hypergraph with maximum edge size \(\ell \) and maximum degree \(\varDelta \). For a given positive integers \(b_v\), \(v\in V\), a set multicover in \(\mathcal {H}\) is a set of edges \(C \subseteq \mathcal {E}\) such that every vertex v in V belongs to at least \(b_v\) edges in C. Set multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that for any fixed \(\varDelta \) and \(b:=\min _{v\in V}b_{v}\), the problem of set multicover is not approximable within a ratio less than \(\delta :=\varDelta -b+1\), unless \(\mathcal {P}=\mathcal {NP}\). Hence it’s a challenge to explore for which classes of hypergraph the conjecture doesn’t hold. We present a polynomial time algorithm for the set multicover problem which combines a deterministic threshold algorithm with conditioned randomized rounding steps. Our algorithm yields an approximation ratio of \(\max \left\{ \frac{148}{149}\delta , \left( 1- \frac{ (b-1)e^{\frac{\delta }{4}}}{94\ell } \right) \delta \right\} \) for \(b\ge 2\) and \(\delta \ge 3\). Our result not only improves over the approximation ratio presented by El Ouali et al. (Algorithmica 74:574, 2016) but it’s more general since we set no restriction on the parameter \(\ell \). Moreover we present a further polynomial time algorithm with an approximation ratio of \(\frac{5}{6}\delta \) for hypergraphs with \(\ell \le (1+\epsilon )\bar{\ell }\) for any fixed \(\epsilon \in [0,\frac{1}{2}]\), where \(\bar{\ell }\) is the average edge size. The analysis of this algorithm relies on matching/covering duality due to Ray-Chaudhuri (1960), which we convert into an approximative form. The second performance disprove the conjecture of Peleg et al. for a large subclass of hypergraphs.

Breaking the rmax Barrier: Enhanced Approximation Algorithms for Partial Set Multicover Problem

Given an element set E of order n, a collection of subsets [Formula: see text], a cost cS on each set [Formula: see text], a covering requirement re for each element [Formula: see text], and an integer k, the goal of a minimum partial set multicover problem (MinPSMC) is to find a subcollection [Formula: see text] to fully cover at least k elements such that the cost of [Formula: see text] is as small as possible and element e is fully covered by [Formula: see text] if it belongs to at least re sets of [Formula: see text]. This problem generalizes the minimum k-union problem (MinkU) and is believed not to admit a subpolynomial approximation ratio. In this paper, we present a [Formula: see text]-approximation algorithm for MinPSMC, in which [Formula: see text] is the maximum size of a set in S. And when [Formula: see text], we present a bicriteria algorithm fully covering at least [Formula: see text] elements with approximation ratio [Formula: see text], where [Formula: see text] is a fixed number. These results are obtained by studying the minimum density subcollection problem with (or without) cardinality constraint, which might be of interest by itself.

A primal-dual algorithm for the minimum partial set multi-cover problem

In a minimum partial set multi-cover problem (MinPSMC), given an element set E, a collection of subsets $${\mathcal {S}} \subseteq 2^E$$, a cost $$w_S$$ on each set $$S\in {\mathcal {S}}$$, a covering requirement $$r_e$$ for each element $$e\in E$$, and an integer k, the goal is to find a sub-collection $${\mathcal {F}} \subseteq {\mathcal {S}}$$ to fully cover at least k elements such that the cost of $${\mathcal {F}}$$ is as small as possible, where element e is fully covered by $${\mathcal {F}}$$ if it belongs to at least $$r_e$$ sets of $${\mathcal {F}}$$. On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor $$O(n^\frac{1}{2(\log \log n)^c})$$ for some constant c under the ETH assumption. Furthermore, assuming $$r_{\max }$$ is a constant, where $$r_{\max } =\max _{e\in E} r_e$$ is the maximum covering requirement and f is the maximum number of sets containing a common element, we present a primal-dual algorithm for MinPSMC and show that its performance ratio is $$O(\sqrt{n})$$. We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure.

Resource Allocation in Highly Dynamic Device-to-Device Communication: An Adaptive Set Multi-Cover Approach

Device-to-device (D2D) communication has recently gained much attention for its potential to boost the capacity of cellular systems. D2D enables direct communication between devices while bypassing a base station (BS), hence decreasing the load of BSs and increasing the network throughput via spatial reuse of radio resources. However, the cellular system is highly dynamic, an optimal allocation plan of radio resource to D2D links at one time point can easily become suboptimal when devices move. Thus, to maximize spatial reuse in cellular systems, it is crucial to update the resource allocation adaptively to reflect the current system status. In this paper, we develop the first adaptive solution framework to the dynamic resource problem for maximizing spatial reuse. At the core of the framework, we present the two algorithms for the adaptive set multicover problem with approximation ratio f and log n, respectively, where f is the frequency of the most frequent element and n is the total number of elements. The experimental results not only show that our solutions have a significant improvement in running time, compared with optimal or approximated offline methods, but also demonstrate their good performance through the resource usage, network throughput, and other metrics.

Approximation algorithm for the partial set multi-cover problem

Partial set cover problem and set multi-cover problem are two generalizations of the set cover problem. In this paper, we consider the partial set multi-cover problem which is a combination of them: given an element set E, a collection of sets \(\mathcal S\subseteq 2^E\), a total covering ratio q, each set \(S\in \mathcal S\) is associated with a cost \(c_S\), each element \(e\in E\) is associated with a covering requirement \(r_e\), the goal is to find a minimum cost sub-collection \({\mathcal {S}}'\subseteq {\mathcal {S}}\) to fully cover at least q|E| elements, where element e is fully covered if it belongs to at least \(r_e\) sets of \({\mathcal {S}}'\). Denote by \(r_{\max }=\max \{r_e:e\in E\}\) the maximum covering requirement. We present an \((O (r_{\max }\log ^2n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q})),1-\varepsilon )\)-bicriteria approximation algorithm, that is, the output of our algorithm has cost \(O(r_{\max }\log ^2 n(1+\ln (\frac{1}{\varepsilon })+\frac{1-q}{\varepsilon q}))\) times of the optimal value while the number of fully covered elements is at least \((1-\varepsilon )q|E|\).

A bicriteria algorithm for the minimum submodular cost partial set multi-cover problem

This paper presents a bicriteria approximation algorithm for the minimum submodular cost partial set multi-cover problem (SCPSMC), the goal of which is to find a minimum cost sub-collection of sets to fully cover q percentage of total profit of all elements, where the cost on sub-collections is a submodular function, and an element e with covering requirement re is fully covered if it belongs to at least re picked sets. Assuming that the maximum covering requirement rmax=maxe∈E⁡re is a constant and the cost function is nonnegative and submodular, we give a deterministic (b/qε,(1−ε))-bicriteria algorithm for SCPSMC, the output of which fully covers at least (1−ε)q-percentage of the total profit and the performance ratio is b/qε, where b=maxe⁡(fere) and fe is the number of sets containing element e.

Panoramic Light Field From Hand-Held Video and Its Sampling for Real-Time Rendering

By providing angular and spatial information of light rays, light field images are widely used in many applications. To capture large field of view light field, existing approaches either stitch small field of view light fields, whose apertures are also small, or leverage specialized equipments, which are not accessible to ordinary users. In this paper, we present a method to extract fully 360° field of view panoramic light fields from densely sampled hand-held videos. Unlike previous works, our method handles the large exposure variation and motion blur problems, which are common in the panoramic capture and hand-held videos. To provide real-time rendering performance, our method applies light field sampling to the extracted panoramic light field. Based on the unstructured representation, our method can sample the light field without explicit parameterization. We formulate the sampling problem into a set multicover problem and solve it globally using integer linear programming. Compared with the greedy based method, our approach can use much less video frames to represent the panoramic light field, but still can get better rendering results.

Direction finder deployment based on the partial set multi-cover problem

ABSTRACTThe military uses direction finders to locate enemy forces by detecting the positions of transmitters that emit radio frequencies. We consider the problem of deploying the minimum number of direction finders to effectively cover an area of interest in which transmitters are likely to be positioned. Emissions may not be detectable if obstacles block the direct line of sight between a direction finder and the transmitter. We formulate this problem as a partial set multicover problem in which at least a p-fraction of the likely transmitter positions must be covered, each by at least k direction finders. We present a set of greedy heuristics with random selection rules for the partial set multicover problem. We perform computational tests to evaluate and compare these heuristics, both on randomly generated instances and a realistic instance.

In silico read normalization using set multi-cover optimization.

MotivationDe Bruijn graphs are a common assembly data structure for sequencing datasets. But with the advances in sequencing technologies, assembling high coverage datasets has become a computational challenge. Read normalization, which removes redundancy in datasets, is widely applied to reduce resource requirements. Current normalization algorithms, though efficient, provide no guarantee to preserve important k-mers that form connections between regions in the graph.ResultsHere, normalization is phrased as a set multi-cover problem on reads and a heuristic algorithm, Optimized Read Normalization Algorithm (ORNA), is proposed. ORNA normalizes to the minimum number of reads required to retain all k-mers and their relative k-mer abundances from the original dataset. Hence, all connections from the original graph are preserved. ORNA was tested on various RNA-seq datasets with different coverage values. It was compared to the current normalization algorithms and was found to be performing better. Normalizing error corrected data allows for more accurate assemblies compared to the normalized uncorrected dataset. Further, an application is proposed in which multiple datasets are combined and normalized to predict novel transcripts that would have been missed otherwise. Finally, ORNA is a general purpose normalization algorithm that is fast and significantly reduces datasets with loss of assembly quality in between [1, 30]% depending on reduction stringency.Availability and implementationORNA is available at https://github.com/SchulzLab/ORNA.Supplementary information Supplementary data are available at Bioinformatics online.

Approximation algorithm for partial set multicover versus full set multicover

In a (full) set multicover (SMC) problem, the goal is to select a subcollection of sets to fully cover all elements, where an element is fully covered if it belongs to at least a required number of sets of the selected subcollection. In a partial set multicover (PSMC) problem, it is sufficient to fully cover a required fraction of elements. In this paper, we present a greedy algorithm for PSMC, and analyze the ratio between the cost of the computed solution and the cost of an optimal solution to the corresponding (full) SMC problem. We shall call this ratio as a partial-versus-full ratio. The motivation for such an analysis is to show that PSMC is fairly economic even when it can only be calculated approximately. It turns out that the partial-versus-full ratio of our algorithm is at most [Formula: see text], where [Formula: see text] is the requirement of an element, and [Formula: see text] is the fraction of elements required to be fully covered. An example is given showing that this ratio is tight up to a constant.

Minimizing the number of max-power users in ad-hoc wireless networks with minimum node degree requirements

We consider the problem of assigning one of two possible transmission ranges to each node of a wireless ad-hoc network, aiming for a predefined degree of redundancy and reliability by requiring that each node has at least Δ∈N neighbors in the resulting network. To conserve the networks limited energy, this goal should be achieved with as few as possible nodes using the higher of the two transmission ranges. We show that this so-called Two Power Level Vertex Degree problem in NP-complete, derive an approximation algorithm that utilizes an arbitrary approximation algorithm for the well-known Min Set Multicover problem and show that the approximation quality of this algorithm transfers to our algorithm.

Relaxation heuristics for the set multicover problem with generalized upper bound constraints

We consider an extension of the set covering problem (SCP) introducing (i) multicover and (ii) generalized upper bound (GUB) constraints. For the conventional SCP, the pricing method has been introduced to reduce the size of instances, and several efficient heuristic algorithms based on such reduction techniques have been developed to solve large-scale instances. However, GUB constraints often make the pricing method less effective, because they often prevent solutions from containing highly evaluated variables together. To overcome this problem, we develop heuristic algorithms to reduce the size of instances, in which new evaluation schemes of variables are introduced taking account of GUB constraints. We also develop an efficient implementation of a 2-flip neighborhood local search algorithm that reduces the number of candidates in the neighborhood without sacrificing the solution quality. In order to guide the search to visit a wide variety of good solutions, we also introduce a path relinking method that generates new solutions by combining two or more solutions obtained so far. According to computational comparison on benchmark instances, the proposed method succeeds in selecting a small number of promising variables properly and performs quite effectively even for large-scale instances having hard GUB constraints.

An approximation algorithm for the partial covering 0–1 integer program

The partial covering 0–1 integer program (PCIP) is a relaxed problem of the covering 0–1 integer program (CIP) such that some fixed number of constraints may not be satisfied. This type of relaxation is also discussed in the partial set multi-cover problem (PSMCP) and the partial set cover problem (PSCP). In this paper, we propose an approximation algorithm for PCIP by extending an approximation algorithm for PSCP by Gandhi et al. (2004).

Online set multicover algorithms for dynamic D2D communications

Motivated by the dynamic resource allocation problem for device-to-device (D2D) communications, we study the online set multicover problem (OSMC). In the online set multicover, the set X of elements to be covered is unknown in advance; furthermore, the coverage requirement of each element $$x \in X$$ is initially unknown. Elements of X together with coverage requirements are presented one at a time in an online fashion; and a feasible solution must be maintained at all times. We provide the first deterministic, online algorithms for OSMC with competitive ratios. We consider two versions of OSMC; in the first, each set may be picked only once, while the second version allows each set to be picked multiple times. For both versions, we present the first deterministic, online algorithms, with competitive ratios $$O( \log n \log m )$$ and $$O( \log n (\log m + \log k) )$$ , repectively, where n is the number of elements, m is the number of sets, and k is the maximum coverage requirement. By simulation, we show the efficacy of these algorithms for resource allocation in the D2D setting by analyzing network throughput and other metrics, obtaining a large improvement in running time over offline methods.

Local ratio method on partial set multi-cover

In this paper, we study the minimum partial set multi-cover problem (PSMC). Given an element set E, a collection of subsets \({\mathcal {S}}\subseteq 2^E\), a cost \(c_S\) on each set \(S\in {\mathcal {S}}\), a covering requirement \(r_e\) for each element \(e\in E\), and an integer k, the PSMC problem is to find a sub-collection \({\mathcal {F}}\subseteq {\mathcal {S}}\) to fully cover at least k elements such that the cost of \({\mathcal {F}}\) is as small as possible, where element e is fully covered by \({\mathcal {F}}\) if it belongs to at least \(r_e\) sets of \({\mathcal {F}}\). This paper presents an approximation algorithm using local ratio method achieving performance ratio \(\max \left\{ \frac{\Delta }{k}\left( \frac{1}{f-r_{\min }}+\frac{r_{\max }}{r_{\min }}\right) ,\frac{1}{\rho }+\frac{f}{r_{\min }}+\frac{1}{r_{\max }}-\frac{1}{\rho r_{\max }}-1,\frac{1}{\rho }\right\} \), where \(\Delta \) is the size of a maximum set, f is the maximum number of sets containing a common element, \(\rho \) is the minimum percentage of elements required to be fully covered during iterations of the algorithm, and \(r_{\max }\) and \(r_{\min }\) are the maximum and the minimum covering requirement, respectively. In particular, when \(r_{\max }\) is a constant, the first term can be omitted. Notice that our ratio coincides with the classic ratio f for both the set multi-cover problem (in which case \(k=|E|\)) and the partial set single-cover problem (in which case \(r_{\max }=1\)). However, when \(k 1\), the ratio might be as large as \(\Theta (n)\). This result shows an interesting “shock wave like” feature of approximating PSMC. The purpose of this paper is trying to arouse some interest in such a feature and attract more work on this challenging problem.

Weighted geometric set multi-cover via quasi-uniform sampling

We give a randomized polynomial time algorithm with approximation ratio $O(\log \phi(n))$ for weighted set multi-cover instances with a shallow cell complexity of at most $f(z,k) =z\phi(z) k^{O(1)}$. Up to constant factors, this matches a recent result of Chan, Grant, Konemann and Sharpe for the set cover case, i.e. when all the covering requirements are 1. One consequence of this is an $O(1)$-approximation for geometric weighted set multi-cover problems when the geometric objects have linear union complexity; for example when the objects are disks, unit cubes or halfspaces in $\mathbb{R}^3$. Another consequence is to show that the real diculty of many natural capacitated set covering problems lies with solving the associated priority cover problem only, and not with the associated multi-cover problem.

Approximation algorithm for partial positive influence problem in social network

Influence problem is one of the central problems in the study of online social networks, the goal of which is to influence all nodes with the minimum number of seeds. However, in the real world, it might be too expensive to influence all nodes. In many cases, it is satisfactory to influence nodes only up to some percent p. In this paper, we study the minimum partial positive influence dominating set (MPPIDS) problem. In fact, we presented an approximation algorithm for a more general problem called minimum partial set multicover problem. As a consequence, the MPPIDS problem admits an approximation with performance ratio $$\gamma H(\Delta )$$źH(Δ), where $$H(\cdot )$$H(·) is the Harmonic number, $$\gamma =1/(1-(1-p)\eta ),\eta \approx \Delta ^2/\delta $$ź=1/(1-(1-p)ź),źźΔ2/ź, and $$\Delta ,\delta $$Δ,ź are the maximum degree and the minimum degree of the graph, respectively. For power-law graphs, we show that our algorithm has a constant performance ratio.

Randomized Approximation for the Set Multicover Problem in Hypergraphs

Let $$b\in {\mathbb {N}}_{\ge 1}$$b?N?1 and let $${\mathcal {H}}=(V,{\mathcal {E}})$$H=(V,E) be a hypergraph with maximum vertex degree $${\varDelta }$$Δ and maximum edge size $$l$$l. A set $$b$$b-multicover in $${\mathcal {H}}$$H is a set of edges $$C \subseteq {\mathcal {E}}$$C⊆E such that every vertex in $$V$$V belongs to at least $$b$$b edges in $$C$$C. $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER is the problem of finding a set $$b$$b-multicover of minimum cardinality, and for $$b=1$$b=1 it is the fundamental set cover problem. Peleg et al. (Algorithmica 18(1):44---66, 1997) gave a randomized algorithm achieving an approximation ratio of $$\delta \cdot \big (1-\big (\frac{c}{n}\big )^\frac{1}{\delta }\big )$$?·(1-(cn)1?), where $$\delta := {\varDelta }- b + 1$$?:=Δ-b+1 and $$c>0$$c>0 is a constant. As this ratio depends on the instance size $$n$$n and tends to $$\delta $$? as $$n$$n tends to $$\infty $$?, it remained an open problem whether an approximation ratio of $$\delta \alpha $$?? with a constant$$\alpha < 1$$?<1 can be proved. In fact, the authors conjectured that for any fixed $${\varDelta }$$Δ and $$b$$b, the problem is not approximable within a ratio smaller than $$\delta $$?, unless $${\mathcal {P}}={\mathcal {NP}}$$P=NP. We present a randomized algorithm of hybrid type for $${\textsc {set }}\, b\text {-}{\textsc {multicover}}$$SETb-MULTICOVER, $$b \ge 2$$b?2, combining LP-based randomized rounding with greedy repairing, and achieve an approximation ratio of $$\delta \cdot \left( 1 - \frac{11({\varDelta }- b)}{72l} \right) $$?·1-11(Δ-b)72l for hypergraphs with maximum edge size $$l \in {\mathcal {O}}\left( \max \big \{(nb)^\frac{1}{5},n^\frac{1}{4}\big \}\right) $$l?Omax{(nb)15,n14}. In particular, for all hypergraphs where $$l$$l is constant, we get an $$\alpha \delta $$??-ratio with constant $$\alpha < 1$$?<1. Hence the above stated conjecture does not hold for hypergraphs with constant $$l$$l and we have identified the boundedness of the maximum hyperedge size as a relevant parameter responsible for approximations below $$\delta $$?.

On integer programming models for the multi-channel PMU placement problem and their solution

Phasor Measurement Units (PMUs) are starting to see increased deployment, enabling accurate measurement of power grid electrical properties to determine system health. Due to the costs associated with PMU acquisition and maintenance, it is practically important to place the minimum number of PMUs in order to achieve system complete observability. In this paper, we consider a variety of optimization models for the PMU placement problem that addresses more realistic assumptions than simple infinite-capacity placement models. Specifically, instead of assuming that a PMU can sense all lines incident to the bus at which it is placed, we impose the more realistic assumption that PMUs have restricted channel capacity, with per-unit cost given as a function of channel capacity. The optimization objective is then to minimize the total cost of placed PMUs, in contrast to their number. Further, we leverage the zero-injection bus properties to reduce the quantity and cost of placed PMUs. In formulating our optimization models, we identify a close relationship between the PMU placement problem (PPP) and a classic combinatorial problem, the set cover problem (SCP). If channel capacity limits are ignored, there is a close relationship between the PPP and the dominating set problem (DSP), a special case of the SCP. Similarly, when measurement redundancy is imposed as a design requirement, there is a close relationship between the PPP and the set multi-cover problem (SMCP), a generalized version of the SCP. These connections to well-studied combinational problems are not well-known in the power systems literature, and can be leveraged to improve solution algorithms. We demonstrate that more realistic, high-fidelity PPP optimization models can be solved to optimality using commercial integer programing solvers such as CPLEX. Specifically, run-times for all test cases, ranging from IEEE 14-bus to 300-bus test systems, are less than a second. This result indicates that the size of system that can be analyzed using state-of-the-art solvers is considerable. Further, our results call into question the need for problem-specific heuristic solution algorithms for the PPP, many of which have been proposed over the past decade. Finally, we analyze cost versus performance tradeoffs using our PPP optimization models on various IEEE test systems.