Showing papers on "Set cover problem published in 2010"

Differentially private combinatorial optimization

Abstract: Consider the following problem: given a metric space, some of whose points are "clients," select a set of at most k facility locations to minimize the average distance from the clients to their nearest facility. This is just the well-studied k-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. This raises the following quandary: what if the locations of the clients are sensitive information that we would like to keep private? Is it even possible to design good algorithms for this problem that preserve the privacy of the clients?In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any non-trivial approximation to even the value of an optimal solution, let alone the entire solution.Apart from the k-median problem, we consider the problems of vertex and set cover, min-cut, k-median, facility location, and Steiner tree, and give approximation algorithms and lower bounds for these problems. We also consider the recently introduced sub-modular maximization problem, "Combinatorial Public Projects" (CPP), shown by Papadimitriou et al. [28] to be inapproximable to subpolynomial multiplicative factors by any efficient and truthful algorithm. We give a differentially private (and hence approximately truthful) algorithm that achieves a logarithmic additive approximation.

Weighted geometric set cover via quasi-uniform sampling

Abstract: There has been much progress on geometric set cover problems, but most known techniques only apply to the unweighted setting. For the weighted setting, very few results are known with approximation guarantees better than that for the combinatorial set cover problem. In this article, we employ the idea of quasi-uniform sampling to obtain improved approximation guarantees in the weighted setting for a large class of problems for which such guarantees were known in the unweighted case. As a consequence of this sampling method, we obtain new results on the fractional set cover packing problem.

Applications of the Location Set‐covering Problem

Abstract: The location set-covering problem is extended to apply to three new situations. (1) The demands are assumed to occur continuously alone arcs of a network. (2) A mobile unit departs from one of the locations to be chosen and picks up the demand, providing service at a still more distant point. (3) New demands and sites occur over time.

Set cover algorithms for very large datasets

Abstract: The problem of Set Cover - to find the smallest subcollection of sets that covers some universe - is at the heart of many data and analysis tasks. It arises in a wide range of settings, including operations research, machine learning, planning, data quality and data mining. Although finding an optimal solution is NP-hard, the greedy algorithm is widely used, and typically finds solutions that are close to optimal. However, a direct implementation of the greedy approach, which picks the set with the largest number of uncovered items at each step, does not behave well when the input is very large and disk resident. The greedy algorithm must make many random accesses to disk, which are unpredictable and costly in comparison to linear scans. In order to scale Set Cover to large datasets, we provide a new algorithm which finds a solution that is provably close to that of greedy, but which is much more efficient to implement using modern disk technology. Our experiments show a ten-fold improvement in speed on moderately-sized datasets, and an even greater improvement on larger datasets.

Interactive Submodular Set Cover

Abstract: We introduce a natural generalization of sub-modular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.

Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks

Abstract: We present a distributed algorithm that finds a maximal edge packing in O(Δ + log* W) synchronous communication rounds in a weighted graph, independent of the number of nodes in the network; here Δ is the maximum degree of the graph and W is the maximum weight. As a direct application, we have a distributed 2-approximation algorithm for minimum-weight vertex cover, with the same running time. We also show how to find an $f$-approximation of minimum-weight set cover in O(f2k2 + fk log* W) rounds; here k is the maximum size of a subset in the set cover instance, f is the maximum frequency of an element, and W is the maximum weight of a subset. The algorithms are deterministic, and they can be applied in anonymous networks.

A constant factor approximation algorithm for generalized min-sum set cover

Abstract: Consider the following generalized min-sum set cover or multiple intents re-ranking problem proposed by Azar et al. (STOC 2009). We are given a universe of elements and a collection of subsets, with each set S having a covering requirement of K(S). The objective is to pick one element at a time such that the average covering time of the sets is minimized, where the covering time of a set S is the first time at which K(S) elements from it have been selected.There are two well-studied extreme cases of this problem: (i) when K(S) = 1 for all sets, we get the min-sum set cover problem, and (ii) when K(S) = |S| for all sets, we get the minimum-latency set cover problem. Constant factor approximations are known for both these problems. In their paper, Azar et al. considered the general problem and gave a logarithmic approximation algorithm for it. In this paper, we improve their result and give a simple randomized constant factor approximation algorithm for the generalized min-sum set cover problem.

Unidirectional graph-based wavelet transforms for efficient data gathering in sensor networks

Abstract: We design lifting-based wavelet transforms for any arbitrary communication graph in a wireless sensor network (WSN). Since transmitting raw data bits along the routing trees in WSN usually requires more bits than transmitting encoded data, we seek to minimize raw data transmissions in the network. We especially focus on unidirectional transforms which are computed as data is forwarded towards the sink on a routing tree. We formalize the problem of minimizing the number of raw data transmitting nodes as a weighted set cover problem and provide greedy approximations. We compare our method with existing distributed wavelet transforms on communication graphs. The results validate that our proposed transforms reduce the total energy consumption in the network with respect to existing designs.

Solving a bi-objective winner determination problem in a transportation procurement auction

Abstract: This paper introduces a bi-objective winner determination problem which arises in the procurement of transportation contracts via combinatorial auctions where bundle bidding is possible. The problem is modelled as a bi-objective extension to the set covering problem. We consider both the minimisation of the total procurement costs and the maximisation of the service-quality level at which the transportation contracts are executed. Taking into account the size of real-world transport auctions, a solution method has to cope with problems of up to some hundred contracts and a few thousand bundle bids. To solve the problem, we propose a bi-objective branch-and-bound algorithm and eight variants of a multiobjective genetic algorithm. Artificial benchmark instances that comply with important economic features of the transport domain are introduced to evaluate the methods. The branch-and-bound approach is able to find the optimal trade-off solutions in reasonable time for very small instances only. The eight variants of the genetic algorithm are compared among each other by means of large instances. The best variant is also evaluated using the small instances with known optimal solutions. The results indicate that the performance largely depends on the initialisation heuristic and suggest also that a well-balanced combination of genetic operators is crucial to obtain good solutions.

Exact solutions and bounds for general art gallery problems

Abstract: The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NP-hard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known. We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and---in case of convergence and integrality---ends with an optimal solution. We describe our implementation and give results for an assortment of polygons, including non-orthogonal polygons with holes.

Combination of Metaheuristic and Exact Algorithms for Solving Set Covering-Type Optimization Problems

Abstract: We propose a new generic framework for solving combinatorial optimization problems that can be modeled as a set covering problem. The proposed algorithmic framework combines metaheuristics with exact algorithms through a guiding mechanism based on diversification and intensification decisions. After presenting this generic framework, we extensively demonstrate its application to the vehicle routing problem with time windows. We then conduct a thorough computational study on a set of well-known test problems, where we show that the proposed approach not only finds solutions that are very close to the best-known solutions reported in the literature, but also improves them. We finally set up an experimental design to analyze the effects of different parameters used in the proposed algorithm.

PTAS for weighted set cover on unit squares

Abstract: We study the planar version of Minimum-Weight Set Cover, where one has to cover a given set of points with a minimum-weight subset of a given set of planar objects. For the unit-weight case, one PTAS (on disks) is known. For arbitrary weights however, the problem appears much harder, and in particular no PTASs are known. We present the first PTAS for Weighted Geometric Set Cover on planar objects, namely on axis-parallel unit squares. By extending the algorithm, we also obtain a PTAS for Minimum-Weight Dominating Set on intersection graphs of unit squares and Geometric Budgeted Maximum Coverage on unit squares. The running time of the developed algorithms is optimal under the exponential time hypothesis. We also show inapproximability results for Geometric Set Cover on various object shapes that are more general than unit squares.

Differentially Private Combinatorial Optimization

Abstract: Consider the following problem: given a metric space, some of whose points are ``clients,'' select a set of at most $k$ facility locations to minimize the average distance from the clients to their nearest facility. This is just the well-studied $k$-median problem, for which many approximation algorithms and hardness results are known. Note that the objective function encourages opening facilities in areas where there are many clients, and given a solution, it is often possible to get a good idea of where the clients are located. This raises the following quandary: what if the locations of the clients are sensitive information that we would like to keep private? emph{Is it even possible to design good algorithms for this problem that preserve the privacy of the clients?} In this paper, we initiate a systematic study of algorithms for discrete optimization problems in the framework of differential privacy (which formalizes the idea of protecting the privacy of individual input elements). We show that many such problems indeed have good approximation algorithms that preserve differential privacy; this is even in cases where it is impossible to preserve cryptographic definitions of privacy while computing any non-trivial approximation to even the emph{value} of an optimal solution, let alone the entire solution. Apart from the $k$-median problem, we consider the problems of vertex and set cover, min-cut, facility location, and Steiner tree, and give approximation algorithms and lower bounds for these problems. We also consider the recently introduced submodular maximization problem, ``Combinatorial Public Projects'' (CPP), shown by Papadimitriou et al. cite{PSS08} to be inapproximable to subpolynomial multiplicative factors by any efficient and emph{truthful} algorithm. We give a differentially private (and hence approximately truthful) algorithm that achieves a logarithmic additive approximation. Joint work with Anupam Gupta, Katrina Ligett, Frank McSherry and Aaron Roth.

Interactive Submodular Set Cover

Abstract: We introduce a natural generalization of submodular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.

Approximating minimum reset sequences

Abstract: We consider the problem of finding minimum reset sequences in synchronizing automata. The well-known Cerný conjecture states that every n-state synchronizing automaton has a reset sequence with length at most (n - 1)2. While this conjecture gives an upper bound on the length of every reset sequence, it does not directly address the problem of finding the shortest reset sequence. We call this the MINIMUM RESET SEQUENCE (MRS) problem. We give an O(kmnk + n4/k)-time ⌈n-1/k-1⌉-approximation for the MRS problem for any k ≥ 2. We also show that our analysis is tight. When k = 2 our algorithm reduces to Eppstein's algorithm and yields an (n-1)-approximation. When k = n our algorithm is the familiar exponential-time, exact algorithm. We define a nontrivial class of MRS which we call STACK COVER. We show that STACK COVER naturally generalizes two classic optimization problems: MIN SET COVER and SHORTEST COMMON SUPERSEQUENCE. Both these problems are known to be hard to approximate, although at present, SET COVER has a slightly stronger lower bound. In particular, it is NP-hard to approximate SET COVER to within a factor of c ċ log n for some c > 0. Thus, the MINIMUM RESET SEQUENCE problem is as least as hard to approximate as SET COVER. This improves the previous best lower bound which showed that it was NP-hard to approximate the MRS on binary alphabets to within any constant factor. Our result requires an alphabet of arbitrary size.

Approximation Analysis of Influence Spread in Social Networks

Abstract: In the context of influence propagation in a social graph, we can identify three orthogonal dimensions - the number of seed nodes activated at the beginning (known as budget), the expected number of activated nodes at the end of the propagation (known as expected spread or coverage), and the time taken for the propagation. We can constrain one or two of these and try to optimize the third. In their seminal paper (12), Kempe, Kleinberg and Tardos (KKT) constrained the budget, left time unconstrained, and maximized the coverage. In this paper, we study alternative optimization problems: MINSEED and MINTIME. In MINSEED, a coverage thresholdis given and the task is to find the minimum size seed set such that by activating it, at leastnodes are eventually activated in the expected sense. In MINTIME, a coverage thresholdand a budget threshold k are given, and the task is to find a seed set of size at most k such that by activating it, at leastnodes are activated in the expected sense, in the minimum possible time. It turns out both these problems are NP-hard. Given the hardness of the problems, we naturally turn to the subject of approximation algo- rithms. For MINSEED, we develop a bicriteria approximation by exploiting its relationship to the problem of Real-valued Submodular Set Cover (RSSC). We prove a generic inapproximabil- ity result for RSSC suggesting that improving this approximation factor is likely to be hard. For MINTIME we show that even bicriteria and tricriteria approximations are hard under several conditions. Our proof exploits the relationship between MINTIME and the problem of Robust Asymmetric k-center (RAKC). We show, however, that if we allow the budget for number of seeds k to be boosted by a logarithmic factor and allow the coverage to fall short, then the problem can be solved exactly in PTIME, i.e., we can achieve the required coverage within the time achieved by the optimal solution to MINTIME with budget k and coverage threshold �. Finally, we show the value of the approximation algorithms, by conducting an experimental comparison of their quality against that achieved by various heuristics.

Integrating facility location and production planning decisions

Abstract: We consider a metric uncapacitated facility location problem where we must assign each customer to a facility and meet the demand of the customer in future time periods through production and inventory decisions at the facility. We show that the problem, in general, is as hard to approximate as the set cover problem. We therefore focus on developing approximation algorithms for special cases of the problem. These special cases come in two forms: (i) specialize the production and inventory cost structure and (ii) specialize the demand pattern of the customers. In the former, we offer reductions to variants of the metric uncapacitated facility location problem that have been previously studied. The latter gives rise to a class of metric uncapacitated facility location problems where the facility cost function is concave in the amount of demand assigned to the facility. We develop a modified greedy algorithm together with the idea of cost-scaling to provide an algorithm for this class of problems with an approximation guarantee of 1.52. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Thresholded covering algorithms for robust and max-min optimization

Abstract: The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow and require coverage, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case covering cost (summed over both days) is minimized? We consider the k-robust model [6,15] where the possible scenarios tomorrow are given by all demand-subsets of size k. We present a simple and intuitive template for k-robust problems. This gives improved approximation algorithms for the k-robust Steiner tree and set cover problems, and the first approximation algorithms for k- robust Steiner forest, minimum-cut and multicut. As a by-product of our techniques, we also get approximation algorithms for k-max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".

ConsNet--A tabu search approach to the spatially coherent conservation area network design problem

Abstract: This paper presents a new approach to the solution of the well-studied conservation area network design problem (CANP), which is closely related to the classical set cover problem (SCP). The goal is to find the smallest amount of land that (when placed under conservation) will contain and protect a specified representation level of biodiversity resources. A new tabu search methodology is applied to an extension of the "basic" CANP which explicitly considers additional spatial requirements for improved conservation planning. The underlying search engine, modular adaptive self-learning tabu search (MASTS), incorporates state-of-the-art techniques including adaptive tabu search, dynamic neighborhood selection, and rule-based objectives. The ability to utilize intransitive orderings within a rule-based objective gives the search flexibility, improving solution quality while saving computation. This paper demonstrates how rule-based objectives can be used to design near optimal conservation area networks in which the individual conservation areas are well connected. The results represent a considerable improvement over classical techniques that do not consider spatial features. This paper provides an initial description of ConsNet, a comprehensive software package for systematic conservation planning.

A Novel Mask-Coding Representation for Set Cover Problems with Applications in Test Suite Minimisation

Abstract: Multi-Objective Set Cover problem forms the basis of many optimisation problems in software testing because the concept of code coverage is based on the set theory. This paper presents Mask-Coding, a novel representation of solutions for set cover optimisation problems that explores the problem space rather than the solution space. The new representation is empirically evaluated with set cover problems formulated from real code coverage data. The results show that Mask-Coding representation can improve both the convergence and diversity of the Pareto-efficient solution set of the multi-objective set cover optimisation.

Fast Discovery of Reliable Subnetworks

Abstract: We present a novel and efficient algorithm, Path Covering, for solving the most reliable subgraph problem. A reliable subgraph gives a concise summary of the connectivity between two given individuals in a social network. Formally, the given network is seen as a Bernoulli random graph G, and the objective is to find a subgraph H with at most B edges such that the probability that a path exists in H between the given two individuals is maximized. The algorithm is based on an efficient stochastic search of candidate paths, and the use of Monte-Carlo simulation to cast the problem as a set cover problem. Experimental evaluation on real graphs derived from DBLP bibliography database indicates superior performance of the proposed algorithm.

A Greedy Algorithm for Target Coverage Scheduling in Directional Sensor Networks.

Abstract: The wireless sensor networks have emerged as a promising tool for monitoring the physical world. Recently directional sensor networks (DSNs) consisting of directional sensors have gained attention. DSNs comprise a large number of sensors equipped with limited angles of sensing range and a limited battery. In DSNs, maximizing network lifetime while covering all the targets in a given area is still a challenge problem. A major technique to save the energy power of sensors is to use a node wake-up scheduling protocol by which some sensor nodes stay active to provide sensing service, while the others are inactive for conserving their energy. In this paper, we first address the MSCD (Maximum Set Cover for DSNs) problem that is known as NP-complete and then present a new target coverage scheduling scheme to solve this problem with a greedy algorithm. To verify and evaluate the proposed scheme, we conduct simulations and show that it can contribute to extending the network lifetime largely. By the simulations, we also present an energy-efficient strategy to choose a sensor in order to organize a scheduling set in the greedy scheme.

Using GIS and K = 3 Central Place Lattices for Efficient Solutions to the Location Set-Covering Problem in a Bounded Plane

Abstract: One of the simplest location models in terms of its constraint structure in locationallocation modeling is the location set-covering problem (LSCP). Although there have been a variety of geographic applications of the set-covering problem (SCP), the use of the SCP as a facility location model is one of the most common. In the early applications of the LSCP, both potential facility sites as well as demand were represented by points discretely located in geographic space. The advent of geographic information systems (GIS), however, has made possible a greater range of object representations that can reduce representation error. The purpose of this article is to outline a methodology using GIS and K = 3 central place lattices to solve the LSCP when demand is continuously distributed over a bounded area and potential facility sites have not been defined a priori. Although, demand is assumed to exist over an area, it is shown how area coverage can be accomplished by the coverage of a point pattern. Potential facility site distributions based on spacings that are powers of one-third the coverage distance are also shown to provide more efficient coverage than arbitrarily chosen spacings. Using GIS to make interactive adjustments to an incomplete coverage also provides an efficient alternative to smaller spacings between potential facility sites for reducing the number of facilities necessary for complete coverage.tgis_1199 331..350 Address for correspondence: Robert G Cromley, Department of Geography U-4148, 215 Glenbrook Road, University of Connecticut, Storrs, CT 06269-4148, USA. E-mail: robert.cromley@ uconn.edu Transactions in GIS, 2010, 14(3): 331–349 © 2010 Blackwell Publishing Ltd doi: 10.1111/j.1467-9671.2010.01199.x

Covering analysis of the greedy algorithm for partial cover

Abstract: The greedy algorithm is known to have a guaranteed approximation performance in many variations of the well-known minimum set cover problem. We analyze the number of elements covered by the greedy algorithm for the minimum set cover problem, when executed for k rounds. This analysis quite easily yields in the p-partial cover problem over a ground set of m elements the harmonic approximation guarantee H(⌈pm⌉) for the number of required covering sets. Thus, we tie together the coverage analysis of the greedy algorithm for minimum set cover and its dual problem partial cover.

Implicit hitting set problems and multi-genome alignment

Abstract: Let U be a finite set and S a family of subsets of U. Define a hitting set as a subset of U that intersects every element of S. The optimal hitting set problem is: given a positive weight for each element of U, find a hitting set of minimum total weight. This problem is equivalent to the classic weighted set cover problem.We consider the optimal hitting set problem in the case where the set system S is not explicitly given, but there is an oracle that will supply members of S satisfying certain conditions; for example, we might ask the oracle for a minimum-cardinality set in S that is disjoint from a given set Q. The problems of finding a minimum feedback arc set or minimum feedback vertex set in a digraph are examples of implicit hitting set problems. Our interest is in the number of oracle queries required to find an optimal hitting set. After presenting some generic algorithms for this problem we focus on our computational experience with an implicit hitting set problem related to multi-genome alignment in genomics. This is joint work with Erick Moreno Centeno.

The conditional covering problem on unweighted interval graphs

Abstract: The conditional covering problem is an important variation of well studied set covering problem. In the set covering problem, the problem is to find a minimum cardinality vertex set which will cover all the given demand points. The conditional covering problem asks to find a minimum cardinality vertex set that will cover not only the given demand points but also one another. This problem is NP-complete for general graphs. In this paper, we present an efficient algorithm to solve the conditional covering problem on interval graphs with n vertices which runs in O(n)time.

The checkpoint problem

Abstract: In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of source-destination pairs {(s1, t1), ..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E′; namely, a set of edges whose removal disconnects every source-destination pair. For each p ∈ P we define cpE′(p) = |p ∩ E′|. In the sum checkpoint (SCP) problem the goal is to minimize Σp∈P cpE′(p), while in the maximum checkpoint (MCP) problem the goal is to minimize maxp∈p cpE′(p). These problem have several natural applications, e.g., in urban transportation and network security. In a sense, they combine the multicut problem and the minimum membership set cover problem. For the sum objective we show that weighted SCP is equivalent, with respect to approximability, to undirected multicut. Thus there exists an O(log n) approximation for SCP in general graphs. Our current approximability results for the max objective have a wide gap: we provide an approximation factor of O(√nlogn/opt) for MCP and a hardness of 2 under the assumption P ≠ NP. The hardness holds for trees, in which case we can obtain an asymptotic approximation factor of 2. Finally we show strong hardness for the well-known problem of finding a path with minimum forbidden pairs, which in a sense can be considered the dual to the checkpoint problem. Despite various works on this problem, hardness of approximation was not known prior to this work. We show that the problem cannot be approximated within cn for some constant c > 0, unless P = NP. This is the strongest type of hardness possible. It carries over to directed acyclic graphs and is a huge improvement over the plain NP-hardness of Gabow (SIAM J. Comp 2007, pages 1648-1671).