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Algorithms and Data Structures

Set Covering with Our Eyes Closed

Profile image of Fabrizio GrandoniFabrizio Grandoni

2013, SIAM Journal on Computing

Abstract

Given a universe U of n elements and a weighted collection S of m subsets of U , the universal set cover problem is to a-priori map each element u ∈ U to a set S(u) ∈ S containing u, such that any set X ⊆ U is covered by S(X) = ∪ u∈X S(u). The aim is to find a mapping such that the cost of S(X) is as close as possible to the optimal set-cover cost for X. (Such problems are also called oblivious or a-priori optimization problems.) Unfortunately, for every universal mapping, the cost of S(X) can be Ω(√ n) times larger than optimal if the set X is adversarially chosen. In this paper we study the performance on average, when X is a set of randomly chosen elements from the universe: we show how to efficiently find a universal map whose expected cost is O(log mn) times the expected optimal cost. In fact, we give a slightly improved analysis and show that this is the best possible. We generalize these ideas to weighted set cover and show similar guarantees to (non-metric) facility location, where we have to balance the facility opening cost with the cost of connecting clients to the facilities. We show applications of our results to universal multi-cut and disc-covering problems, and show how all these universal mappings give us algorithms for the stochastic online variants of the problems with the same competitive factors.

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In this article, we consider a situation in which a group of facilities need to be constructed in order to serve a given set of customers. However, the facilities cannot guarantee an absolute coverage to any of the customers. Hence, we formulate this problem as one of maximizing the total service reliability of the system subject to a budgetary constraint. For this problem, we develop and test suitable branch-and-bound algorithms and study the effect of problem parameters on solution difficulty. Some generalizations of this problem are also mentioned as possible extensions.

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The Covert Set-Cover Problem with Application to Network Discovery
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Lecture Notes in Computer Science, 2010

We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OP T within O(log N ) factor with high probability using O(OP T • log 2 N ) queries where N is the number of element in the universal set. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log 2 n)competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω( √ n log n) and therefore our result achieves an exponential improvement.

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Given a finite set E and a family F ={E 1 ,. . . ,E m } of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.

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The Covert Set Cover Problem with Applications to Network Discovery§
Sandeep Sen

Proceedings of the Indian National Science Academy, 2014

We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OP T within O(log N ) factor with high probability using O(OP T •log 2 N ) queries where N is the input size. We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log 2 n)-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm has a competitive ratio of Ω( √ n log n) and therefore our result achieves an exponential improvement. * A Preliminary version of the results have appeared earlier in the 4th Workshop on Algorithms and Computation 2010 1 We have chosen n ′ , m ′ as notations to keep them distinct from graphs with n vertices and m edges.

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Taming the set covering problem: the value of dual information
S. Ilker Birbil

2010

We offer a fresh perspective on solving the set covering problem to near optimality with off-the-shelf methods. We formulate minimizing the gap of a generic primal-dual heuristic for the set covering problem as an integer program and analyze its performance. The empirical insights from this analysis lead to a simple and powerful primal-dual approach for solving the set covering problem to near optimality with a state-of-the-art standard solver.

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Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems with Applications
Piotr Berman

arXiv (Cornell University), 2009

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We study the following on-line model for set-covering: elements of a ground set of size n arrive one-by-one and with any such element ci, arrives also the name of some set Si0 containing ci and covering the most of the uncovered ground set-elements (obviously, these elements have not been yet revealed). For this model we analyze a simple greedy algorithm consisting of taking Si0 into the cover, only if ci is not already covered. We prove that the competitive ratio of this algorithm is p n and that it is asymptotically optimal for the model dealt, since no on-line algorithm can do better than p n=2. We next show that this model can also be used for solving minimum dominating set with competitive ratio bounded above by the square root of the size of the input graph. We nally deal with the maximum budget saving problem. Here, an initial budget is allotted that is destined to cover the cost of an algorithm for solving set-covering. The objective is to maximize the savings on the initial...

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Allocating costs in set covering problems
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  • Mathematics
  • Computer Science
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  • Approximation Algorithms
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  • Online Algorithms
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