Set cover problem

About: Set cover problem is a research topic. Over the lifetime, 1470 publications have been published within this topic receiving 33905 citations. The topic is also known as: set covering.

Reducibility Among Combinatorial Problems

Abstract: A large class of computational problems involve the determination of properties of graphs, digraphs, integers, arrays of integers, finite families of finite sets, boolean formulas and elements of other countable domains. Through simple encodings from such domains into the set of words over a finite alphabet these problems can be converted into language recognition problems, and we can inquire into their computational complexity. It is reasonable to consider such a problem satisfactorily solved when an algorithm for its solution is found which terminates within a number of steps bounded by a polynomial in the length of the input. Many problems with wide applicability – e.g., set cover, knapsack, hitting set, max cut, and satisfiability – lack a polynomial algorithm for solving them, but also lack a proof that no such polynomial algorithm exists. Hence, they remain “open problems.” This paper references the recent work, “On the Reducibility of Combinatorial Problems” [1]. BODY A large class of open problems are mutually convertible via poly-time reduc- tions. Hence, either all can be solved in poly-time, or none can. REFERENCES [1] R. Karp. Reducibility Among Combinatorial Problems. In Complexity of Computer Computations, 1972. ∗With apologies to Professor Richard Karp. Volume X of Tiny Transactions on Computer Science This content is released under the Creative Commons Attribution-NonCommercial ShareAlike License. Permission to make digital or hard copies of all or part of this work is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. CC BY-NC-SA 3.0: http://creativecommons.org/licenses/by-nc-sa/3.0/.

Greedy strikes back: improved facility location algorithms

Abstract: A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as theuncapacitated facility locationproblem. Application to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser, and Wolsey. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos, and Aardal. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos, and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is max SNP-hard. However, the inapproximability constants derived from the max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio, assumingNP?DTIMEnO(loglogn)].

Almost optimal set covers in finite VC-dimension

Abstract: We give a deterministic polynomial-time method for finding a set cover in a set system (X, ?) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal size,c. For constant VC-dimensional set systems, which are common in computational geometry, our method gives anO(logc) approximation factor. This improves the previous ?(log?X?) bound of the greedy method and challenges recent complexity-theoretic lower bounds for set covers (which do not make any assumptions about the VC-dimension). We give several applications of our method to computational geometry, and we show that in some cases, such as those arising in three-dimensional polytope approximation and two-dimensional disk covering, we can quickly findO(c)-sized covers.