Abstract
An instance of the generalized partial cover problem consists of a ground set U and a family of subsets \({\cal S} \subseteq 2^U\). Each element e ∈U is associated with a profit p(e), whereas each subset \(S \in {\cal S}\) has a cost c(S). The objective is to find a minimum cost subcollection \({\cal S}' \subseteq {\cal S}\) such that the combined profit of the elements covered by \({\cal S}'\) is at least P, a specified profit bound. In the prize-collecting version of this problem, there is no strict requirement to cover any element; however, if the subsets we pick leave an element e ∈U uncovered, we incur a penalty of π(e). The goal is to identify a subcollection \({\cal S}' \subseteq {\cal S}\) that minimizes the cost of \({\cal S}'\) plus the penalties of uncovered elements.
Although problem-specific connections between the partial cover and the prize-collecting variants of a given covering problem have been explored and exploited, a more general connection remained open. The main contribution of this paper is to establish a formal relationship between these two variants. As a result, we present a unified framework for approximating problems that can be formulated or interpreted as special cases of generalized partial cover. We demonstrate the applicability of our method on a diverse collection of covering problems, for some of which we obtain the first non-trivial approximability results.
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References
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: A general approach to online network optimization problems. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 577–586 (2004)
Balas, E., Padberg, M.: Set partitioning: A survey. SIAM Review 18(4), 710–760 (1976)
Bar-Yehuda, R.: Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms 39(2), 137–144 (2001)
Bshouty, N.H., Burroughs, L.: Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science, pp. 298–308 (1998)
Chawla, S., Krauthgamer, R., Kumar, R., Rabani, Y., Sivakumar, D.: On the hardness of approximating multicut and sparsest-cut. In: Proceedings of the 20th Annual IEEE Conference on Computational Complexity, pp. 144–153 (2005)
Chudak, F.A., Roughgarden, T., Williamson, D.P.: Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation. Mathematical Programming 100(2), 411–421 (2004)
Cook, W.J., Cunningham, W.H., Pulleyblank, W.R., Schrijver, A.: Combinatorial Optimization. John Wiley and Sons, New York (1997)
Edmonds, J., Johnson, E.L.: Matching: A well-solved class of integer linear programs. In: Combinatorial Structures and their Applications, pp. 89–92. Gordon and Breach, New York (1970)
Even, G., Feldman, J., Kortsarz, G., Nutov, Z.: A 3/2-approximation algorithm for augmenting the edge-connectivity of a graph from 1 to 2 using a subset of a given edge set. In: Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 90–101 (2001)
Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)
Fujito, T.: On approximation of the submodular set cover problem. Operations Research Letters 25(4), 169–174 (1999)
Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. Journal of Algorithms 53(1), 55–84 (2004)
Garfinkel, R.S., Nemhauser, G.L.: Optimal set covering: A survey. In: Geoffrion, A.M. (ed.) Perspectives on Optimization, pp. 164–183 (1972)
Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing 25(2), 235–251 (1996)
Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)
Gaur, D.R., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. Journal of Algorithms 43(1), 138–152 (2002)
Golovin, D., Nagarajan, V., Singh, M.: Approximating the k-multicut problem. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 621–630 (2006)
Hochbaum, D.S.: Approximating covering and packing problems: Set cover, vertex cover, independent set, and related problems. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problems, ch. 3, pp. 94–143. PWS Publishing Company (1997)
Hochbaum, D.S.: The t-vertex cover problem: Extending the half integrality framework with budget constraints. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 111–122. Springer, Heidelberg (1998)
Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9(3), 256–278 (1974)
Kearns, M.J.: The Computational Complexity of Machine Learning. MIT Press, Cambridge (1990)
Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 767–775 (2002)
Könemann, J., Parekh, O., Segev, D.: A unified approach to approximating partial covering problems (2006), available at: http://www.math.tau.ac.il/~segevd
Levin, A., Segev, D.: Partial multicuts in trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 320–333. Springer, Heidelberg (2006)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)
Murty, K.G., Perin, C.: A 1-matching blossom type algorithm for edge covering problems. Networks 12, 379–391 (1982)
Nagamochi, H.: An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree. Discrete Applied Mathematics 126(1), 83–113 (2003)
Padberg, M.W.: Covering, packing and knapsack problems. Annals of Discrete Mathematics 4, 265–287 (1979)
Parekh, O.: Polyhedral Techniques for Graphic Covering Problems. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University (2002)
Plesník, J.: Constrained weighted matchings and edge coverings in graphs. Discrete Applied Mathematics 92(2–3), 229–241 (1999)
Räcke, H.: Minimizing congestion in general networks. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pp. 43–52 (2002)
Slavík, P.: Improved performance of the greedy algorithm for partial cover. Information Processing Letters 64(5), 251–254 (1997)
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Könemann, J., Parekh, O., Segev, D. (2006). A Unified Approach to Approximating Partial Covering Problems. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_43
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