Approximation Algorithm for Vertex Cover with Multiple Covering Constraints
We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G=(V,E) with a maximum edge size f, a cost function w: V - > Z^+, and edge subsets P_1,P_2,...,P_r of E along with covering requirements k_1,k_2,...,k_r for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset P_i, at least k_i edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al.
We present a primal-dual algorithm yielding an (f * H_r + H_r)-approximation for this problem, where H_r is the r^{th} harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality.
Vertex cover
multiple cover constraints
Approximation algorithm
Mathematics of computing~Approximation algorithms
43:1-43:11
Regular Paper
This work is supported in part by Ministry of Science and Technology (MOST), Taiwan, under Grants MOST107-2218-E-194-015-MY3 and MOST106-2221-E-001-006-MY3.
Eunpyeong
Hong
Eunpyeong Hong
Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan
Mong-Jen
Kao
Mong-Jen Kao
Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan
10.4230/LIPIcs.ISAAC.2018.43
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Eunpyeong Hong and Mong-Jen Kao
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