TY - GEN

T1 - Semi-streaming set cover

AU - Emek, Yuval

AU - Rosén, Adi

PY - 2014

Y1 - 2014

N2 - This paper studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph G = (V, E) whose edges arrive one-by-one and the goal is to construct an edge cover F ⊆ E with the objective of minimizing the cardinality (or cost in the weighted case) of F. We consider a parameterized relaxation of this problem, where given some 0 ≤ ε < 1, the goal is to construct an edge (1-ε)-cover, namely, a subset of edges incident to all but an ε-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight trade-off between ε and the approximation ratio: We design a semi-streaming algorithm that on input graph G, constructs a succinct data structure D such that for every 0 ≤ ε < 1, an edge (1 - ε)-cover that approximates the optimal edge (1-)cover within a factor of f(ε, n) can be extracted from D (efficiently and with no additional space requirements), where (Equation Presented) In particular for the traditional set cover problem we obtain an O(√n)-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by ε) of matching lower bounds.

AB - This paper studies the set cover problem under the semi-streaming model. The underlying set system is formalized in terms of a hypergraph G = (V, E) whose edges arrive one-by-one and the goal is to construct an edge cover F ⊆ E with the objective of minimizing the cardinality (or cost in the weighted case) of F. We consider a parameterized relaxation of this problem, where given some 0 ≤ ε < 1, the goal is to construct an edge (1-ε)-cover, namely, a subset of edges incident to all but an ε-fraction of the vertices (or their benefit in the weighted case). The key limitation imposed on the algorithm is that its space is limited to (poly)logarithmically many bits per vertex. Our main result is an asymptotically tight trade-off between ε and the approximation ratio: We design a semi-streaming algorithm that on input graph G, constructs a succinct data structure D such that for every 0 ≤ ε < 1, an edge (1 - ε)-cover that approximates the optimal edge (1-)cover within a factor of f(ε, n) can be extracted from D (efficiently and with no additional space requirements), where (Equation Presented) In particular for the traditional set cover problem we obtain an O(√n)-approximation. This algorithm is proved to be best possible by establishing a family (parameterized by ε) of matching lower bounds.

UR - http://www.scopus.com/inward/record.url?scp=84904196917&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-43948-7_38

DO - 10.1007/978-3-662-43948-7_38

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AN - SCOPUS:84904196917

SN - 9783662439470

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 453

EP - 464

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

T2 - 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014

Y2 - 8 July 2014 through 11 July 2014

ER -