TY - GEN

T1 - Hitting and covering partially

AU - Agrawal, Akanksha

AU - Choudhary, Pratibha

AU - Jain, Pallavi

AU - Kanesh, Lawqueen

AU - Sahlot, Vibha

AU - Saurabh, Saket

N1 - Funding Information:
P. Jain—The author acknowledges DST, India for SERB-NPDF fellowship [PDF/2016/003508].
Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - d-Hitting Set and d-Set Cover are among the classical NP-hard problems. In this paper, we study variants of d-Hitting Set and d-Set Cover, which are called Partial d -Hitting Set (Partial d -HS) and Partial d -Exact Set Cover (Partial d -Exact SC), respectively. In Partial d -HS, given a universe U, a family F, of sets of size at most d over U, and integers k and t, the objective is to decide if there exists a (Formula Presented) of size at most k such that S intersects with at least t sets in F. We obtain a kernel for Partial d -HS in which the size of the universe is bounded by O(dt) and the size of the family is bounded by O(dt2). Using this result, we obtain a kernel for Partial Vertex Cover (PVC) with O(t) vertices, where t is the number of edges to be covered. Next, we study the Partial d -Exact SC problem, where, given a universe U, a family F, of sets of size exactly d over U, and integers k and t, the objective is to decide if there is (Formula Presented) of size at most k, such that S covers at least t elements in U. We design a kernel for Partial d -Exact SC in which sizes of the universe and the family are bounded by (Formula Presented). Finally, we study a special case of Partial d -HS, when d=2, and design an exact exponential time algorithm with running time (Formula Presented).

AB - d-Hitting Set and d-Set Cover are among the classical NP-hard problems. In this paper, we study variants of d-Hitting Set and d-Set Cover, which are called Partial d -Hitting Set (Partial d -HS) and Partial d -Exact Set Cover (Partial d -Exact SC), respectively. In Partial d -HS, given a universe U, a family F, of sets of size at most d over U, and integers k and t, the objective is to decide if there exists a (Formula Presented) of size at most k such that S intersects with at least t sets in F. We obtain a kernel for Partial d -HS in which the size of the universe is bounded by O(dt) and the size of the family is bounded by O(dt2). Using this result, we obtain a kernel for Partial Vertex Cover (PVC) with O(t) vertices, where t is the number of edges to be covered. Next, we study the Partial d -Exact SC problem, where, given a universe U, a family F, of sets of size exactly d over U, and integers k and t, the objective is to decide if there is (Formula Presented) of size at most k, such that S covers at least t elements in U. We design a kernel for Partial d -Exact SC in which sizes of the universe and the family are bounded by (Formula Presented). Finally, we study a special case of Partial d -HS, when d=2, and design an exact exponential time algorithm with running time (Formula Presented).

KW - Exact algorithm

KW - Kernel

KW - Partial Vertex Cover

KW - Partial d-Hitting Set

KW - Partial d-Set Cover

KW - k-Maximum Coverage

UR - http://www.scopus.com/inward/record.url?scp=85049665790&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-94776-1_62

DO - 10.1007/978-3-319-94776-1_62

M3 - Conference contribution

AN - SCOPUS:85049665790

SN - 9783319947754

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

SP - 751

EP - 763

BT - Computing and Combinatorics - 24th International Conference, COCOON 2018, Proceedings

A2 - Zhu, Daming

A2 - Wang, Lusheng

PB - Springer Verlag

T2 - 24th International Conference on Computing and Combinatorics Conference, COCOON 2018

Y2 - 2 July 2018 through 4 July 2018

ER -