TY - GEN

T1 - Analyzing the optimal neighborhood

T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

AU - Khuller, Samir

AU - Purohit, Manish

AU - Sarpatwar, Kanthi K.

PY - 2014

Y1 - 2014

N2 - We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem (PCDS), we are given an undirected graph G = (V,E) and an integer n', and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n' vertices. We obtain the first polynomial time algorithm with an O(ln δ) approximation factor for this problem, thereby significantly extending the results of Guha and Khuller (Algorithmica, Vol. 20(4), Pages 374-387, 1998) for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem (BCDS), there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a -1/13(1 - 1/e) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a "special" submodular function. This general-ization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies a O(lnq) approximation (where q denotes the quota) and an O( 1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function.

AB - We study partial and budgeted versions of the well studied connected dominating set problem. In the partial connected dominating set problem (PCDS), we are given an undirected graph G = (V,E) and an integer n', and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n' vertices. We obtain the first polynomial time algorithm with an O(ln δ) approximation factor for this problem, thereby significantly extending the results of Guha and Khuller (Algorithmica, Vol. 20(4), Pages 374-387, 1998) for the connected dominating set problem. We note that none of the methods developed earlier can be applied directly to solve this problem. In the budgeted connected dominating set problem (BCDS), there is a budget on the number of vertices we can select, and the goal is to dominate as many vertices as possible. We obtain a -1/13(1 - 1/e) approximation algorithm for this problem. Finally, we show that our techniques extend to a more general setting where the profit function associated with a subset of vertices is a "special" submodular function. This general-ization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies a O(lnq) approximation (where q denotes the quota) and an O( 1) approximation algorithms for the partial and budgeted versions of these problems. While the algorithms are simple, the results make a surprising use of the greedy set cover framework in defining a useful profit function.

UR - http://www.scopus.com/inward/record.url?scp=84902095958&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84902095958&partnerID=8YFLogxK

U2 - 10.1137/1.9781611973402.123

DO - 10.1137/1.9781611973402.123

M3 - Conference contribution

AN - SCOPUS:84902095958

SN - 9781611973389

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1702

EP - 1713

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery

Y2 - 5 January 2014 through 7 January 2014

ER -