TY - JOUR

T1 - Analyzing the optimal neighborhood

T2 - Algorithms for partial and budgeted connected dominating set problems

AU - Khuller, Samir

AU - Purohit, Manish

AU - Sarpatwar, Kanthi K.

N1 - Publisher Copyright:
© 2020 Society for Industrial and Applied Mathematics.

PY - 2020

Y1 - 2020

N2 - We study partial and budgeted versions of the well-studied connected dominating set problem. In the partial connected dominating set (PCDS) problem, we are given an undirected graph G = (V;E) and an integer n0, and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n0 vertices. We obtain the first polynomial time algorithm with an O(ln Δ) approximation guarantee for this problem, thereby significantly extending the results of Guha and Khuller [Algorithmica, 20(1998), pp. 374-387] 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, 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 12 (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 generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies an O(ln q) approximation (where q denotes the quota) and 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. Finally, we prove that (both edge and node) weighted versions of the PCDS problem are as hard as the more general group Steiner tree problem.

AB - We study partial and budgeted versions of the well-studied connected dominating set problem. In the partial connected dominating set (PCDS) problem, we are given an undirected graph G = (V;E) and an integer n0, and the goal is to find a minimum subset of vertices that induces a connected subgraph of G and dominates at least n0 vertices. We obtain the first polynomial time algorithm with an O(ln Δ) approximation guarantee for this problem, thereby significantly extending the results of Guha and Khuller [Algorithmica, 20(1998), pp. 374-387] 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, 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 12 (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 generalization captures the connected dominating set problem with capacities and/or weighted profits as special cases. This implies an O(ln q) approximation (where q denotes the quota) and 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. Finally, we prove that (both edge and node) weighted versions of the PCDS problem are as hard as the more general group Steiner tree problem.

KW - Approximation algorithms

KW - Connected dominating set

KW - Partial and budgeted connected dominating set

KW - Sub-modular optimization

UR - http://www.scopus.com/inward/record.url?scp=85079753572&partnerID=8YFLogxK

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U2 - 10.1137/18M1212094

DO - 10.1137/18M1212094

M3 - Article

AN - SCOPUS:85079753572

VL - 34

SP - 251

EP - 270

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -