Analysis of greedy approximations with nonsubmodular potential functions

In this paper, we present two techniques to analyze greedy approximation with nonsubmodular functions restricted submodularity and shifted submodularity. As an application of the restricted submodularity, we present a worst-case analysis of a greedy algorithm for Network Steiner tree adapted from a heuristic originally proposed by Chang in 1972 for Euclidean Steiner tree. The performance ratio of Chang's heuristic is a longstanding open problem due to the nonsubmodularity of its potential function. As an application of the shifted submodularity, we present a worst-case analysis of a greedy algorithm for Connected Dominating Set generalized from a greedy algorithm proposed by Ruan et al. Such generalized greedy algorithm is shown to have performance ratio at most (1 + )(1 + ln( - 1)), which matches the well-known lower bound (1-) ln , where is the maximum vertex-degree of input graph and is any positive constant.

• Type: Conference
• Year: 2008
• Where: SODA
• Authors: Ding-Zhu Du, Ronald L. Graham, Panos M. Pardalos, Peng-Jun Wan, Weili Wu, Wenbo Zhao
• Keywords: Algorithms, Greedy Algorithm, Greedy Approximation, SODA 2008, Worst-case Analysis