Analysis of greedy approximations with nonsubmodular potential functions

Authors:
Ding-Zhu Du;Ronald L. Graham;Panos M. Pardalos;Peng-Jun Wan;Weili Wu;Wenbo Zhao
Affiliations:
University of Texas at Dallas, Richardson, TX and Xi'an Jiaotong University, Xi'an, China;University of California at San Diego, La Jolla, CA;University of Florida, Gainsville, FL;Illinois Institute of Technology, Chicago, IL;University of Texas at Dallas, Richardson, TX;University of California at San Diego La Jolla, CA
Venue:
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2008

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Abstract

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 + ε)(1 + ln(Δ - 1)), which matches the well-known lower bound (1-ε)ln Δ, where Δ is the maximum vertex-degree of input graph and ε is any positive constant.