Abstract
An accelerated continuous greedy algorithm is proposed for maximization of a special class of non-decreasing submodular functions \(f:2^{X} \rightarrow \mathfrak {R}_{+}\) subject to a matroid constraint with a \(\frac{1}{c} (1 - e^{-c} - \varepsilon ) \) approximation for any \(\varepsilon > 0\), where \(c\) is the curvature with respect to the optimum. Functions in the special class of submodular functions satisfy the criterion \(\forall A, B \subseteq X,\, \forall j \in X {\setminus } (A \cup B)\), \(\triangle f_j(A,B) \mathop {=}\limits ^{\Delta } f(A \cup \{j\}) + f(B \cup \{j\}) - f((A \cap B) \cup \{j\}) - f(A \cup B \cup \{j\}) - [f(A) + f(B) - f(A \cap B) - f(A \cup B)] \le 0\). As an alternative to the standard continuous greedy algorithm, the proposed algorithm can substantially reduce the computational expense by removing redundant computational steps and, therefore, is able to efficiently handle the maximization problems for this special class of submodular functions. Examples of such functions are presented.
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Notes
A polytope \(P\) is down-monotone if for all \(\varvec{x},\varvec{y} \in [0,1]^X\) satisfying \(\varvec{x} \preceq \varvec{y}\) and \(\varvec{y} \in P\), it holds that \(\varvec{x} \in P\); \(P\) is solvable if we can maximize linear functions over \(P\) in polynomial time (Dughmi 2009)
An independent set \(Q_i = \{s_1, \ldots ,s_{i-1}, r_i,s_{i+1},\ldots ,s_q \}\) is submax-weight, if for all \(r_i^\prime \in X {\setminus } Q \setminus \{r_i\}\), \(\nu ^{\prime \prime } \cdot \nabla F \le \nu ^\prime \cdot \nabla F < \nu \cdot \nabla F\), where \(\nu = \mathbf 1 _Q, \nu ^{\prime } = \mathbf 1 _{Q_i}, \nu ^{\prime \prime } = \mathbf 1 _{Q_i^{\prime } = \{s_1, \ldots ,s_{i-1}, r_i^{\prime }, s_{i+1},\ldots ,s_q \} }\)
This proof was kindly provided by a referee.
This simplified proof was kindly provided by a referee.
The idea of this proof was kindly provided by a referee.
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Acknowledgments
The first author is thankful for having communicated with Dr Jan Vondrák to discuss the continuous greedy algorithm. The authors also wish to acknowledge the anonymous reviwers/referees for their constructive comments and suggestions for this work. This work was supported in part by the NSFC (No.61135001) and the AFOSR Grant (FA2386-13-1-4080).
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Wang, Z., Moran, B., Wang, X. et al. An accelerated continuous greedy algorithm for maximizing strong submodular functions. J Comb Optim 30, 1107–1124 (2015). https://doi.org/10.1007/s10878-013-9685-x
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DOI: https://doi.org/10.1007/s10878-013-9685-x