TY - GEN
T1 - An efficient branch-and-cut algorithm for approximately submodular function maximization
AU - Uematsu, Naoya
AU - Umetani, Shunji
AU - Kawahara, Yoshinobu
N1 - Publisher Copyright:
© 2019 IEEE.
PY - 2019/10
Y1 - 2019/10
N2 - When approaching problems in computer science, we often encounter situations where a subset of a finite set maximizing some utility function needs to be selected. Some of such utility functions are known to be approximately submodular. For the problem of maximizing an approximately submodular function (ASFM problem), a greedy algorithm quickly finds good feasible solutions for many instances while guaranteeing (1-e{-gamma})-approximation ratio for a given submodular ratio gamma. However, we still encounter its applications that ask more accurate or exactly optimal solutions within a reasonable computation time. In this paper, we present an efficient branch-and-cut algorithm for the non-decreasing ASFM problem based on its binary integer programming (BIP) formulation with an exponential number of constraints. To this end, we first derive a BIP formulation of the ASFM problem, and then we develop an improved constraint generation algorithm that starts from a reduced BIP problem with a small subset of constraints and repeats solving the reduced BIP problem while adding a promising set of constraints at each iteration. Moreover, we incorporate it into a branch-and-cut algorithm to attain good upper bounds while solving a smaller number of nodes of a search tree. The computational results for three types of well-known benchmark instances show that our algorithm performs better than the conventional exact algorithms.
AB - When approaching problems in computer science, we often encounter situations where a subset of a finite set maximizing some utility function needs to be selected. Some of such utility functions are known to be approximately submodular. For the problem of maximizing an approximately submodular function (ASFM problem), a greedy algorithm quickly finds good feasible solutions for many instances while guaranteeing (1-e{-gamma})-approximation ratio for a given submodular ratio gamma. However, we still encounter its applications that ask more accurate or exactly optimal solutions within a reasonable computation time. In this paper, we present an efficient branch-and-cut algorithm for the non-decreasing ASFM problem based on its binary integer programming (BIP) formulation with an exponential number of constraints. To this end, we first derive a BIP formulation of the ASFM problem, and then we develop an improved constraint generation algorithm that starts from a reduced BIP problem with a small subset of constraints and repeats solving the reduced BIP problem while adding a promising set of constraints at each iteration. Moreover, we incorporate it into a branch-and-cut algorithm to attain good upper bounds while solving a smaller number of nodes of a search tree. The computational results for three types of well-known benchmark instances show that our algorithm performs better than the conventional exact algorithms.
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U2 - 10.1109/SMC.2019.8913989
DO - 10.1109/SMC.2019.8913989
M3 - Conference contribution
AN - SCOPUS:85076726900
T3 - Conference Proceedings - IEEE International Conference on Systems, Man and Cybernetics
SP - 3160
EP - 3167
BT - 2019 IEEE International Conference on Systems, Man and Cybernetics, SMC 2019
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 2019 IEEE International Conference on Systems, Man and Cybernetics, SMC 2019
Y2 - 6 October 2019 through 9 October 2019
ER -