TY - GEN

T1 - An efficient branch-and-cut algorithm for approximately submodular function maximization

AU - Uematsu, Naoya

AU - Umetani, Shunji

AU - Kawahara, Yoshinobu

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.

UR - http://www.scopus.com/inward/record.url?scp=85076726900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85076726900&partnerID=8YFLogxK

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 -