TY - JOUR

T1 - Integer linear programming formulations for double roman domination problem

AU - Cai, Qingqiong

AU - Fan, Neng

AU - Shi, Yongtang

AU - Yao, Shunyu

N1 - Funding Information:
Q. Cai is partially supported by National Natural Science Foundation of China (No. 11701297), Natural Science Foundation of Tianjin (No. 19JCQNJC14400) and Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018005). Y. Shi and S. Yao are partially supported by National Natural Science Foundation of China (No. 11771221 and 11811540390), Natural Science Foundation of Tianjin, China (No. 17JCQNJC00300) and China-Slovenia bilateral project ‘Some topics in modern graph theory’ (No. 12-6).
Publisher Copyright:
© 2019 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2022

Y1 - 2022

N2 - For a graph (Formula presented.), a double Roman dominating function (DRDF) is a function (Formula presented.) having the property that if (Formula presented.), then vertex v must have at least two neighbours assigned 2 under f or at least one neighbour u with (Formula presented.), and if (Formula presented.), then vertex v must have at least one neighbour u with (Formula presented.). In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF f such that (Formula presented.) is minimum. We propose five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations. Further, we prove that the first four models indeed solve the double Roman domination problem, and the last two models are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an (Formula presented.) -approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

AB - For a graph (Formula presented.), a double Roman dominating function (DRDF) is a function (Formula presented.) having the property that if (Formula presented.), then vertex v must have at least two neighbours assigned 2 under f or at least one neighbour u with (Formula presented.), and if (Formula presented.), then vertex v must have at least one neighbour u with (Formula presented.). In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF f such that (Formula presented.) is minimum. We propose five integer linear programming (ILP) formulations and one mixed integer linear programming formulation with polynomial number of constraints for this problem. Some additional valid inequalities and bounds are also proposed for some of these formulations. Further, we prove that the first four models indeed solve the double Roman domination problem, and the last two models are equivalent to the others regardless of the variable relaxation or usage of a smaller number of constraints and variables. Additionally, we use one ILP formulation to give an (Formula presented.) -approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

KW - Double roman domination

KW - approximation algorithm

KW - integer linear programming

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U2 - 10.1080/10556788.2019.1679142

DO - 10.1080/10556788.2019.1679142

M3 - Article

AN - SCOPUS:85074501881

VL - 37

SP - 1

EP - 22

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 1

ER -