Integer linear programming formulations for double roman domination problem

ABSTRACT

For a graph $G=(V,E)$, a double Roman dominating function (DRDF) is a function $f:V\to \{0,1,2,3\}$ having the property that if $f(v)=0$, then vertex v must have at least two neighbours assigned 2 under f or at least one neighbour u with $f(u)=3$, and if $f(v)=1$, then vertex v must have at least one neighbour u with $f(u)\ge 2$. In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF f such that $\sum _{v\in V}f(v)$ 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 $H(2(\mathrm{\Delta }+1))$-approximation algorithm. All proposed formulations and approximation algorithm are evaluated on randomly generated graphs to compare the performance.

Keywords:

• Double roman domination
• integer linear programming
• approximation algorithm

2010 Mathematics Subject Classifications:

• 90C10
• 68W25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

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).

Notes on contributors

Qingqiong Cai

Qingqiong Cai, a lecturer at College of Computer Science, Nankai University, has research interests in graph theory and combinatorial optimization.

Neng Fan

Neng Fan, is an associate professor at Department of Systems and Industrial Engineering at the University of Arizona (UA), Tucson, Arizona. His research focuses on development of various optimization methodologies, and their applications in energy systems, healthcare, sustainable agriculture and data analytics.

Yongtang Shi

Yongtang Shi, a full Professor at Center for Combinatorics, Nankai University, has research interests in graph Theory and its applications and combinatorial optimization.

Shunyu Yao

Shunyu Yao, will obtain his master degree in Mathematics from Center for Combinatorics, Nankai University in 2020.