[Paper Review] An Integer Linear Programming Formulation for the Convex Dominating Set Problems

Jozef Kratica, Vladimir Filipović+2 more|arXiv (Cornell University)|Apr 4, 2019

Advanced Graph Theory Research8 references4 citations

TL;DR

This paper presents two integer linear programming (ILP) formulations for the convex dominating set problem (CVXDSP) and one for the weakly convex dominating set problem (WCVXDSP), proving their correctness by showing equivalence between the ILP models and the original combinatorial problems. The key contribution is a rigorous ILP framework that enables exact solution of these NP-hard graph optimization problems using standard solvers.

ABSTRACT

Due to their importance in practice, dominating set problems in graphs have been greatly studied in past and different formulations of these problems are presented in literature. This paper's focus is on two problems: weakly convex dominating set problem (WCVXDSP) and convex dominating set problem (CVXDSP). It introduces two integer linear programming (ILP) formulation for CVXDSP and one ILP mode for WCVXDSP, as well as proof for equivalency between ILP models for CVXDSP. The proof of correctness for all introduced ILP formulations is provided by showing that optimal solution to the each ILP formulation is equal to the optimal solution of the original problem.

Motivation & Objective

• To develop exact ILP formulations for the convex dominating set problem (CVXDSP) and weakly convex dominating set problem (WCVXDSP).
• To establish the equivalence between the proposed ILP models and the original combinatorial optimization problems.
• To prove that the optimal solution of each ILP formulation matches the optimal solution of the corresponding graph problem.
• To provide a foundation for exact solution of these NP-hard problems using standard optimization solvers.

Proposed method

• Formulates CVXDSP using two distinct ILP models with binary variables indicating vertex inclusion in the dominating set.
• Introduces constraints ensuring that every vertex is dominated and that all shortest paths between selected vertices remain within the set.
• Employs distance-based constraints to enforce weak convexity: for each pair of vertices in the set, at least one neighbor on a shortest path must also be included.
• Uses mathematical induction and case analysis to prove that the ILP constraints correctly enforce weak convexity and domination.
• Demonstrates equivalence between the ILP models and the original problems by proving that any feasible ILP solution corresponds to a valid weakly or convex dominating set.
• Validates correctness via proof that optimal ILP solutions yield the minimal cardinality dominating sets for both CVXDSP and WCVXDSP.

Experimental results

Research questions

• RQ1 Can the convex dominating set problem be accurately modeled using integer linear programming?
• RQ2 Are the two proposed ILP formulations for CVXDSP equivalent in terms of optimal solution?
• RQ3 Can the weakly convex dominating set problem be effectively formulated as an ILP model?
• RQ4 Does the ILP formulation for WCVXDSP correctly enforce both domination and weak convexity?
• RQ5 Is the optimal solution of the ILP formulation guaranteed to match the optimal solution of the original graph problem?

Key findings

• The two ILP formulations for CVXDSP are proven to be equivalent, meaning they yield the same optimal solution.
• The ILP model for WCVXDSP correctly enforces both domination and weak convexity, ensuring feasible solutions correspond to valid weakly convex dominating sets.
• Proof of correctness is established by showing that any feasible solution to the ILP corresponds to a valid dominating set with the required convexity property.
• The optimal solution value of each ILP formulation is guaranteed to equal the corresponding graph domination number (γ_cvx(G) or γ_wcvx(G)).
• The ILP models are mathematically sound and can be used as a basis for exact solution via standard ILP solvers.
• The work provides a solid foundation for future experimental evaluation and meta-heuristic development on large-scale instances.