Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

QUBO Formulations for Variation of Domination Problem

Haoqian Pan, Changhong Lu

TL;DR

This work tackles solving the Domination Problem (DP) and its variants on quantum hardware by formulating them as Quadratic Unconstrained Binary Optimization (QUBO) problems. It introduces a systematic workflow that starts from 0-1 integer programming formulations and converts the DP constraints into quadratic penalties, achieving a qubit-efficient representation with at most $|V|+2|E|$ variables. The authors provide explicit QUBO expressions for DP variants such as independent, total, perfect, clique, and k-domination (via a $G'$ construction), and demonstrate concrete examples on a 4-node graph where the proposed approach uses fewer qubits than prior work. This work thus lowers the barrier to solving DP and its variants on near-term quantum devices and lays the groundwork for applying quantum algorithms like QAOA or QA to these problems.

Abstract

With the development of quantum computing, the use of quantum algorithms to solve combinatorial optimization problems on quantum computers has become a major research focus. The Quadratic Unconstrained Binary Optimization (QUBO) model serves as a bridge between combinatorial optimization problems and quantum computers, and is a prerequisite for these studies. In combinatorial optimization problems, the Domination Problem (DP) is related to many practical issues in the real world, such as the fire station problem, social network theory, and so on. Additionally, the DP has numerous variants, such as independent DP, total DP, k-domination, and so forth. However, there is a scarcity of quantum computing research on these variant problems. A possible reason for this is the lack of research on QUBO modeling for these issues. This paper investigates the QUBO modeling methods for the classic DP and its variants. Compared to previous studies, the QUBO modeling method we propose for the classic DP can utilize fewer qubits. This will lower the barrier for solving DP on quantum computers. At the same time, for many variants of DP problems, we provide their QUBO modeling methods for the first time. Our work will accelerate the entry of DP into the quantum era.

QUBO Formulations for Variation of Domination Problem

TL;DR

This work tackles solving the Domination Problem (DP) and its variants on quantum hardware by formulating them as Quadratic Unconstrained Binary Optimization (QUBO) problems. It introduces a systematic workflow that starts from 0-1 integer programming formulations and converts the DP constraints into quadratic penalties, achieving a qubit-efficient representation with at most variables. The authors provide explicit QUBO expressions for DP variants such as independent, total, perfect, clique, and k-domination (via a construction), and demonstrate concrete examples on a 4-node graph where the proposed approach uses fewer qubits than prior work. This work thus lowers the barrier to solving DP and its variants on near-term quantum devices and lays the groundwork for applying quantum algorithms like QAOA or QA to these problems.

Abstract

With the development of quantum computing, the use of quantum algorithms to solve combinatorial optimization problems on quantum computers has become a major research focus. The Quadratic Unconstrained Binary Optimization (QUBO) model serves as a bridge between combinatorial optimization problems and quantum computers, and is a prerequisite for these studies. In combinatorial optimization problems, the Domination Problem (DP) is related to many practical issues in the real world, such as the fire station problem, social network theory, and so on. Additionally, the DP has numerous variants, such as independent DP, total DP, k-domination, and so forth. However, there is a scarcity of quantum computing research on these variant problems. A possible reason for this is the lack of research on QUBO modeling for these issues. This paper investigates the QUBO modeling methods for the classic DP and its variants. Compared to previous studies, the QUBO modeling method we propose for the classic DP can utilize fewer qubits. This will lower the barrier for solving DP on quantum computers. At the same time, for many variants of DP problems, we provide their QUBO modeling methods for the first time. Our work will accelerate the entry of DP into the quantum era.

Paper Structure

This paper contains 8 sections, 1 theorem, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Method
  4. Definitions of DP and its variation
  5. Mathematical model
  6. Quadratic penalties
  7. QUBO formulations examples
  8. Conclusion

Key Result

Theorem 4

The upper bound of the number of variables used by the method proposed in this paper to form the DP is $V + 2E$.

Figures (2)

  • Figure 1: The illustration of DP
  • Figure 3: A graph $G$ with 4 nodes.

Theorems & Definitions (2)

  • Theorem 4: Theorem 4
  • proof