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Scientific Reports

Biomolecular and quantum algorithms for the dominating set problem in arbitrary networks

Renata Wong, Weng-Long Chang, Wen‐Yu Chung, Athanasios V. Vasilakos

Published Mar 14, 2023• 9 citations
Publisher

Abstract

A dominating set of a graph [Formula: see text] is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in [Formula: see text] queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum's qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.

DOI
10.1038/s41598-023-30600-4

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A dominating set of a graph [Formula: see text] is a subset U of its vertices V, such that any vertex of G is either in U, or has a neighbor in U. The dominating-set problem is to find a minimum dominating set in G. Dominating sets are of critical importance for various types of networks/graphs, and find therefore potential applications in many fields. Particularly, in the area of communication, dominating sets are prominently used in the efficient organization of large-scale wireless ad hoc and sensor networks. However, the dominating set problem is also a hard optimization problem and thus currently is not efficiently solvable on classical computers. Here, we propose a biomolecular and a quantum algorithm for this problem, where the quantum algorithm provides a quadratic speedup over any classical algorithm. We show that the dominating set problem can be solved in [Formula: see text] queries by our proposed quantum algorithm, where n is the number of vertices in G. We also demonstrate that our quantum algorithm is the best known procedure to date for this problem. We confirm the correctness of our algorithm by executing it on IBM Quantum's qasm simulator and the Brooklyn superconducting quantum device. And lastly, we show that molecular solutions obtained from solving the dominating set problem are represented in terms of a unit vector in a finite-dimensional Hilbert space.
In this paper, it is shown that the proposed quantum algorithm for implementing Boolean circuits generated from the DNA-based algorithm solving the vertex-cover problem of any graph G with m edges and n vertices is the optimal quantum algorithm. Next, it is also demonstrated that mathematical solutions of the same biomolecular solutions are represented in terms of a unit vector in the finite-dimensional Hilbert space. Furthermore, for testing our theory, a nuclear magnetic resonance (NMR) experiment of three quantum bits to solve the simplest vertex-cover problem is completed.
Recent advancements in quantum computing have spurred substantial research into the application of quantum algorithms to combinatorial optimization problems. Among these challenges, the Total Domination Problem (TDP) emerges as a classic and critical paradigm in the field. For a graph G(V, E), TDP entails finding a minimal subset D subset of V that contains no isolated vertices, where every vertex not in D has at least one neighbor in D. TDP finds extensive applications across domains such as computer networks, social networks, and communications. Since the latter half of the last century, research efforts have focused on establishing its NP-completeness and developing solution algorithms, which have become foundational to combinatorial mathematics. Despite this rich history, the application of quantum algorithms to TDP remains largely underexplored. In this study, we present a pioneering application of the Quantum Approximate Optimization Algorithm (QAOA) to tackle TDP, evaluating its efficacy across a diverse set of parameters. This paper proves that the upper bound on the number of qubits required to solve TDP is 2|V| + |V| log_2( (2|E|)/|V| - 1 ). Our experimental findings demonstrate that QAOA is effective in addressing TDP: under most parameter combinations, it successfully computes a valid total dominating set (TDS). However, the algorithm's performance in identifying the optimal TDS is contingent upon specific parameter choices, revealing a significant bias in the distribution of effective parameter points. This research contributes valuable insights into the potential of quantum algorithms for solving TDP and lays a solid groundwork for future investigations in this area.
Grover's algorithm is a well-known unstructured quantum search algorithm run on quantum computers. It constructs an oracle and calls the oracle O(N\sqrt NN​) times to locate specific data out of N unsorted data. This represents a quadratic speedup compared to the classical unstructured data sequential search algorithm, which requires to call the oracle O(N) times. We are currently in the noisy intermediate-scale quantum (NISQ) era in which quantum computers have a limited number of qubits, short decoherence time, and low gate fidelity. It is thus desirable to design quantum components with three good properties: (i) a reduced number of qubits, (ii) shorter quantum depth, and (iii) fewer gates. This paper utilizes novel quantum counters with the above-mentioned three good properties to construct the oracle of Grover's algorithm to efficiently solve the dominating set problem (DSP), as defined below. For a given graph G=(V, E), a dominating set (DS) D is a subset of the vertex set V, such that every vertex is in D or has an adjacent vertex in D. The DSP is to decide for a given graph G and an integer k whether there exists a DS with size k. Algorithms solving the DSP have many applications. For example, they can be applied to check whether k routers suffice to connect all computers in a computer network. The DSP is an NP-complete problem, indicating that no classical algorithm exists to solve the DSP with polynomial time complexity in the worst case. Therefore, using quantum algorithms, such as Grover's algorithm, to exploit the potent computational capabilities of quantum computers to solve the DSP is highly promising. We execute the whole quantum circuit of Grover's algorithm using novel quantum counters through the IBM Quantum Lab service to validate that the circuit can solve the DSP efficiently and correctly.
In this paper we design a quantum algorithm for the NP-complete problem of finding the domatic number.The DOMATIC NUMBER problem asks to determine the largest integer k, such that a given undirected graph with n vertices can be partitioned into k pairwise disjoint dominating sets.This problem finds significant applications, such as in wireless sensor networks, where the selection of multiple dominating sets balances energy consumption and extends network lifetime.Each node communicates exclusively with designated nodes, and dominant sets ensure network resilience by enabling seamless replacement in case of node cluster failure.The importance of this problem lies in its implications for network optimization, highlighting the advantages of quantum computing in addressing complex combinatorial challenges.We present a quantum algorithm that solves this problem in time O(2.4143 n ), which we further improve to O(2.3845 n ).
In this paper, we propose a bio-molecular algorithm with O( n2+ m) biological operations, O( 2n) DNA strands, O( n) tubes and the longest DNA strand, O( n), for solving the independent-set problem for any graph G with m edges and n vertices. Next, we show that a new kind of the straightforward Boolean circuit yielded from the bio-molecular solutions with m NAND gates, ( m +n × ( n + 1)) AND gates and (( n × ( n + 1))/2) NOT gates can find the maximal independent-set(s) to the independent-set problem for any graph G with m edges and n vertices. We show that a new kind of the proposed quantum-molecular algorithm can find the maximal independent set(s) with the lower bound Ω( 2n/2) queries and the upper bound O( 2n/2) queries. This work offers an obvious evidence for that to solve the independent-set problem in any graph G with m edges and n vertices, bio-molecular computers are able to generate a new kind of the straightforward Boolean circuit such that by means of implementing it quantum computers can give a quadratic speed-up. This work also offers one obvious evidence that quantum computers can significantly accelerate the speed and enhance the scalability of bio-molecular computers. Next, the element distinctness problem with input of n bits is to determine whether the given 2nreal numbers are distinct or not. The quantum lower bound of solving the element distinctness problem is Ω( 2n×(2/3)) queries in the case of a quantum walk algorithm. We further show that the proposed quantum-molecular algorithm reduces the quantum lower bound to Ω(( 2n/2)/( 21/2)) queries. Furthermore, to justify the feasibility of the proposed quantum-molecular algorithm, we successfully solve a typical independent set problem for a graph G with two vertices and one edge by carrying out experiments on the backend ibmqx4 with five quantum bits and the backend simulator with 32 quantum bits on IBM's quantum computer.
We provide efficient quadratic unconstrained binary optimization (QUBO) formulations for the Dominating Set and Edge Cover combinatorial problems suitable for adiabatic quantum computers, which are viewed as a real-world enhanced model of simulated annealing (e.g. a type of genetic algorithm with quantum tunneling). The number of qubits (dimension of QUBO matrices) required to solve these set cover problems are O(n + n lg n) and O(m + n lg n) respectively, where n is the number of vertices and m is the number of edges. We also extend our formulations for the Minimum Vertex-Weighted Dominating Set problem and the Minimum Edge-Weighted Edge Cover problem. Experimental results for the Dominating Set and Edge Cover problems using a D-Wave Systems quantum computer with 1098 active qubit-coupled processors are also provided for a selection of known common graphs.