Quantum algorithms and the maximum clique problem in graphs

Matheus Augusto Condé Dias

Published Sep 14, 2022• 0 citations

Abstract

Determining the cardinality of the maximum clique on a simple graph is a NP-hard problem with several applications of interest. The best know classical algorithm for this problem has 𝒪(3𝑛/3) time complexity. Classical computing uses phenomena from classical physics to model, store and process information. Consequently, the entire model of classical computing is bounded superiorly by the limits of classical physics. As an alternative, Quantum Computing has been developed to take advantage of quantum phenomena and provide the constructions of circuits capable of performing certain computations more efficiently. In this paper, two quantum solutions to the maximum clique problem will be proposed and analyzed, based on the Grover Quantum Search algorithms and the Quantum Minimal Search Algorithm, comparing their respective time complexities with the best known classical solution.

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Quantum algorithms and the maximum clique problem in graphs

Matheus Augusto Condé Dias

Published Sep 14, 2022• 0 citations

OpenAlex

Abstract

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O\left( {\left| V \right|\sqrt {{2^{\left| V \right|}}} } \right)

time complexity rather than the current classical time complexity of O(2|V|) of the best known algorithm, where |V| is the number of vertices in the graph. The full circuit implementation is presented using the open-source Qiskit environment [1]. The paper also analyzes the growth of the hardware resources requirements with the number of vertices in the targeted graph.

O(2^n)

). In this paper, Maximum Clique Problem has been solved using combinatorial approach of Grover's search algorithm. An algorithm has been proposed that auto-generates the circuit for any given undirected and unweighted graph which makes the approach generalized in nature for Maximum Clique Problem. Computational speed up is achieved by using our approach with the help of quantum mechanical effect that is quantum superposition. We have simulated the proposed circuit using IBM's QISkit platform and verified its correctness.

The k-plex model, which allows each vertex to miss connections with up to

k

neighbors, serves as a relaxation of the clique model. Its adaptability makes it more suitable for analyzing graphs from real-world applications, where noise and imperfect data are common and the stringent clique model is often impractical. The challenge of identifying maximum k-plex (MKP, an NP-hard problem) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent research efforts have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivial

O^{*}(2^{n})

O^{*}(c_{k}^{n})

, with

c_{k} > 1.94

for

k

> 3, where

n

denotes the number of vertices. In this paper, we demonstrate that MKP can be solved in

O^{*}(1.42^{n})

and propose the first two quantum algorithms, qTKP and qMKP, to achieve this complexity. qTKP employs quantum search integrated with graph encoding, degree count, degree comparison, and size determination to find a k-plex of a given size; qMKP uses a binary search to progressively identify the maximum solution. To validate the practical performance and effectiveness of our algorithms, proof-of-principle experiments were conducted using the latest IBM quantum simulator currently available. This work holds potential to be applied to a wide range of clique relaxations, e.g., n-clan and n-club.

\boldsymbol{n}

-fold improvement over state-of-the-art Grover-based methods, which require

O(n\sqrt{2^n})

iterations and

O(n)

measurements for

n

-vertex graphs. We validate algorithmic correctness through simulations on IBM's Qiskit platform and benchmark qubit/gate efficiency against existing Grover-based MCP solvers.