QUBO formulations of the longest path problem

McCollum, Joey; Krauss, Thomas F.

doi:10.1016/j.tcs.2021.02.021

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…If we look at QUBO-related research, there is the research conducted by McCollumn et al [12] that deals with finding the max-weighted path in a graph using k vertices. For this, they utilized two different approaches, the order-based and the degree-based formulation.…”

Section: Literature Reviewmentioning

confidence: 99%

“…If there exists a subgraph H ⊆ G that contains a cycle, then the average degree of H ad(H) ≥ 2. The average degree of a path with k − 1 vertices, so excluding vertex v 1 , however, is [12]. To ensure that the maximum average degree is less than 2 in the solution without v 1 , flow variables are used.…”

Section: Finding the Maximum Weighted Cycle With Length K And Known S...mentioning

confidence: 99%

“…Here, it does not matter that the given graph is directed, i.e., there is some flow in both directions over arc a if a is part of the solution. To make the solution acyclic, the flow on each arc a ∈ A has to be equal to k − 1, while the incoming flow for vertex v ∈ V has to be equal to k − 2 [12]. This leads to Equations ( 5) and ( 6).…”

Section: Finding the Maximum Weighted Cycle With Length K And Known S...mentioning

confidence: 99%

“…The aim of this paper is to provide multiple QUBO formulations for this problem and give an indication of the potential benefit that (hybrid) quantum computing can bring using the currently available hardware. McCollum and Krauss [12] have defined QUBO formulations for the longest path problem. However, we are, to the best of our knowledge, the first to propose this for the MWC problem.…”

Section: Introductionmentioning

confidence: 99%

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Finding Debt Cycles: QUBO Formulations for the Maximum Weighted Cycle Problem Solved Using Quantum Annealing

Künnemann

Phillipson

2023

Mathematics

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The problem of finding the maximum weighted cycle in a directed graph map to solve optimization problems is NP-hard, implying that approaches in classical computing are inefficient. Here, Quantum computing might be a promising alternative. Many current approaches to the quantum computer are based on a Quadratic Unconstrained Binary Optimization (QUBO) problem formulation. This paper develops four different QUBO approaches to this problem. The first two take the starting vertex and the number of vertices used in the cycle as given, while the latter two loosen the second assumption of knowing the size of the cycle. A QUBO formulation is derived for each approach. Further, the number of binary variables required to encode the maximum weighted cycle problem with one or both assumptions for the respective approach is made explicit. The problem is motivated by finding the maximum weighted debt cycle in a debt graph. This paper compares classical computing versus currently available (hybrid) quantum computing approaches for various debt graphs. For the classical part, it investigated the Depth-First-Search (DFS) method and Simulated Annealing. For the (hybrid) quantum approaches, a direct embedding on the quantum annealer and two types of quantum hybrid solvers were utilized. Simulated Annealing and the usage of the hybrid CQM (Constrained Quadratic Model) had promising functionality. The DFS method, direct QPU, and hybrid BQM (Binary Quadratic Model), on the other hand, performed less due to memory issues, surpassing the limit of decision variables and finding the right penalty values, respectively.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Finding the Maximum Weighted Cycle With Length K And Known S...mentioning

confidence: 99%

Section: Finding the Maximum Weighted Cycle With Length K And Known S...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Finding Debt Cycles: QUBO Formulations for the Maximum Weighted Cycle Problem Solved Using Quantum Annealing

Künnemann

Phillipson

2023

Mathematics

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“…Here, we can distinguish optimization problems in graphs, such as the Hamiltonian Cycle 3 in [30], [60] or Graph Traversal problems (such as eulerian tours or optimal postman tours) in [34], [36]. Other works encompass studies dealing with the well-known shortest path problem in [53], [75] or the less studied longest path problem [57]. Finally, it is interesting to highlight those works which combine several techniques or even present the problem from a mixed technical perspective.…”

Section: B Types Of Problems Studiedmentioning

confidence: 99%

A Systematic Literature Review of Quantum Computing for Routing Problems

2022

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Quantum Computing is drawing a significant attention from the current scientific community. The potential advantages offered by this revolutionary paradigm has led to an upsurge of scientific production in different fields such as economics, industry, or logistics. The main purpose of this paper is to collect, organize and systematically examine the literature published so far on the application of Quantum Computing to routing problems. To do this, we embrace the well-established procedure named as Systematic Literature Review. Specifically, we provide a unified, self-contained, and end-to-end review of 18 years of research (from 2004 to 2021) in the intersection of Quantum Computing and routing problems through the analysis of 53 different papers. Several interesting conclusions have been drawn from this analysis, which has been formulated to give a comprehensive summary of the current state of the art by providing answers related to the most recurrent type of study (practical or theoretical), preferred solving approaches (dedicated or hybrid), detected open challenges or most used Quantum Computing device, among others.

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