We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal), motivated by recent advances in near-term quantum computing hardware and the related embedding problems. We assemble a preprocessing suite of fast input reduction routines from the Odd Cycle Transversal (OCT) and Vertex Cover (VC) literature, allowing algorithm implementations to be compared using Quadratic Unconstrained Binary Optimization problems from the quantum literature. These problems represent a wide array of graph properties, leading to harder OCT instances than in previous benchmarks.
In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We compare our results to those on a corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data.
Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal), motivated by recent advances in near-term quantum computing hardware and the related embedding problems. We assemble a preprocessing suite of fast input reduction routines from the Odd Cycle Transversal (OCT) and Vertex Cover (VC) literature, allowing algorithm implementations to be compared using Quadratic Unconstrained Binary Optimization problems from the quantum literature. These problems represent a wide array of graph properties, leading to harder OCT instances than in previous benchmarks.
In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We compare our results to those on a corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data.
Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
We experimentally evaluate the practical state-of-the-art in graph bipartization (Odd Cycle Transversal (OCT)), motivated by the need for good algorithms for embedding problems into near-term quantum computing hardware. We assemble a preprocessing suite of fast input reduction routines from the OCT and Vertex Cover (VC) literature and compare algorithm implementations using Quadratic Unconstrained Binary Optimization problems from the quantum literature. We also generate a corpus of frustrated cluster loop graphs, which have previously been used to benchmark quantum annealing hardware. The diversity of these graphs leads to harder OCT instances than in existing benchmarks. In addition to combinatorial branching algorithms for solving OCT directly, we study various reformulations into other NP-hard problems such as VC and Integer Linear Programming (ILP), enabling the use of solvers such as CPLEX. We find that for heuristic solutions with time constraints under a second, iterative compression routines jump-started with a heuristic solution perform best, after which point using a highly tuned solver like CPLEX is worthwhile. Results on exact solvers are split between using ILP formulations on CPLEX and solving VC formulations with a branch-and-reduce solver. We extend our results with a large corpus of synthetic graphs, establishing robustness and potential to generalize to other domain data. In total, over 8,000 graph instances are evaluated, compared to the previous canonical corpus of 100 graphs. Finally, we provide all code and data in an open source suite, including a Python API for accessing reduction routines and branching algorithms, along with scripts for fully replicating our results.
Quantum annealing (QA) can be used to quickly obtain near-optimal solutions for quadratic unconstrained binary optimization (QUBO) problems. In QA hardware, each decision variable of a QUBO should be mapped to one or more adjacent qubits in such a way that pairs of variables defining a quadratic term in the objective function are mapped to some pair of adjacent qubits. However, qubits have limited connectivity in existing QA hardware. This has spurred work on preprocessing algorithms for embedding the graph representing problem variables with quadratic terms into the hardware graph representing qubits adjacencies, such as the Chimera graph in hardware produced by D-Wave Systems. In this paper, we use integer linear programming to search for an embedding of the problem graph into certain classes of minors of the Chimera graph, which we call template embeddings. One of these classes corresponds to complete bipartite graphs, for which we show the limitation of the existing approach based on minimum odd cycle transversals (OCTs). One of the formulations presented is exact and thus can be used to certify the absence of a minor embedding using that template. On an extensive test set consisting of random graphs from five different classes of varying size and sparsity, we can embed more graphs than a state-of-the-art OCT-based approach, our approach scales better with the hardware size, and the runtime is generally orders of magnitude smaller. Summary of Contribution: Our work combines classical and quantum computing for operations research by showing that integer linear programming can be successfully used as a preprocessing step for adiabatic quantum optimization. We use it to determine how a quadratic unconstrained binary optimization problem can be solved by a quantum annealer in which the qubits are coupled as in a Chimera graph, such as in the quantum annealers currently produced by D-Wave Systems. The paper also provides a timely introduction to adiabatic quantum computing and related work on minor embeddings.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
The recent availability of the first commercial quantum computers has provided a promising tool to tackle NP hard problems which can only be solved heuristically with present techniques. However, it is unclear if the current state of quantum computing already provides a quantum advantage over the current state of the art in classical computing. This article assesses the performance of the D-Wave 2X quantum annealer on two NP hard graph problems, in particular clique finding and graph partitioning. For this, we provide formulations as Qubo and Ising Hamiltonians suitable for the quantum annealer and compare a variety of quantum solvers (Sapi, QBSolv, QSage provided by D-Wave Sys, Inc.) to current classical algorithms (METIS, Simulated Annealing, third-party clique finding and graph splitting heuristics) on certain test sets of graphs. We demonstrate that for small graph instances, classical methods still outperform the quantum annealer in terms of computing time, even though the quality of the best solutions obtained is comparable. Nevertheless, due to the limited problem size which can be embedded on the D-Wave 2X chip, the aforementioned finding applies to most of problems of general nature solvable on the quantum annealer. For instances specifically designed to fit the D-Wave 2X architecture, we observe substantial speed-ups in computing time over classical approaches.
This thesis explores quantum computing for combinatorial optimization through the Traveling Salesman Problem (TSP), which aims to find a minimum-cost Hamiltonian cycle visiting each city exactly once. Using Qiskit, we implement the Quantum Approximate Optimization Algorithm (QAOA) on both simulators and quantum hardware, and compare its performance with that of classical exact optimization via the Gurobi solver. TSP instances are encoded as Quadratic Unconstrained Binary Optimization (QUBO) problems derived from the Miller–Tucker–Zemlin formulation, with degree and subtour constraints embedded using quadratic penalties. Results indicate that QAOA consistently generates feasible tours, but classical branch-and-bound solvers outperform it in both solution quality and runtime. On Noisy Intermediate-Scale Quantum (NISQ) devices, the optimality gap grows with problem size, reflecting sensitivity to variational parameter tuning, penalty scaling, and hardware noise. Among classical optimizers for QAOA, COBYLA offers the most effective tradeoff between efficiency and noise tolerance. Overall, this work establishes a reproducible quantum-classical workflow, provides performance baselines, and identifies critical factors such as encoding design, optimizer choice, and error mitigation for improving near-term quantum heuristic performance.
In many complex networking systems, identifying critical nodes whose removal maximally disrupts network connectivity remains an important yet computationally challenging problem for network vulnerability analysis. Finding near-optimal solutions is known to be NP-hard. In this paper, we explore the potential of near-term quantum computing devices to efficiently solve the k-Critical Node Detection (k-CND) problem. We formulate the problem as a quadratic unconstrained binary optimization (QUBO), a mathematical optimization over binary variables amenable to solution on quantum annealers. We present a novel integer linear programming (ILP) formulation and its conversion into QUBOs and provide benchmarking results on D-Wave's quantum annealers. Our theoretical analysis proves that our proposed formulation, ILP2, generates substantially smaller QUBO than the state-of-the-art ILP1 formulation. Experimentally, our efficient QUBO yields a 59.7% decrease in QUBO variables on a graph with 10 vertices and 40 edges, and an 11.7% reduction in qubits on a 15-vertex, 22-edge graph compared to that of QUBO1. We analyze the solution quality and running time across quantum, classical, and hybrid solvers to assess the potential for quantum advantage. Our work showcases the promise and challenges of tackling this important graph problem on near-term quantum hardware.