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Characterization of QUBO reformulations for the maximum k-colorable subgraph problem

Authors:
Rodolfo Alexander Quintero Ospina at Lehigh University
Rodolfo Alexander Quintero Ospina
David E. Bernal Neira at Purdue University West Lafayette
David E. Bernal Neira
Tamás Terlaky at Lehigh University
Tamás Terlaky
Luis Zuluaga at Lehigh University
Luis Zuluaga
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Abstract and Figures

Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem’s constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently solve these constrained COPT problems. This efficiency is the key toward the goal of using quantum computers to solve constrained COPT problems more efficiently than with classical computers. Along these lines, we consider an important constrained COPT problem, namely the maximum k-colorable subgraph (MkCS) problem, in which the aim is to find an induced k-colorable subgraph with maximum cardinality in a given graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design, human genetic research, and cybersecurity. We derive two QUBO reformulations for the MkCS problem and fully characterize the range of the penalty parameters that can be used in the QUBO reformulations. Further, one of the QUBO reformulations of the MkCS problem is obtained without the need to introduce additional binary variables. To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we benchmark different QUBO reformulations of the MkCS problem by performing numerical tests on D-Wave’s quantum annealing devices. These tests also illustrate the numerical power gained by using the latest D-Wave’s quantum annealing device.
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Quantum Information Processing (2022) 21:89
https://doi.org/10.1007/s11128-022-03421-z
Characterization of QUBO reformulations for the maximum
k-colorable subgraph problem
Rodolfo Quintero1·David Bernal2·Tamás Terlaky1·Luis F. Zuluaga1
Received: 15 July 2021 / Accepted: 11 January 2022 / Published online: 18 February 2022
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
Abstract
Quantum devices can be used to solve constrained combinatorial optimization (COPT)
problems thanks to the use of penalization methods to embed the COPT problem’s
constraints in its objective to obtain a quadratic unconstrained binary optimization
(QUBO) reformulation of the COPT. However, the particular way in which this penal-
ization is carried out affects the value of the penalty parameters, as well as the number
of additional binary variables that are needed to obtain the desired QUBO reformu-
lation. In turn, these factors substantially affect the ability of quantum computers to
efficiently solve these constrained COPT problems. This efficiency is the key toward
the goal of using quantum computers to solve constrained COPT problems more
efficiently than with classical computers. Along these lines, we consider an impor-
tant constrained COPT problem, namely the maximum k-colorable subgraph (MkCS)
problem, in which the aim is to find an induced k-colorable subgraph with maximum
cardinality in a given graph. This problem arises in channel assignment in spectrum
sharing networks, VLSI design, human genetic research, and cybersecurity. We derive
two QUBO reformulations for the MkCS problem and fully characterize the range
of the penalty parameters that can be used in the QUBO reformulations. Further, one
of the QUBO reformulations of the MkCS problem is obtained without the need to
introduce additional binary variables. To illustrate the benefits of obtaining and charac-
terizing these QUBO reformulations, we benchmark different QUBO reformulations
of the MkCS problem by performing numerical tests on D-Wave’s quantum anneal-
ing devices. These tests also illustrate the numerical power gained by using the latest
D-Wave’s quantum annealing device.
Keywords Quantum Computing ·NISQ devices ·QUBO reformulations ·
Combinatorial Optimization ·Chimera versus Pegasus D-Wave Annealer
Mathematics Subject Classification 68Q09 ·68Q12 ·81P68 ·90C27
Extended author information available on the last page of the article
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... This proliferation has reignited interest in reformulation techniques tailored to this type of problems [see, e.g. [12][13][14][15][16][17]. ...
... 21] and gate-based [see, e.g. 22] quantum devices can address the solution of QUBO problems, and potentially have quantum supremacy [see, e.g., 23] over classical computers on this task, it is clearly of interest to study whether or how combinatorial optimization problems that do not have a natural QUBO formulation (e.g., the stable set problem) can be reformulated as a QUBO [see, e.g., [12][13][14][15]24] A classical approach to solving optimization problems with complex or computationally intensive constraints is the use of Lagrangian relaxations [see, e.g., 20,25]. By relaxing (or dualizing) the constraints and solving the resulting simpler problem iteratively, Lagrangian relaxation provides a systematic approach to finding near-optimal solutions efficiently. ...
... On the one hand, general Lagrangian duality results for nonconvex optimization [see, e.g., 34-38, among many others] often lack constructive dual attainment characterizations necessary for deriving the practical reformulations that are of interest here. On the other hand, Lagrangian reformulation results exist for specific optimization problem classes, such as linearly constrained pure binary quadratic optimization [see, e.g., [12][13][14][15][16][17], mixed-integer linear optimization [39], and mixed-integer convex quadratic optimization [40,41]. By deriving constructive strong duality and dual attainment results, we provide Lagrangian reformulation results with practical applications for a broad class of nonconvex optimization problems, including the latter problems metioned. ...
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... However, it is important to prepare the ground for the time when bigger, more resilient to noise quantum computers will become available, by exploring which formulations of combinatorial optimization problems are more suitable for each task. This is a very active area of research, and for instance, in the last few years, several authors have studied the merits of different quantum formulations of the Travelling Salesperson Problem and its variants [23][24][25], of graph-coloring problems [26,27], of workflow scheduling problems [28,29], and of vehicle routing problems [30,31]. ...
... In practice, however, choosing the right formulation can be of paramount importance in practical applications of quantum optimization techniques. In recent years, several researchers have explored the performance of quantum optimization techniques under different formulations for famous problems such as the Travelling Salesperson Problem [23][24][25], graph coloring [26,27], vehicle routing [30,31], and workflow scheduling [28,29], showing that there can be significant differences among them. ...
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... Indeed, QAOA derives from the Quantum Annealing algorithm of Kadowaki and Nishimori (1998), algorithm that takes as input an Ising Model, a model of ferromagnetism in statistical mechanics, which is equivalent to a QUBO problem. Thus, natively, the comparison of different formulations of QUBO models for a given problem recently raised interest in the community, for instance for the graph coloring problem (Tabi et al., 2020) or for Max-k-colorable subgraph problem (Quintero et al., 2022). However, there are in fact no inherent degree limitations of the objective function. ...
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... Therefore, many optimization problems can be formulated as Ising models to find the ground state, or the lowest energy configuration. So that, solving the Ising model becomes a general method for solving many NP problems, like partitioning problems [2], linear programming [1], [3], [5], inequality problems [6], coloring problems [2], [7] and so on. However, the Ising model is known to be NP-hard (Non-deterministic Polynomial Hard) problem [8]. ...
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The maximum k-colorable subgraph (MkCS) problem is to find an induced k-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the MkCS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. To simplify these relaxations, we exploit the symmetry arising from permuting the colors, as well as the symmetry of the given graphs when applicable. We also show how to exploit invariance under permutations of the subsets for other partition problems and how to use the MkCS problem to derive bounds on the chromatic number of a graph. Our numerical results verify that the proposed relaxations provide strong bounds for the MkCS problem and that those outperform existing bounds for most of the test instances. Summary of Contribution: The maximum k-colorable subgraph (MkCS) problem is to find an induced k-colorable subgraph with maximum cardinality in a given graph. The MkCS problem has a number of applications, such as channel assignment in spectrum sharing networks (e.g., Wi-Fi or cellular), very-large-scale integration design, human genetic research, and so on. The MkCS problem is also related to several other optimization problems, including the graph partition problem and the max-k-cut problem. The two mentioned problems have applications in parallel computing, network partitioning, floor planning, and so on. This paper is an in-depth analysis of the MkCS problem that considers various semidefinite programming relaxations, including their theoretical and numerical comparisons. Further, our analysis relates the MkCS results with the stable set and the chromatic number problems. We provide extended numerical results that verify that the proposed bounding approaches provide strong bounds for the MkCS problem and that those outperform existing bounds for most of the test instances. Moreover, our lower bounds on the chromatic number of a graph are competitive with existing bounds in the literature.
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  • Lect Notes Comput Sci
A major limitation of current generations of quantum annealers is the sparse connectivity of manufactured qubits in the hardware graph. This technological limitation has generated considerable interest, motivating efforts to design efficient and adroit minor-embedding procedures that bypass sparsity constraints. In this paper, starting from a previous equational formulation by Dridi et al. (arXiv:1810.01440), we propose integer programming (IP) techniques for solving the minor-embedding problem. The first approach involves a direct translation from the previous equational formulation to IP, while the second decomposes the problem into an assignment master problem and fiber condition checking subproblems. The proposed methods are able to detect instance infeasibility and provide bounds on solution quality, capabilities not offered by currently employed heuristic methods. We demonstrate the efficacy of our methods with an extensive computational assessment involving three families of random graphs of varying sizes and densities. The direct translation as a monolithic IP model can be solved with existing commercial solvers yielding valid minor-embeddings but it is outperformed, overall, by the decomposition approach. Our results demonstrate the promise of our methods for the studied benchmarks, highlighting the advantages of using IP technology for minor-embedding problems.
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Embedding Algorithms for Quantum Annealers with Chimera and Pegasus Connection Topologies
Chapter
  • Jun 2020
  • Lect Notes Comput Sci
We propose two new algorithms – Spring-Based MinorMiner (SPMM) and Clique-Based MinorMiner (CLMM) – which take as input the connectivity graph of a Quadratic Unconstrained Binary Optimization (QUBO) problem and produce as output an embedding of the input graph on a host graph that models the topology of a quantum computing device. As host graphs, we take the Chimera graph and the Pegasus graph, which are the topology graphs of D-Wave’s 2000 qubit (first introduced in 2017) and 5000 qubit (expected 2020) quantum annealer devices, respectively. We evaluate our algorithms on a large set of random graph QUBO inputs (Erdős-Rényi Gn,pG_{n,p}, Barabási-Albert and d-regular graphs) on both host topologies against other embedding algorithms. For the Pegasus topology, we find that CLMM outperforms all other algorithms at edge densities larger than 0.08, while SPMM wins at edge densities smaller than 0.08 for Erdős-Rényi graphs, with very similar transition densities for the other graph classes. Surprisingly, the standard D-Wave MinorMiner embedding algorithm – while also getting slightly outperformed by SPMM for sparse and very dense graphs on Chimera – does not manage to extend its overall good performance on Chimera to Pegasus as it fails to embed even medium-density graphs on 175–180 nodes which are known to have clique embeddings on Pegasus.
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