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arXiv:2410.14111 · ACM/IEEE Design Automation Conference (DAC)

HyCiM: A Hybrid Computing-in-Memory QUBO Solver for General Combinatorial Optimization Problems with Inequality Constraints

Yu Qian, Zeyu Yang, Kai Ni, Alptekin Vardar, Thomas Kämpfe, Xunzhao Yin

Published Jun 23, 2024• 7 citations
arXivHTMLPDF

Abstract

Computationally challenging combinatorial optimization problems (COPs) play a fundamental role in various applications. To tackle COPs, many Ising machines and Quadratic Unconstrained Binary Optimization (QUBO) solvers have been proposed, which typically involve direct transformation of COPs into Ising models or equivalent QUBO forms (D-QUBO). However, when addressing COPs with inequality constraints, this D-QUBO approach introduces numerous extra auxiliary variables, resulting in a substantially larger search space, increased hardware costs, and reduced solving efficiency. In this work, we propose HyCiM, a novel hybrid computing-inmemory (CiM) based QUBO solver framework, designed to overcome aforementioned challenges. The proposed framework consists of (i) an innovative transformation method (first to our known) that converts COPs with inequality constraints into an inequality-QUBO form, thus eliminating the need of expensive auxiliary variables and associated calculations; (ii) "inequality filter", a ferroelectric FET (FeFET)-based CiM circuit that accelerates the inequality evaluation, and filters out infeasible input configurations; (iii) a FeFET-based CiM annealer that is capable of approaching global solutions of COPs via iterative QUBO computations within a simulated annealing process. The evaluation results show that HyCiM drastically narrows down the search space, eliminating 2100 to 22536 infeasible input configurations compared to the conventional D-QUBO approach. Consequently, the narrowed search space, reduced to 2100 feasible input configurations, leads to a substantial hardware area overhead reduction, ranging from 88.06% to 99.96%. Additionally, HyCiM consistently exhibits a high solving efficiency, achieving a remarkable average success rate of 98.54%, whereas D-QUBO implementatoin shows only 10.75%.

DOI
10.1145/3649329.3655989

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Computationally challenging combinatorial optimization problems (COPs) play a fundamental role in various applications. To tackle COPs, many Ising machines and Quadratic Unconstrained Binary Optimization (QUBO) solvers have been proposed, which typically involve direct transformation of COPs into Ising models or equivalent QUBO forms (D-QUBO). However, when addressing COPs with inequality constraints, this D-QUBO approach introduces numerous extra auxiliary variables, resulting in a substantially larger search space, increased hardware costs, and reduced solving efficiency. In this work, we propose HyCiM, a novel hybrid computing-inmemory (CiM) based QUBO solver framework, designed to overcome aforementioned challenges. The proposed framework consists of (i) an innovative transformation method (first to our known) that converts COPs with inequality constraints into an inequality-QUBO form, thus eliminating the need of expensive auxiliary variables and associated calculations; (ii) "inequality filter", a ferroelectric FET (FeFET)-based CiM circuit that accelerates the inequality evaluation, and filters out infeasible input configurations; (iii) a FeFET-based CiM annealer that is capable of approaching global solutions of COPs via iterative QUBO computations within a simulated annealing process. The evaluation results show that HyCiM drastically narrows down the search space, eliminating 2100 to 22536 infeasible input configurations compared to the conventional D-QUBO approach. Consequently, the narrowed search space, reduced to 2100 feasible input configurations, leads to a substantial hardware area overhead reduction, ranging from 88.06% to 99.96%. Additionally, HyCiM consistently exhibits a high solving efficiency, achieving a remarkable average success rate of 98.54%, whereas D-QUBO implementatoin shows only 10.75%.
In this paper, we propose a hybrid framework to solve large-scale permutation-based combinatorial problems effectively using a high-performance quadratic unconstrained binary optimization (QUBO) solver. To do so, transformations are required to change a constrained optimization model to an unconstrained model that involves parameter tuning. We propose techniques to overcome the challenges in using a QUBO solver that typically comes with limited numbers of bits. First, to smooth the energy landscape, we reduce the magnitudes of the input without compromising optimality. We propose a machine learning approach to tune the parameters for good performance effectively. To handle possible infeasibility, we introduce a polynomial-time projection algorithm. Finally, to solve large-scale problems, we introduce a divide-and-conquer approach that calls the QUBO solver repeatedly on small sub-problems. We tested our approach on provably hard Euclidean Traveling Salesman (E-TSP) instances and Flow Shop Problem (FSP). Optimality gap that is less than 10%10\%10% and 11%11\%11% are obtained respectively compared to the best-known approach.
Quantum annealing (QA) is used to solve combinatorial optimization problems (COPs). When COPs are implemented on quantum annealers, they are typically encoded as quadratic unconstrained binary optimization (QUBO) problems, but constraint encodings often increase the number of qubits and the embedding overhead. This issue is particularly important for COPs with inequality constraints, where standard slack-variable formulations introduce additional binary variables. Unbalanced penalization (UP) avoids slack variables, but the original UP formulation requires tuning two penalty coefficients and contains a squared residual term that can increase the number of quadratic couplings. In this paper, we propose the unbalanced penalization Ohzeki method (UPOM), which combines UP with the Ohzeki method for inequality-constrained COPs. UPOM replaces the two static penalty coefficients of original UP with an auxiliary-variable update and removes the squared residual term from the Hamiltonian used for sampling. We further propose the deep-unfolded unbalanced penalization Ohzeki method (DU-UPOM), which learns the step-size schedule of the UPOM update from training instances. Numerical experiments on random knapsack problems show that UPOM improves over original UP and that DU-UPOM reaches optimal solutions in fewer iterations than fixed-step UPOM and other baseline. These results demonstrate that the proposed framework reduces the tuning and embedding burdens of UP while making the Ohzeki method trainable for inequality constraints.
This study proposes a novel method for simplifying inequality constraints in Higher-Order Binary Optimization (HOBO) formulations. The proposed method addresses challenges associated with Quadratic Unconstrained Binary Optimization (QUBO) formulations, specifically the increased computational complexity and reduced solution accuracy caused by the introduction of slack variables and the resulting growth in auxiliary qubits. By efficiently integrating constraints, the method enhances the computational efficiency and accuracy of both quantum and classical solvers. The effectiveness of the proposed approach is demonstrated through numerical experiments applied to combinatorial optimization problems. The results indicate that this method expands the applicability of quantum algorithms to high-dimensional problems and improves the practicality of classical optimization solvers for optimization problems involving inequality constraints.
This paper presents a digital annealing engine capable of high-speed solving the Binary Quadratic Programming Problem (BQP) with equality and inequality constraints. While many Quadratic Unconstrained Binary Optimization (QUBO) solvers have been developed for combinatorial optimization problems, they are specialized for “unconstrained” BQP and struggle to solve “constrained” BQP at high speed. To address this issue, we developed a digital annealing engine on GPUs that leverages constraints to narrow the search space and directly evaluates inequality violations during Markov Chain Monte Carlo based searches. We evaluated the performance of the Digital Annealer (DA) system with the developed engine on benchmark problems with one-hot (a type of equality constraint) and inequality constraints. The results showed that the developed DA system achieved superior solving performance compared to the previous generation of DA. It also showed comparable performance to state-of-the-art dedicated solvers. Our results suggest that this digital annealing engine is a promising solution for high-speed solving constrained BQP.
Quantum annealing is a promising approach for solving combinatorial optimization problems. However, its performance is often limited by the overhead of additional qubits required for embedding logical QUBO models onto quantum annealers. This overhead becomes severe when logical QUBO models have dense connectivity. Such dense structures frequently arise when formulating equality and inequality constraints. To address this issue, we propose a method to construct a significantly sparser logical QUBO model for these constraints. By adding auxiliary variables based on specific network structures, our approach decomposes the original constraint into smaller, more manageable constraints. We demonstrate that this method reduces the number of edges (quadratic terms) from O(N2)O(N^2)O(N2) to O(N)O(N)O(N) for the one-hot constraint and to O(Nlog⁡N)O(N\log N)O(NlogN) in the worst case for general equality constraints, where NNN is the number of variables. Experimental results on D-Wave's hardware show that our formulation leads to substantial reductions in the number of qubits required for embedding, shorter average chain lengths, lower chain break rates, and higher feasible solution rates compared to conventional methods. This work provides a practical tool for efficiently solving constrained optimization problems on current and future quantum annealers.
Ising Machines are emerging hardware architectures that efficiently solve NP-Hard combinatorial optimization problems. Generally, combinatorial problems are transformed into quadratic unconstrained binary optimization (QUBO) form, but this transformation often complicates the solution landscape, degrading performance, especially for multi-state problems. To address this challenge, we model spin interactions as generalized boolean logic function to significantly reduce the exploration space. We demonstrate the effectiveness of our approach on graph coloring problem using probabilistic Ising solvers, achieving similar accuracy compared to state-of-the-art heuristics and machine learning algorithms. It also shows significant improvement over state-of-the-art QUBO-based Ising solvers, including probabilistic Ising and simulated bifurcation machines. We also design 1024-neuron all-to-all connected probabilistic Ising accelerator on FPGA with the proposed approach that shows ~10000x performance acceleration compared to GPU-based Tabucol heuristics and reducing physical neurons by 1.5-4x over baseline Ising frameworks. Thus, this work establishes superior efficiency, scalability and solution quality for multi-state optimization problems.