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

TL;DR

HyCiM is proposed, a novel hybrid computing-in-memory (CiM) based QUBO solver framework, designed to overcome aforementioned challenges, and drastically narrows down the search space, eliminating 2100 to 22536 infeasible input configurations.

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-in-memory (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) %When feasible solutions are detected, 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 $2^{100} \text{ to } 2^{2536}$ infeasible input configurations compared to the conventional D-QUBO approach.

Figures (10)

• Figure 1: (a) A general COP with an inequality constraint involves a search space of $2^{n}$ variable configurations; (b) D-QUBO introduces extra scaled auxiliary variable vector $\Vec{y}$, expanding the search space to $2^{n+C}$.
• Figure 2: (a) By applying different write pulses, (b) multi-level $I_D$-$V_G$ characteristics of FeFETs can be programmed storing $q_0$ to $q_3$. (c) FeFET storing a binary bit can naturally perform consecutive multiplications $i = x\times q_i \times y$ by simultaneously applying the inputs $x$ and $y$ to gate and drain, respectively.
• Figure 3: Overview of HyCiM. (a) Inequality-QUBO transformation method; (b) FeFET-based CiM inequality filter for input configuration filtering; (c)(d) FeFET-based CiM annealer for approaching optimal solutions.
• Figure 4: (a) Cell schematic of inequality filter array; (b) Transfer curves of cells storing five distinct weights and corresponding $V_{\text{read}}$'s; (c) Transient waveforms of filter cells storing five weights during the evaluation operation.
• Figure 5: (a) Schematic of an $m \times n$ working array. The output $\text{ML} \propto - \Vec{w}\Vec{x}$; (b) The inequality filter architecture, containing one working array, one replica array and a voltage comparator; (c) Symbol of 2-stage voltage comparator; (d) First-stage differential pre-amplifier; (e) second-stage dynamic latched voltage comparator; (f) Transient waveforms of an exemplary inequality evaluation. Infeasible configurations are filtered out.