Linearizing Binary Optimization Problems Using Variable Posets for Ising Machines

TL;DR

This paper tackles the practical limitations of Ising machines on solving QUBO problems by introducing a linearization technique based on variable posets. It extracts valid partial orders to impose auxiliary penalties that reduce quadratic terms while preserving the optimum, thereby narrowing the energy landscape and sparsifying the problem to ease minor-embedding. The authors provide a concrete algorithm to extract the partial order and a linearization procedure, and demonstrate the approach on synthetic QUBO instances and multi-dimensional knapsack problems, showing improved embedding feasibility and lower solution energies. The work suggests that linearization can significantly enhance the utility of Ising machines for large-scale, constraint-rich combinatorial optimization tasks and opens directions for extension to other ILP domains and hardware platforms.

Abstract

Ising machines are next-generation computers expected to efficiently sample near-optimal solutions of combinatorial optimization problems. Combinatorial optimization problems are modeled as quadratic unconstrained binary optimization (QUBO) problems to apply an Ising machine. However, current state-of-the-art Ising machines still often fail to output near-optimal solutions due to the complicated energy landscape of QUBO problems. Furthermore, the physical implementation of Ising machines severely restricts the size of QUBO problems to be input as a result of limited hardware graph structures. In this study, we take a new approach to these challenges by injecting auxiliary penalties preserving the optimum, which reduces quadratic terms in QUBO objective functions. The process simultaneously simplifies the energy landscape of QUBO problems, allowing the search for near-optimal solutions, and makes QUBO problems sparser, facilitating encoding into Ising machines with restriction on the hardware graph structure. We propose linearization of QUBO problems using variable posets as an outcome of the approach. By applying the proposed method to synthetic QUBO instances and to multi-dimensional knapsack problems, we empirically validate the effects on enhancing minor-embedding of QUBO problems and the performance of Ising machines.

Figures (5)

• Figure 1: Conceptual figure of effect of auxiliary penalty on energy landscape. Dashed and solid lines (black) represent original landscape and modified landscape after adding auxiliary penalty, respectively. Area crossing contour (dashed gray line) is made narrow by modification, restricting search space of Ising machines. One local minimum (dashed circle) is removed by an auxiliary penalty, and a linearized problem has only one local minimum (red circle).
• Figure 2: Running time of Algorithm \ref{['alg:extract_order']} averaged over 10 randomly generated QUBO instances for each problem size $n$ with power curve fitting.
• Figure 3: Success counts of finding minor-embedding of linearized QUBO problems on fixed hardware graphs with increasing problem size.
• Figure 4: Success counts of finding minor-embedding of linearized QUBO problems to $K_{256,256}.$
• Figure 5: Results on tuning penalty coefficient $\lambda$ on MKPs. Bar charts represent numbers of feasible solutions and lines represent scores. As $\lambda$ becomes smaller, numbers of feasible solutions tend to decrease and scores tend to increase (as long as sufficient number of feasible solutions are obtained).

Theorems & Definitions (21)

• Definition 1
• Proposition 1
• Definition 2
• Example 1
• Theorem 3
• proof
• Theorem 4
• Example 2
• Theorem 5
• Proposition : = Proposition \ref{['prop:aux_penalty']}